Statistics

Quick reference

• Statistics is the science of collecting, organizing, summarizing, and analyzing information to answer questions.
• Data are the information collected from individuals. A variable is the characteristic being recorded, and the data are the observed values of that variable.
• The process of statistics usually starts with a research objective, then data are collected, summarized, and used for inference when appropriate.
• Statistics has two main branches.
  • Descriptive statistics organizes and summarizes data using tables, graphs, or numerical summaries.
  • Inferential statistics uses sample data to draw conclusions about a population and reports uncertainty.
• Three terms identify who or what is being studied.
  • A population is the entire group being studied.
  • A sample is a subset of that population.
  • An individual is one person or object in the group being studied.
• A numerical summary is named according to whether it describes a population or a sample.
  • A parameter is a numerical summary of a population.
  • A statistic is a numerical summary of a sample.
• Variables can be qualitative or quantitative.
  • Qualitative variables describe categories.
  • Quantitative variables are numerical values where arithmetic has meaning, so number labels such as ZIP codes or coded weekdays are not treated as quantitative.
• Quantitative variables can be discrete or continuous.
  • Discrete quantitative variables come from counting.
  • Continuous quantitative variables come from measurement and can take values between two measured values.
• The four levels of measurement describe how the values of a variable can be compared.
  • Nominal means names only.
  • Ordinal means categories with an order.
  • Interval means differences can be compared, but zero is not a true none.
  • Ratio means differences and ratios make sense because zero is a true none.

Quick reference

• Two common ways to collect data are observational studies and designed experiments.
  • An observational study measures characteristics without trying to change the subjects.
  • A designed experiment assigns treatments and observes the response, while controlling other conditions when possible.
• Two variables describe a possible relationship in a study.
  • The explanatory variable may help explain changes in the response variable.
  • The response variable is the outcome being measured.
• Experiments use treatments and experimental units.
  • A treatment is the condition applied in an experiment.
  • An experimental unit is the person or object receiving the treatment.
• Common types of observational studies differ in when and how information is collected.
  • Cross-sectional studies collect information at one point in time or over a short time period.
  • Case-control studies are retrospective: researchers look back or use existing records to compare individuals with a condition to similar individuals without it.
  • Cohort studies are prospective: researchers follow a group over time and record what happens.
• Other variables can make a relationship difficult to interpret.
  • Confounding happens when the effects of two or more explanatory variables are mixed together.
  • A lurking variable was not included in the study but may affect the response and may be related to an explanatory variable in the study.
  • A confounding variable was included in the study, but its effect cannot be separated from the effect of another explanatory variable.
• No type of observational study is always best. The research question, available data, time, and cost help determine which type is most appropriate.
• Observational studies can show association, but they do not prove cause and effect by themselves. A well-designed experiment gives stronger evidence for cause and effect.

Quick reference

• Data may be collected from a census or a sample.
  • A census collects data from every individual in the population.
  • A sample collects data from only part of the population.
• Random sampling uses chance to select individuals. A simple random sample of size \(n\) goes further: every possible group of \(n\) individuals has the same chance of being selected.
• A sampling frame is the list of all individuals available for selection. Give each individual one unique number; for a random number table, use the same number of digits, such as 01 through 40.
• Sampling may be done with or without replacement.
  • Sampling without replacement means a selected individual cannot be chosen again.
  • Sampling with replacement means the individual is returned and may be chosen again.
• A random number table can be used to select a simple random sample.
  • Choose a starting position and direction, read digits in groups that match the assigned numbers, and keep valid numbers until the sample is complete.
  • Skip 00, numbers outside the population range, and repeated numbers when sampling without replacement.
• Random selection helps reduce selection bias.
• TI-84: press MATH → PRB to access random-integer commands.
  • Use randInt( to generate random integers. For sampling without replacement, ignore repeated values.
  • Use randIntNoRep( when available to generate different values without replacement.

Quick reference

• Stratified sampling separates the population into nonoverlapping groups, called strata, based on a shared characteristic. A random sample is selected from every stratum.
  • The number selected from each stratum may be the same or may be proportional to the stratum's size.
  • In a proportional sample, a group containing 30% of the population supplies 30% of the sample.
  • Stratified sampling guarantees representation from every stratum.
• Cluster sampling separates the population into groups, randomly selects some entire groups, then includes everyone in the selected groups.
  • Cluster sampling can reduce travel, time, or cost when the groups are spread out.
• To distinguish the methods, remember: stratified sampling selects some individuals from every group; cluster sampling selects everyone from some groups.
• Systematic sampling selects every \(k\)th individual after a random starting position.
  • To obtain a sample of size \(n\) from a population of size \(N\), use \(k=\lfloor N/n\rfloor\), randomly choose \(p\) from 1 through \(k\), and select positions \(p\), \(p+k\), \(p+2k\), and so on through \(p+(n-1)k\).
  • A systematic sample uses a random start, but it is not a simple random sample because only samples following the every-\(k\)th pattern can be selected.
• Convenience sampling uses individuals who are easy to reach and is not a random sampling method.
  • A voluntary response sample is a common convenience sample in which people choose whether to participate.

Quick reference

• Bias occurs when a method systematically produces results that do not accurately represent the population.
• The three main sources of bias are sampling bias, nonresponse bias, and response bias.
• Sampling bias occurs when the method used to select individuals tends to favor one part of the population over another.
  • Undercoverage occurs when a group in the population is left out or represented too little in the sample.
  • Voluntary response bias occurs when people choose whether to participate. People with strong opinions may be more likely to respond.
• Nonresponse bias occurs when selected individuals who do not respond differ in an important way from those who do respond. A low response rate creates concern, but nonresponse causes bias when the two groups differ.
• Response bias occurs when people respond, but their answers do not reflect their true beliefs or behavior.
  • Wording bias is caused by leading, loaded, or unbalanced wording. Survey questions should be written in a neutral, balanced way.
  • Interviewer influence or pressure can affect how a person answers.
  • Sensitive questions can lead people to misrepresent their beliefs or behavior.
  • The order of questions or answer choices can influence responses.
• To distinguish the main sources, ask: Who had a chance to be selected? Who was selected but did not respond? Did the responses reflect the truth? These questions point to sampling bias, nonresponse bias, and response bias, respectively.
• Callbacks, alternate contact methods, and rewards or incentives can reduce nonresponse. Simply increasing the number contacted does not correct a biased response pattern.
• Neutral wording, impartial administration, anonymity when appropriate, and rotating question or answer order can reduce response bias.
• Sampling error is the natural sample-to-sample difference that occurs because only part of the population is observed; it is not the same as sampling bias. Undercoverage, nonresponse, response bias, and data-entry mistakes are nonsampling errors and can occur even in a census.

Quick reference

• An experiment deliberately applies one or more treatments to experimental units and measures a response.
  • An experimental unit is the person or object receiving a treatment. A person in an experiment may also be called a subject.
  • A factor is an explanatory variable whose effect is being studied. Its levels are the values chosen for the experiment.
  • A treatment is a specific condition applied to an experimental unit. When an experiment has several factors, a treatment is a combination of factor levels.
  • The response variable is the outcome measured after the treatment is applied.
• Good experimental designs use control, random assignment, and replication to make treatment comparisons more reliable.
  • Control means keeping other conditions as similar as possible so the treatments are the main planned difference between groups.
  • Some variables can be controlled by holding them constant, such as giving every plant the same amount of water. Other variables, such as natural differences among people or plants, cannot be held constant; random assignment helps distribute their effects across treatment groups.
  • Random assignment uses chance to place experimental units into treatment groups. It helps balance other variables across the groups.
  • Replication means applying each treatment to multiple experimental units. Repeatedly measuring the same unit does not provide the same replication as using additional units.
• Confounding occurs when the effect of a treatment cannot be separated from the effect of another variable. Control and random assignment help reduce confounding.
• A control group gives a baseline for comparison, and a placebo can help separate a treatment effect from the effect of expecting a treatment.
  • A control group may receive a placebo, an existing treatment, or no treatment, depending on the experiment.
  • A placebo looks like the treatment being tested but does not contain its active ingredient.
  • The placebo effect is a response caused by a subject's expectation of receiving a treatment rather than by the treatment itself.
• Blinding reduces the chance that knowledge of the assigned treatment will influence a subject's behavior or an evaluator's judgment.
  • In a single-blind experiment, the subjects usually do not know which treatment they receive.
  • In a double-blind experiment, neither the subjects nor the researchers who interact with them or evaluate the response know the treatment assignments.
• Three common experimental designs differ in how the experimental units are organized before treatments are assigned.
  • Completely randomized design: randomly assign all experimental units directly to the treatments.
  • Randomized block design: first separate similar experimental units into blocks, then randomly assign treatments within every block. Blocking reduces variation caused by the characteristic used to form the blocks.
  • Matched-pairs design: compare two treatments using closely matched units or the same unit under both conditions, then compare responses within each pair.
• Random sampling and random assignment serve different purposes.
  • Random sampling helps produce a sample that can represent the population.
  • Random assignment helps create comparable treatment groups and supports cause-and-effect conclusions from a well-designed experiment.

Quick reference

• For qualitative data, a frequency distribution lists each category and its count.
  • Raw data are the original observations before they are organized or summarized.
  • Frequency is the number of observations in a category.
  • Relative frequency is the proportion of observations in a category. Find it by dividing the category frequency by the total frequency.
  • A relative-frequency distribution lists each category with its relative frequency, written as a decimal or percent.
• Check the totals and interpret relative frequencies in context.
  • The frequencies should total the sample size \(n\).
  • The relative frequencies should total 1, or 100%, except for a small difference caused by rounding.
  • A relative frequency of 0.36 means 36% of the observations are in that category; it does not mean 36 observations.
• A bar graph displays qualitative categories using separated bars of equal width.
  • Place the categories on one axis and frequency or relative frequency on the other. The height of each bar gives the category's value.
  • Frequency and relative-frequency bar graphs for the same data have the same overall pattern but different vertical scales.
  • Horizontal bars are useful when category names are long.
• Special bar graphs organize bars for a particular comparison.
  • A Pareto chart orders categories from greatest to least frequency or relative frequency.
  • A side-by-side bar graph compares the same categories across two or more groups.
  • Use relative frequencies when comparing groups of different sizes so the comparison is based on proportions rather than counts.
• A pie chart divides one whole into sectors whose sizes represent category proportions.
  • The categories must represent nonoverlapping parts of the same whole, and their percentages should total 100%.
  • To construct a sector by hand, multiply the relative frequency by 360 degrees to find the sector angle.
  • A pie chart is not appropriate when the percentages overlap or do not represent parts of one whole.
• Category definitions matter. A graph can give a misleading comparison when one category combines several subcategories but the others do not.

Quick reference

• Quantitative frequency distributions may use individual values or grouped intervals.
  • For discrete data with relatively few possible values, each value can be its own class.
  • Discrete data with many values and continuous data are usually grouped into nonoverlapping intervals.
  • Frequency is the count in a class. Relative frequency is the class frequency divided by the total frequency.
  • Frequencies should total the sample size \(n\); relative frequencies should total 1, or 100%, except for a small difference caused by rounding.
• Class terms describe the intervals in a grouped frequency distribution.
  • The lower class limit is the smallest value that can belong to a class; the upper class limit is the largest value that can belong to that class.
  • Class width is the difference between consecutive lower class limits. For the classes 10-19 and 20-29, the width is 20 - 10 = 10, not 19 - 10 = 9.
  • Classes should cover all observations without overlapping. Unless an open-ended class is needed, use equal class widths.
  • If the class setup is not given, begin at the smallest observation or a convenient value below it. To estimate the class width, subtract the minimum from the maximum, divide by the desired number of classes, and round up to a convenient width.
  • Different valid class widths can produce somewhat different-looking summaries. Use enough classes to show the pattern without making the display unnecessarily detailed.
• A histogram displays a quantitative frequency distribution.
  • Place the classes on the horizontal axis and frequency or relative frequency on the vertical axis.
  • The bars have equal widths and touch because the classes represent adjoining numerical intervals. In a bar graph for qualitative data, the bars are separated.
  • Frequency and relative-frequency histograms for the same classes have the same shape and proportional bar heights, but different vertical scales.
  • A histogram shows how many observations fall in each class, but it usually does not show the exact values within a class.
• A dot plot displays every observation in a small quantitative data set.
  • Place each possible value on a number line and draw one dot for each observation. Stack repeated observations above the same value.
  • Dot plots preserve individual values, making clusters, gaps, and repeated values easy to see. They become crowded for large data sets.
• When a question includes several classes, add the relevant frequencies or relative frequencies.
  • At least includes the stated value and all larger values.
  • At most includes the stated value and all smaller values.
• Describe a quantitative distribution using its overall pattern and the direction of any tail.
  • Symmetric: the left and right sides are approximately mirror images.
  • Bell-shaped: symmetric with one central peak and frequencies that taper in both directions.
  • Uniform: the bar heights are approximately equal.
  • Skewed right: the longer tail points to the right.
  • Skewed left: the longer tail points to the left.
  • Real data may only approximate these shapes, so reasonable judgments can differ. Do not use these shape terms for qualitative data.
• TI-84 histogram workflow:
  • Enter raw data in L1 using STAT → EDIT.
  • Open 2nd Y= (STAT PLOT), turn Plot1 on, choose the histogram icon, and set Xlist:L1 and Freq:1. Use ZoomStat for an initial window.
  • For specified classes, set Xmin to the first lower class limit and Xscl to the class width.
  • For a grouped table, enter class midpoints in L1, frequencies in L2, and use Freq:L2. StatCrunch is convenient for larger data sets.

Quick reference

• A dot plot displays each observation as a dot above its value.
  • Repeated values form vertical stacks. Count all dots to find the sample size, and use the tallest stack to identify the value that occurs most often.
  • Dot plots work best for relatively small raw data sets because they preserve individual values.
  • Look for the overall shape, clusters, gaps, and observations separated from the rest of the data.
• A stem-and-leaf plot organizes quantitative data while preserving the original values.
  • The stem contains the leading digit or digits, and the leaf contains the final digit. A key such as 13 | 5 = 13.5 explains the place value.
  • The same 13 | 5 notation could represent 135, 13.5, or another value under a different place-value setup, so always use the plot's key.
  • Write stems in increasing order, place every leaf with its stem, and order the leaves from least to greatest.
  • Split stems when one row is crowded. Repeat each stem, using the first row for leaves 0-4 and the second row for leaves 5-9.
  • A back-to-back stem-and-leaf plot uses shared stems to compare the distributions of two groups.
• A frequency polygon displays grouped frequencies using connected points.
  • Find each class midpoint by adding the lower and upper class limits and dividing by 2.
  • Plot each class midpoint on the horizontal axis and its frequency on the vertical axis, then connect consecutive points.
  • Add one midpoint before the first class and one after the last class, each with frequency zero, so the graph meets the horizontal axis.
  • Frequency polygons show the shape of grouped data and can compare multiple distributions on the same axes.
• Cumulative distributions use running totals.
  • Cumulative frequency is the number of observations at or below a class's upper endpoint.
  • Cumulative relative frequency is the proportion or percentage of observations at or below that endpoint.
  • The final cumulative frequency should equal \(n\), and the final cumulative relative frequency should equal 1, or 100%.
• An ogive graphs cumulative frequency or cumulative relative frequency.
  • Plot upper class endpoints on the horizontal axis and cumulative values on the vertical axis. Class boundaries may be used when the classes are written with whole-number limits.
  • Begin at zero using the lower boundary before the first class, then connect the points. Because cumulative totals cannot decrease, an ogive must stay level or rise.
  • Use an ogive to find the amount at or below a cutoff, estimate a cutoff for a given cumulative percentage, or find the amount above a cutoff by subtracting the cumulative amount from the total.
• A time-series plot displays measurements recorded over time.
  • Place time on the horizontal axis and the measured values on the vertical axis, then connect the points chronologically.
  • Describe the overall trend, peaks, lows, changes in direction, and unusual changes.
  • When two series share the same axes, compare their values, trends, and the times when they are closest or farthest apart.
• Choose the display that matches the data and the question.
  • Use a dot plot for a small raw data set and a stem-and-leaf plot when retaining exact values is especially useful.
  • Use a frequency polygon for grouped frequencies, an ogive for cumulative amounts or cutoff questions, and a time-series plot for values recorded over time.
• TI-84 workflows for connected statistical plots:
  • For a frequency polygon, enter class midpoints in L1 and frequencies in L2, including the two zero-frequency endpoints. Open 2nd Y=, choose the xyLine plot, and set Xlist:L1 and Ylist:L2.
  • For cumulative totals, enter frequencies in L2 and calculate cumSum(L2)→L3. Use L3/sum(L2)→L4 for cumulative relative frequencies.
  • For an ogive, graph upper class endpoints against L3 or L4 with an xyLine plot. Include the starting point with cumulative value zero.
  • For a time series, enter times in L1 and values in L2, then use an xyLine plot. Turn on a second Stat Plot to compare another series.
  • The TI-84 has no dedicated dot-plot or stem-and-leaf command. Construct those displays by hand or use StatCrunch when the data set is large.

Quick reference

• Check the scale before interpreting a graph.
  • Axis values should be clearly labeled and increase by consistent amounts.
  • A truncated vertical axis can exaggerate or hide differences. Bar graphs should generally begin at zero because the length of each bar represents the amount.
  • If a scale is truncated for a valid reason, the break or shortened scale should be clearly shown.
• Check what each bar or class represents.
  • Bars used to compare categories should have equal visual widths.
  • Percentages or counts from time intervals of different lengths do not provide a fair comparison of how often something occurs.
  • For introductory histograms, use equal class widths unless the graph clearly explains how unequal widths are being handled.
• Check pictures and visual effects for distortion.
  • If a picture is enlarged in both height and width, its area grows much faster and can exaggerate the difference.
  • Decorative pictures can make it unclear where a bar begins or ends.
  • Three-dimensional graphs can make categories closer to the viewer appear larger. A clear two-dimensional graph is usually easier to compare.
• Check the labels and context.
  • Use an informative title, clearly labeled axes, units, and enough context to identify the data.
  • Include exact values or a data source when they help the reader judge the display.
• Decide whether the graph is actually misleading.
  • A graph is not misleading simply because one category is much larger or smaller than another. The display is honest when its lengths, areas, scales, and labels accurately represent the data.
  • To correct a misleading graph, preserve the data while using a clear scale, comparable intervals, simple shapes, and complete labels.

Quick reference

• Measures of central tendency describe the center or typical value of a distribution.
  • The mean uses every numerical value. Add the values and divide by the number of observations.
  • For a sample, \(\bar{x}=\frac{\sum x}{n}\). For a population, \(\mu=\frac{\sum x}{N}\).
  • Unless a problem gives another instruction, round the mean to one more decimal place than the original data.
• The median, \(M\), is the middle of the ordered data.
  • First arrange the values from least to greatest.
  • If \(n\) is odd, the median is the single middle observation.
  • If \(n\) is even, the median is the mean of the two middle observations. The result does not have to be one of the original data values.
• The mode is the value or category that occurs most often.
  • A data set may have no mode, one mode, two modes (bimodal), or more than two modes (multimodal).
  • The mode can describe qualitative data because categories can be counted even when arithmetic is not meaningful.
  • The TI-84 1-Var Stats screen does not report the mode; inspect the list or a frequency table.
• A resistant measure is not changed substantially by an extreme value.
  • The median is resistant, but the mean is pulled toward unusually large or small values.
  • For a roughly symmetric distribution without strong outliers, the mean and median are usually close and the mean is commonly reported.
  • For a strongly skewed distribution or one with extreme values, the median usually better represents a typical observation.
  • Skewed left: mean less than median. Roughly symmetric: mean approximately equal to median. Skewed right: mean greater than median.
• Interpret a measure of center in the context and units of the variable.
  • The word average is sometimes used loosely, so identify whether a report means the mean, median, or mode.
  • Do not calculate a mean for qualitative labels or numerical codes whose arithmetic has no meaning.
• TI-84 1-Var Stats workflow:
  • Press STAT → 1:Edit and enter the raw data in L1.
  • Press STAT → CALC → 1:1-Var Stats, set List:L1, and choose Calculate.
  • Read \(\bar{x}\) for the sample mean and \(n\) for the number of observations. Scroll down to Med for the median.

Quick reference

• Measures of dispersion describe how spread out the data are.
  • The range is maximum - minimum. It uses only the two extreme values and is not resistant to outliers.
  • Standard deviation measures a typical distance of the observations from the mean. It is reported in the same units as the data.
  • Variance is the square of the standard deviation, so its units are squared.
• Standard deviation is never negative.
  • A standard deviation of 0 means every observation is equal.
  • A larger standard deviation means more spread when comparing the same variable measured in the same units.
  • A value more than 2 standard deviations from the mean is often considered far from the mean.
• Use population measures when the data contain the entire population and sample measures when the data are a sample.
  • Population standard deviation is \(\sigma=\sqrt{\frac{\sum(x-\mu)^2}{N}}\), and population variance is \(\sigma^2\).
  • Sample standard deviation is \(s=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}\), and sample variance is \(s^2\).
  • The formulas show how spread is calculated, but use calculator output for most data sets. A sample standard deviation can be smaller or larger than the population standard deviation.
• Range and standard deviation are not resistant measures.
  • An extreme value can increase the range immediately and can pull the mean away from the other values, increasing the standard deviation.
  • Compare both the numerical measures and the distribution's shape when describing spread.
• For a roughly bell-shaped distribution, the Empirical Rule estimates how much data lie near the mean.
  • About 68% lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.
  • On either side of the mean, the regions are about 34%, 13.5%, 2.35%, and 0.15% as you move outward.
  • Check that the distribution is roughly bell-shaped before applying the rule.
• TI-84 1-Var Stats workflow:
  • Enter the data in L1, then choose STAT → CALC → 1:1-Var Stats and calculate using L1.
  • Read Sx for the sample standard deviation and \(\sigma_x\) for the population standard deviation.
  • Square the appropriate standard deviation to obtain the variance.

Quick reference

• A frequency table summarizes repeated observations.
  • For a table of individual values, treat each value as if it appeared the stated number of times.
  • The total number of observations is the sum of the frequencies: \(n=\sum f\) for a sample or \(N=\sum f\) for a population.
  • The mean is \(\bar{x}=\frac{\sum xf}{\sum f}\) for a sample or \(\mu=\frac{\sum xf}{\sum f}\) for a population.
• For data grouped into class intervals, use each class midpoint as a representative value.
  • Class midpoint = (lower class limit + upper class limit) / 2.
  • Classes do not need to have equal widths. Find the midpoint of each class separately.
  • Because the original values within each class are unknown, a grouped-data mean, variance, or standard deviation is an approximation. Use raw data when they are available.
• A weighted mean gives observations different amounts of influence.
  • Use \(\bar{x}_w=\frac{\sum xw}{\sum w}\), where \(x\) is a value and \(w\) is its weight.
  • Common weights include course credit hours, category percentages, quantities purchased, and frequencies.
  • If weights are percentages that total 100%, multiply each value by its decimal weight and add the products.
• Use the grouped summary in context.
  • Choose Sx and \(s^2\) for sample data; choose \(\sigma_x\) and \(\sigma^2\) for an entire population.
  • Interpret the mean and standard deviation in the original units, and label grouped results as approximate.
  • When comparing groups measured in the same units, compare both center and spread rather than relying on only one statistic.
• TI-84 grouped-data workflow:
  • Enter the values or class midpoints in L1 and their frequencies or weights in L2.
  • Choose STAT → CALC → 1:1-Var Stats, set List:L1 and FreqList:L2, then calculate.
  • Confirm that the reported \(n\) equals the sum of the frequencies, then read \(\bar{x}\), Sx, or \(\sigma_x\) as required.

Quick reference

• A z-score gives a value's distance from the mean in standard-deviation units.
  • Use \(z=\frac{x-\mu}{\sigma}\) for a population or \(z=\frac{x-\bar{x}}{s}\) for a sample.
  • A positive z-score is above the mean, a negative z-score is below the mean, and \(z=0\) is at the mean. A z-score has no units.
  • To recover a raw value, use \(x=\mu+z\sigma\) or \(x=\bar{x}+zs\).
• Z-scores allow fair relative comparisons between different distributions.
  • When larger values are better, the larger z-score is the stronger relative result.
  • When smaller values are better, such as race times, the more negative z-score is the stronger relative result.
  • After standardizing every value in a population, the z-scores have mean 0 and population standard deviation 1.
• The kth percentile is a value with about k% of the observations at or below it.
  • The remaining observations are above it. For example, the 80th percentile has about 80% at or below and 20% above.
  • On an ogive, the vertical cumulative relative frequency can be read as a percentile. Read upward from a value to find its percentile or across from a percentile to estimate its value.
• Quartiles divide ordered data into four parts.
  • \(Q_1\) is the 25th percentile, \(Q_2\) is the median or 50th percentile, and \(Q_3\) is the 75th percentile.
  • About 25% of observations are at or below \(Q_1\), about 50% are at or below \(Q_2\), and about 75% are at or below \(Q_3\).
  • The interquartile range is \(IQR=Q_3-Q_1\). It measures the spread of the middle 50% and is resistant to extreme values.
• Use the 1.5(IQR) rule to identify potential outliers.
  • Lower fence: \(Q_1-1.5(IQR)\). Upper fence: \(Q_3+1.5(IQR)\).
  • A value below the lower fence or above the upper fence is an outlier. A value equal to a fence is not outside it.
  • An outlier may be a valid unusual observation or may come from measurement, entry, or sampling error. Investigate it rather than automatically deleting it.
• Choose resistant summaries when the distribution is skewed or has outliers.
  • Median and IQR are resistant; mean, range, and standard deviation are not.
  • For roughly symmetric data without strong outliers, mean and standard deviation are commonly reported. For skewed data or data with outliers, median and IQR usually give a more stable summary.
• TI-84 quartile workflow:
  • Enter the data in L1 and run 1-Var Stats with List:L1.
  • Scroll down to read minX, \(Q_1\), Med, \(Q_3\), and maxX.
  • The TI-84 reports quartiles but does not label outliers; calculate the fences and compare the original observations with them.

Quick reference

• The five-number summary lists minimum, \(Q_1\), median, \(Q_3\), and maximum in ascending order.
  • Use ordered data or the lower portion of TI-84 1-Var Stats output to find the five values.
  • The five-number summary describes position but does not show every observation, the mean, or the standard deviation.
• A modified boxplot displays quartiles, non-outlier extremes, and outliers on a number line.
  • The box runs from \(Q_1\) to \(Q_3\), so its length represents the IQR. The line inside the box marks the median.
  • Whiskers extend to the smallest and largest observations that are not outliers; they do not extend to the fences unless an observation occurs there.
  • Plot observations outside the 1.5(IQR) fences separately.
• To construct a modified boxplot:
  • Find \(Q_1\), the median, \(Q_3\), the IQR, and both outlier fences.
  • Draw a scaled number line, make the box from \(Q_1\) to \(Q_3\), and mark the median inside it.
  • Draw whiskers to the most extreme non-outliers and plot each outlier separately.
• A boxplot gives clues about distribution shape.
  • Roughly symmetric: the median is near the center of the box and the whiskers have similar lengths.
  • Skewed right: the median is usually left of the box's center and the right side or right whisker is longer.
  • Skewed left: the median is usually right of the box's center and the left side or left whisker is longer. Treat these as visual clues rather than absolute rules.
• Use side-by-side boxplots to compare groups on the same scale.
  • Compare medians for center, IQRs and overall ranges for spread, and separately plotted points for outliers.
  • A group with a smaller IQR or range is more consistent. A longer box and whiskers usually suggest greater spread, but a boxplot does not give an exact standard deviation.
  • A boxplot does not show clusters, gaps, or sample size, so use the available evidence without claiming more than the display supports.
• TI-84 modified-boxplot workflow:
  • Enter the data in a list. Open 2nd → Y= (STAT PLOT), turn one plot on, and choose the modified boxplot icon with separate outlier marks.
  • Set Xlist to the data list and Freq to 1, then use ZOOM → 9:ZoomStat.
  • Press TRACE and move left or right to read the quartiles, whisker endpoints, and outliers.

Quick reference

• A scatterplot displays paired quantitative data measured on the same individuals or objects.
  • Plot the explanatory or predictor variable on the horizontal axis and the response variable on the vertical axis.
  • The explanatory variable may help explain or predict the response, but the roles are not always determined by the data alone. Use the study's question and context.
• Describe a scatterplot by direction, form, strength, and unusual features.
  • Direction may be positive, negative, or neither. Form may be linear, curved, or show no clear pattern.
  • Strength describes how closely the points follow the form. Also note clusters, gaps, and points that do not follow the overall pattern.
  • Always inspect the scatterplot. A numerical summary alone can hide curvature or an influential unusual point.
• The linear correlation coefficient measures the direction and strength of a linear relationship.
  • Use \(r\) for a sample and \(\rho\) for a population. The long formula is \(r=\frac{1}{n-1}\sum\left(\frac{x-\bar{x}}{s_x}\right)\left(\frac{y-\bar{y}}{s_y}\right)\), but technology is used for computation.
  • The value is between -1 and 1. The sign gives direction; the closer \(|r|\) is to 1, the stronger the linear relationship.
  • A value near 0 means little evidence of a linear relationship; a strong curved relationship can still have \(r\) near 0.
• Correlation is unitless and is not resistant.
  • Changing inches to centimeters or applying another positive linear unit conversion does not change \(r\).
  • An unusual point can change the direction or strength of \(r\), so investigate it with the scatterplot.
• Use the homework critical-value table to decide whether the sample shows a linear relation.
  • Find \(|r|\) and the critical value for the sample size \(n\).
  • If \(|r|\) is greater than the critical value, conclude that a linear relation exists; use the sign of \(r\) to call it positive or negative.
  • If \(|r|\) is not greater than the critical value, conclude that the data do not provide evidence of a linear relation.
• Correlation does not by itself establish causation.
  • With observational data, say the variables are associated, not that one causes the other.
  • A lurking variable may be related to both variables and may help explain the observed association.
• TI-84 scatterplot and correlation workflow:
  • Enter explanatory values in L1 and matching response values in L2. Keep each pair in the same row.
  • Use STAT PLOT to turn on a scatterplot with Xlist:L1 and Ylist:L2, then use ZoomStat.
  • Run LinReg(ax+b) to read \(r\). In this form, \(a\) is the slope and \(b\) is the intercept. If \(r\) is hidden, run DiagnosticOn from the calculator's catalog first.

Quick reference

• The least-squares regression line predicts a quantitative response from a quantitative explanatory variable.
  • Write the line as \(\hat{y}=b_0+b_1x\), where \(\hat{y}\) is the predicted response.
  • The line minimizes the sum of the squared vertical residuals. For formula context, \(b_1=r\frac{s_y}{s_x}\) and \(b_0=\bar{y}-b_1\bar{x}\), but use technology to fit the model.
  • Fit and use a linear model only after the scatterplot and correlation support a linear relationship.
• Interpret the slope and intercept in context.
  • The slope is the predicted change in the response for each one-unit increase in the explanatory variable, on average. Include both variables and their units.
  • The intercept is the predicted response when \(x=0\). Interpret it only when zero is meaningful and reasonably within the scope of the data.
  • A negative slope predicts a decrease in \(y\) as \(x\) increases; a positive slope predicts an increase.
• Use the regression equation for prediction and comparison.
  • Substitute an explanatory value for \(x\) to find the predicted response \(\hat{y}\). The same prediction can estimate a mean response at that \(x\) or an individual response, though individual values vary around the line.
  • A residual is observed - predicted: \(e=y-\hat{y}\). A positive residual means the observation is above the line; a negative residual means it is below the line.
  • Keep more digits in the regression equation during calculations, then round the final prediction as requested.
• Limit predictions to the model's scope.
  • Interpolation predicts within the observed range of \(x\); extrapolation predicts outside it and may be unreliable because the pattern may change.
  • Do not automatically apply a model to a different population or type of subject, even when the new \(x\)-value lies in the numerical range.
• The coefficient of determination measures explained variation.
  • For simple linear regression, \(R^2=r^2\). Convert it to a percent for interpretation.
  • Say: 'R-squared percent of the variation in the response is explained by the linear model with the explanatory variable.' Do not say that percent of the response itself is explained.
• Use a residual plot to check whether a linear model is appropriate.
  • A useful linear model has residuals scattered around zero without a clear pattern and with reasonably similar vertical spread across \(x\).
  • A curved pattern suggests a nonlinear relationship. A fan shape suggests changing residual variation. A point far above or below the others may be an outlier.
• TI-84 regression workflow:
  • Enter explanatory values in L1 and paired responses in L2. Turn on DiagnosticOn when \(r\) and \(r^2\) are needed.
  • Run LinReg(ax+b) using L1,L2. This course uses that form consistently, so \(a\) is the slope and \(b\) is the intercept.
  • The TI-84 also lists LinReg(a+bx), which reverses the coefficient roles: \(a\) is the intercept and \(b\) is the slope. Check the displayed equation form before interpreting output from another source.
  • To graph the fitted line, store the regression equation to Y1, turn on the scatterplot, and use ZoomStat.

Quick reference

• Probability measures the likelihood of an outcome or event.
  • A probability experiment is a repeatable process with uncertain results. An outcome is one possible result, the sample space is the collection of all possible outcomes, and an event contains one or more outcomes.
  • Probabilities are between 0 and 1 inclusive. Probability 0 represents an impossible event, and probability 1 represents a certain event.
  • An event is often called unusual when \(P(E)\le0.05\), meaning its probability is 0.05 or less, unless the problem sets another cutoff.
• A probability model lists each outcome and its probability.
  • Every probability must be between 0 and 1 inclusive, and all probabilities must sum to 1.
  • When outcomes are equally likely, \(P(E)=\frac{\text{number of outcomes in }E}{\text{number of outcomes in the sample space}}\).
• Probability can come from different sources.
  • Theoretical probability uses a known model with equally likely outcomes, such as a fair die.
  • Empirical probability uses observed relative frequency: number of times the event occurred / number of trials.
  • Subjective probability is an informed judgment based on experience or available information rather than equally likely outcomes or repeated data.
• Long-run relative frequency tends to settle near the true probability as the number of trials grows.
  • Short runs can vary substantially, so an empirical result does not need to match a theoretical probability exactly.
  • When a theoretical probability is known, compare it with the empirical probability. A difference may be random variation and does not by itself prove that the model is wrong; more trials provide stronger evidence.
  • A probability can be used to predict an approximate long-run count: expected count = number of trials x probability.
• The complement of event \(A\), written \(A^c\), contains all outcomes where \(A\) does not occur.
  • Complement rule: \(P(A^c)=1-P(A)\).
  • The event and its complement are mutually exclusive and together cover the entire sample space.

Quick reference

• The union \(A\text{ or }B\) contains outcomes in \(A\), in \(B\), or in both.
  • In probability, or is inclusive unless the wording explicitly says exactly one.
  • The intersection \(A\text{ and }B\) contains outcomes shared by both events.
• Mutually exclusive or disjoint events cannot happen together.
  • For disjoint events, \(P(A\text{ and }B)=0\), so \(P(A\text{ or }B)=P(A)+P(B)\).
  • Do not assume events are disjoint merely because their labels differ; decide whether one outcome can belong to both.
• Use the general addition rule whenever events may overlap.
  • \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\).
  • Subtract the overlap once because it was included in both individual probabilities. The same idea appears visually in a Venn diagram.
• Use the complement rule for not, none, or neither.
  • \(P(A^c)=1-P(A)\).
  • For neither \(A\) nor \(B\), find the complement of \(A\text{ or }B\): \(1-P(A\text{ or }B)\).
• Use the structure of a standard 52-card deck when card events appear.
  • The deck has four suits with 13 cards in each suit. Hearts and diamonds are red; clubs and spades are black.
  • Each rank appears four times, once in each suit. There are four aces, four twos, and so on, including four jacks, four queens, and four kings.
  • Events involving different ranks are disjoint for one draw, but a rank and a suit overlap in one card, such as the six of diamonds.
• Read a two-way table by identifying the correct cells and the grand total.
  • For an and event, use the single intersection cell. For an or event, combine the relevant row and column without double-counting their shared cell.
  • For an unconditional probability, divide the desired count by the table's grand total. Conditional probabilities use a restricted denominator and are covered in Section 5.4.

Quick reference

• Independent events do not change each other's probabilities; dependent events do.
  • Use the multiplication check: the events are independent when \(P(A\text{ and }B)=P(A)P(B)\).
  • Independence is an assumption that must fit the context. Repeated real-world outcomes are not automatically independent.
• Mutually exclusive and independent do not mean the same thing.
  • To check mutual exclusivity, ask: Can both events happen in the same trial? If not, they are mutually exclusive.
  • To check independence, ask: Does knowing that one event occurred change the probability of the other? If not, they are independent; if it does, they are dependent.
  • Mutually exclusive events with positive probabilities are dependent. Once one occurs, the probability of the other becomes 0.
  • Mutually exclusive events can also be independent only in the special case where at least one event has probability 0.
• For independent events, multiply probabilities to find an and probability.
  • \(P(A\text{ and }B)=P(A)P(B)\). For repeated independent trials with the same probability \(p\), the probability of success every time is \(p^n\).
  • Interpret a small probability as a long-run frequency when requested: expected repetitions = number of repetitions x probability.
• With replacement usually keeps selections independent; without replacement usually makes them dependent.
  • With replacement, restore the selected item before the next draw, so both the numerator and denominator return to their original values.
  • Without replacement, update the remaining favorable items and total items after each draw. A tree diagram helps display the changing branches.
• Use a complement for at least one.
  • \(P(\text{at least one})=1-P(\text{none})\).
  • For independent trials, multiply the failure probabilities to find none, then subtract from 1.

Quick reference

• Conditional probability restricts the sample space to known information.
  • \(P(A\mid B)\) means the probability of \(A\) given that \(B\) has occurred.
  • Use \(P(A\mid B)=\frac{P(A\text{ and }B)}{P(B)}\) or, with counts, number in both \(A\) and \(B\) / number in \(B\).
  • The event after the vertical bar supplies the denominator.
• The order in conditional notation matters.
  • \(P(A\mid B)\) and \(P(B\mid A)\) usually have different restricted sample spaces and different denominators.
  • Translate the condition into words before selecting numbers from a table.
• In a two-way table, condition on a row or column total.
  • Locate the row or column named after given. Its total is the denominator.
  • The numerator is the intersection cell that also satisfies the event before the vertical bar.
• The general multiplication rule works for independent or dependent events.
  • \(P(A\text{ and }B)=P(A)P(B\mid A)=P(B)P(A\mid B)\).
  • For independent events, the conditional probability equals the original probability, reducing the rule to \(P(A)P(B)\).
  • With replacement or replay allowed, the original probabilities return before the next selection. Without replacement, multiply along the desired path while updating the favorable count and total after every selection.
• Use addition when an event can occur through more than one disjoint path.
  • For exactly one of two types, calculate first type then second plus second type then first.
  • For one red and one yellow without replacement, add \(P(R\text{ then }Y)+P(Y\text{ then }R)\).
• Check independence by comparing conditional and unconditional probabilities.
  • For independent events, \(P(A\mid B)=P(A)\) whenever \(P(B)>0\). This is equivalent to \(P(A\text{ and }B)=P(A)P(B)\).
  • Compare the overall rate for an event with the rate inside the given group. These are \(P(A)\) and \(P(A\mid B)\).
  • If the rates are the same, allowing for normal rounding differences, the events are independent. If knowing the group changes the rate meaningfully, the events are dependent.

Quick reference

• The Fundamental Counting Principle multiplies the number of choices across successive stages.
  • If one stage has \(a\) choices and the next has \(b\), there are \(ab\) ordered outcomes.
  • When repetition is allowed, the same choice count may be reused at each stage, such as \(5^7\) seven-digit keypad codes using five keys.
• A factorial counts arrangements of distinct objects.
  • \(n!=n(n-1)(n-2)\cdots1\), and \(0!=1\).
  • Arranging all \(n\) distinct objects uses \(n!\).
• Use a permutation when order or assigned roles matter.
  • \({}_nP_r=\frac{n!}{(n-r)!}\) counts ordered selections of \(r\) distinct objects from \(n\).
  • Examples include race finishes, officer positions, and ordered playlists without repetition.
• Use a combination when only the selected group matters.
  • \({}_nC_r=\frac{n!}{r!(n-r)!}\) counts unordered selections of \(r\) objects from \(n\).
  • Examples include committees, card hands, lottery selections, and simple random samples.
• For arrangements with identical repeated objects, divide out duplicate orders.
  • If \(n\) objects include repeated groups of sizes \(n_1,n_2,\ldots\), the number of distinguishable arrangements is \(\frac{n!}{n_1!n_2!\cdots}\).
  • Use this for repeated letters, repeated tree types, or birth-order categories with fixed counts.
• Counting can supply the numerator and denominator of a probability.
  • For equally likely outcomes, \(P(E)=\frac{\text{favorable outcomes}}{\text{total outcomes}}\). Count both with methods that match the selection rules.
  • When selecting without replacement from two groups, count a specific composition by multiplying combinations. For example, choose the desired number of liked tracks and the remaining number of unliked tracks, then divide by the number of all possible selections.
  • On a TI-84, enter \(n\), press MATH → PRB, select nPr or nCr, enter \(r\), and press ENTER. Factorial is in the same menu.

Quick reference

• A random variable assigns a numerical value to each outcome of a probability experiment.
  • A discrete random variable has a finite or countable set of possible values, often counts such as 0, 1, 2, and so on.
  • A continuous random variable can take any value in an interval, often measurements such as time, weight, area, or distance.
  • State realistic possible values when classifying a variable; counts use whole numbers, while measurements may include decimals.
• A discrete probability distribution lists each possible value and its probability.
  • Every probability must satisfy \(0\le P(x)\le1\), and \(\sum P(x)=1\).
  • To find a missing probability, subtract the sum of the known probabilities from 1.
  • A probability histogram uses separated bars because the random variable takes discrete values.
• Expected value is the probability-weighted long-run mean.
  • \(\mu_X=E(X)=\sum xP(x)\). Multiply each value by its probability and add the products.
  • Interpret expected value as an average over many repetitions, not as a prediction for one trial. It does not need to be a possible individual outcome.
  • For games, insurance, or investments, a positive expected value is an average gain and a negative expected value is an average loss from the stated point of view.
  • Distinguish a payout or prize from net gain. Subtract the cost to play when the listed values are payouts, but do not subtract it again when the problem already gives net gains and losses.
• The standard deviation describes the typical distance of outcomes from the expected value.
  • For formula context, \(\sigma_X=\sqrt{\sum (x-\mu_X)^2P(x)}\).
  • Report the standard deviation in the same units as the random variable. A larger value means more variable outcomes.
• TI-84 weighted-statistics workflow:
  • Enter possible values in L1 and probabilities in L2.
  • Run 1-Var Stats with List:L1 and FreqList:L2.
  • Read \(\bar{x}\) as the expected value and \(\sigma_x\) as the distribution's standard deviation. The displayed \(n\) may equal 1 when probabilities are used as weights; it is not the number of trials.

Quick reference

• A binomial experiment must satisfy four conditions.
  • There is a fixed number of trials, \(n\).
  • Each trial has two outcomes labeled success and failure, the trials are independent, and the success probability \(p\) remains constant.
  • Sampling without replacement can be treated as approximately independent when the sample is no more than 5% of a large population.
• Define the binomial variables before calculating.
  • \(n\) is the number of trials, \(p\) is the success probability, \(q=1-p\) is the failure probability, and \(X\) counts successes.
  • State what success means in context; it is simply the outcome being counted, not necessarily a desirable result.
• The binomial probability formula shows the probability structure.
  • \(P(X=x)=\binom{n}{x}p^xq^{n-x}\). The combination counts where the successes can occur.
  • Use the formula for context and the TI-84 for routine probability calculations.
• Translate probability wording before choosing a calculator command.
  • binompdf finds the probability of one exact count. binomcdf accumulates probabilities from 0 through its final argument.
  • Exactly \(k\): \(P(X=k)\), so use binompdf(n,p,k).
  • At most or no more than \(k\): \(P(X\le k)\), so use binomcdf(n,p,k).
  • Fewer than or less than \(k\): \(P(X\lt k)=P(X\le k-1)\), so use binomcdf(n,p,k-1).
  • More than or greater than \(k\): \(P(X\gt k)=1-P(X\le k)\), so use 1-binomcdf(n,p,k).
  • At least \(k\): \(P(X\ge k)=1-P(X\le k-1)\), so use 1-binomcdf(n,p,k-1).
  • Between \(a\) and \(b\), inclusive: use binomcdf(n,p,b)-binomcdf(n,p,a-1).
  • Strictly between \(a\) and \(b\): use binomcdf(n,p,b-1)-binomcdf(n,p,a).
• A binomial distribution has mean \(\mu=np\) and standard deviation \(\sigma=\sqrt{npq}\).
  • The mean is the expected number of successes over many repetitions of the whole experiment; it need not be a whole number.
  • A count more than about 2 standard deviations from the mean may be considered unusual. Use probabilities when the question gives a specific unusual-event cutoff.
• The binomial shape depends on \(n\) and \(p\).
  • When \(p=0.5\), the distribution is symmetric. Small \(p\) tends to skew right, and large \(p\) tends to skew left.
  • As \(n\) grows and both \(np\) and \(nq\) grow, the distribution becomes more bell-shaped.

Quick reference

• A continuous probability density curve represents probabilities by area.
  • The total area under the curve is 1, and the area over an interval equals the probability that the variable falls in that interval.
  • For a continuous variable, a single exact value has area and probability 0, so \(P(X\lt a)=P(X\le a)\).
• A continuous uniform distribution has constant height over its interval.
  • Every equal-width subinterval has the same probability. Probability = interval length / total length.
  • For a uniform distribution from \(a\) to \(b\), the rectangle's height is \(1/(b-a)\).
• A normal distribution is bell-shaped, symmetric, and determined by \(\mu\) and \(\sigma\).
  • The mean, median, and mode are equal at the center. Half the area lies on each side of the mean, and the curve approaches but never touches the horizontal axis.
  • Changing \(\mu\) shifts the center. A larger \(\sigma\) makes the curve wider and lower; a smaller \(\sigma\) makes it narrower and taller.
  • The inflection points occur at \(\mu-\sigma\) and \(\mu+\sigma\), where the curve changes bending direction.
• A z-score standardizes a normal value.
  • \(z=\frac{x-\mu}{\sigma}\) tells how many standard deviations \(x\) is above or below the mean.
  • The standard normal distribution has \(\mu=0\) and \(\sigma=1\). Standardizing changes the scale, not the corresponding area.
• Normal distributions satisfy the bell-shaped condition for the Empirical Rule reviewed in Section 3.2.
  • Refer to the Section 3.2 visual and examples for the 68-95-99.7 estimates.
  • Use exact normal calculator methods in Section 7.2 when cutoffs are not whole numbers of standard deviations from the mean.

Quick reference

• Start by sketching or naming the region under the normal curve.
  • Less than shades left, greater than shades right, and between shades the region between two cutoffs.
  • Probability is area, so the answer must be between 0 and 1. Use symmetry and the cutoff's location to check whether the result is reasonable.
• Use normalcdf(lower,upper,mean,standard deviation) to find normal probabilities.
  • For less than \(a\), use a very large negative lower bound and upper bound \(a\). For greater than \(a\), use lower bound \(a\) and a very large positive upper bound.
  • For between \(a\) and \(b\), enter the two cutoffs in increasing order. For two tails, calculate each tail and add or use a complement when easier.
  • For standard-normal z-scores, use mean 0 and standard deviation 1.
• Use invNorm(left-tail area,mean,standard deviation) to find a cutoff from an area.
  • A percentile is already a left-tail area. For a right-tail area \(r\), enter \(1-r\).
  • For the middle \(C\)% of a symmetric normal distribution, split the remaining area equally between the two tails, then find both cutoff percentiles.
  • Use mean 0 and standard deviation 1 for z-cutoffs, or enter the original \(\mu\) and \(\sigma\) to obtain raw-value cutoffs directly.
• The notation \(z_\alpha\) identifies a standard-normal right-tail cutoff.
  • \(z_\alpha\) is the z-score with area \(\alpha\) to its right, so the area to its left is \(1-\alpha\).
  • On a TI-84, use invNorm(1-alpha,0,1). For \(z_{\alpha/2}\), use left-tail area \(1-\alpha/2\); the middle area between \(-z_{\alpha/2}\) and \(z_{\alpha/2}\) is \(1-\alpha\).
• Interpret the calculator result in context.
  • A probability describes one randomly selected observation; the same decimal can be interpreted as an approximate long-run proportion or percentage.
  • A percentile cutoff is a value with the stated percentage of observations at or below it.

Quick reference

• Assess normality with graphs rather than expecting perfect normal data.
  • A roughly bell-shaped, symmetric histogram with one main peak supports a normal model.
  • Strong skewness, multiple peaks, large gaps, or clear outliers are warnings against a normal model. A boxplot can help reveal skewness and outliers but does not show the full shape.
• A normal probability plot compares ordered observations with expected normal z-scores.
  • Points that follow an approximately straight line support a normal model.
  • Systematic curvature suggests skewness or another nonnormal shape. A point far from the line may indicate an outlier.
  • To construct the plot, order the observations, calculate \(f_i=(i-0.375)/(n+0.25)\), find each expected score with invNorm(fi,0,1), and plot observed values against expected z-scores.
• Use the normal-probability-plot critical-value table for the correlation check.
  • Find the correlation between the ordered observations and their expected z-scores, then find the critical value for \(n\).
  • For more than 30 observations, use the \(n=30\) critical value, following the homework convention.
  • If the correlation is greater than the critical value, the plot is sufficiently linear to support normality. If it is not greater, the data do not pass that normality check.
• Use all available evidence and context.
  • A small sample can make a histogram jagged, while a very large sample can reveal minor departures from normality. Focus on departures that matter for the intended method.
  • Normality checks decide whether normal-based methods are reasonable; they do not prove that the population is exactly normal.
• Technology can create the required displays.
  • StatCrunch is useful for normal probability plots, histograms, and larger data sets.
  • On a TI-84, enter the ordered observations in L1 and expected z-scores in L2. A normal probability plot can be selected in STAT PLOT; LinReg(ax+b) L1,L2 gives the correlation when diagnostics are on. Compare that correlation with the quick-reference critical value.

Critical Values for Normal Probability Plots

Quick reference

• The sampling distribution of \(\bar{x}\) is the distribution of sample means from all possible random samples of the same size \(n\).
  • The population distribution describes individual values; one sample has its own data distribution; the sampling distribution describes how \(\bar{x}\) varies from sample to sample.
  • A simulation approximates the sampling distribution by repeatedly taking samples, computing each mean, and graphing those means.
• The sampling distribution is centered at the population mean and has less spread than individual values.
  • \(\mu_{\bar{x}}=\mu\). The sample mean is an unbiased estimator because its long-run mean equals the population mean.
  • \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). This standard deviation of sample means is called the standard error.
  • As \(n\) increases, standard error decreases, so larger samples produce means that cluster more tightly around \(\mu\).
• Check independence and shape before using a normal model for \(\bar{x}\).
  • Use a simple random sample or a design that produces representative, approximately independent observations. When sampling without replacement, check \(n\le0.05N\).
  • If the population is normal, the sampling distribution of \(\bar{x}\) is normal for any sample size.
  • If the population is not normal, the Central Limit Theorem says the sampling distribution becomes approximately normal as \(n\) grows. \(n\ge30\) is a common introductory guideline, but strong skewness or outliers may require a larger sample.
• Use the sampling-distribution parameters, not the population spread, in probability calculations.
  • For probabilities involving one individual value \(X\), use mean \(\mu\) and standard deviation \(\sigma\).
  • For probabilities involving a sample mean \(\bar{x}\), use mean \(\mu\) and standard error \(\sigma/\sqrt{n}\).
  • On the TI-84, use normalcdf(lower,upper,mean,standard deviation), entering \(\mu\) for the mean and \(\sigma/\sqrt{n}\) for the standard deviation. Interpret the result as a proportion of samples or a probability for a randomly selected sample.

Quick reference

• The sample proportion is \(\hat{p}=x/n\), where \(x\) is the number of sampled individuals with the characteristic.
  • The population proportion \(p\) is a fixed parameter; \(\hat{p}\) varies from sample to sample and estimates \(p\).
  • The sampling distribution of \(\hat{p}\) describes the proportions from all possible random samples of the same size.
• The sampling distribution is centered at \(p\) and becomes less variable as \(n\) increases.
  • \(\mu_{\hat{p}}=p\), so the sample proportion is an unbiased estimator of the population proportion.
  • \(\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}\). This is the standard error when the population proportion is used in a sampling-distribution problem.
• Check the sampling conditions before using a normal model for \(\hat{p}\).
  • Use a simple random sample or another representative random process. For sampling without replacement, check \(n\le0.05N\).
  • For this course's homework, check \(np(1-p)\ge10\). If this value is below 10, the sampling distribution may be too skewed for the normal approximation.
  • When solving for a required sample size, isolate \(n\), round the result up, and subtract the current sample size if the question asks how many additional individuals are needed.
• Use the proportion mean and standard error in normal probability calculations.
  • On the TI-84, use normalcdf(lower,upper,p,standard deviation), entering \(\sqrt{p(1-p)/n}\) for the standard deviation.
  • Interpret the result as the probability that a random sample produces a proportion in the stated range, or as the long-run proportion of such samples.

Quick reference

• A point estimate is a sample statistic used to estimate a population parameter.
  • For a population proportion \(p\), the point estimate is \(\hat{p}=x/n\). The variable must have two outcomes after defining success and failure.
  • A confidence interval has the form point estimate +/- margin of error. Its midpoint is the point estimate, and \(E=(\text{upper}-\text{lower})/2\).
• The confidence level describes the long-run success rate of the interval method.
  • The population proportion is fixed but unknown. Different random samples produce different sample proportions and therefore different intervals.
  • If many random samples were taken and an interval were built from each, about \(C\)% of those intervals would capture the fixed population proportion.
  • Once one interval is computed, the fixed proportion either is or is not inside it. Say, 'We are \(C\)% confident that the interval captures the population proportion,' not that there is a \(C\)% probability the parameter is inside.
• Check conditions for a one-proportion z-interval.
  • Use a random or representative sample and, when sampling without replacement, check \(n\le0.05N\).
  • For this course's homework, check \(n\hat{p}(1-\hat{p})\ge10\) using the observed sample proportion.
• For formula context, \(\hat{p}\pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\).
  • Higher confidence uses a larger critical value and produces a wider interval when other information is fixed.
  • A larger sample size reduces standard error and produces a narrower interval when the confidence level and sample proportion are fixed.
• The critical value \(z^*=z_{\alpha/2}\) captures the middle \((1-\alpha)100\%\) of the standard normal distribution.
  • On the TI-84, use invNorm(1-alpha/2,0,1). Common values are 1.645 for 90%, 1.960 for 95%, and 2.576 for 99% confidence.
  • Each tail has area \(\alpha/2\), so the positive critical value has left-tail area \(1-\alpha/2\).
• Choose a sample size large enough to achieve the requested margin of error \(E\).
  • With a prior estimate \(\tilde{p}\), use \(n=\tilde{p}(1-\tilde{p})(z^*/E)^2\). Without a prior estimate, use \(n=0.25(z^*/E)^2\).
  • Write a percent margin of error as a decimal and always round the calculated sample size up to the next whole number.
• TI-84 1-PropZInt workflow:
  • Open STAT → TESTS → 1-PropZInt. Enter the number of successes \(x\), sample size \(n\), and confidence level, then calculate.
  • Read the interval and \(\hat{p}\), then write a contextual interpretation naming the population and the characteristic.

Quick reference

• The sample mean \(\bar{x}\) is the point estimate for a population mean \(\mu\).
  • A mean confidence interval has the form \(\bar{x}\pm E\). Its midpoint is \(\bar{x}\), and its margin of error is half the interval width.
  • Use a one-sample t-interval when the population standard deviation \(\sigma\) is unknown and is estimated with the sample standard deviation \(s\).
• The t-distribution accounts for the additional uncertainty from estimating \(\sigma\).
  • Degrees of freedom tell how many sample values can vary freely after using the sample to estimate its mean. For one sample, \(df=n-1\): once the mean and \(n-1\) values are fixed, the last value is forced to make the deviations balance around the mean.
  • Degrees of freedom select the appropriate t-distribution and critical value. Smaller \(df\) means more uncertainty, heavier tails, and a larger critical value.
  • The t-distribution is symmetric and bell-shaped. As degrees of freedom increase, it approaches the standard normal distribution.
• The critical value \(t^*=t_{\alpha/2}\) depends on the confidence level and degrees of freedom.
  • For a \((1-\alpha)100\%\) interval, use the t-table or invT(1-alpha/2,df) with \(df=n-1\).
  • The interval extends \(t^*\) standard errors below and above the sample mean.
• Check conditions for a one-sample t-interval.
  • Use a random or representative sample and check \(n\le0.05N\) when sampling without replacement.
  • For a small sample, the population should be approximately normal with no strong skewness or outliers. Larger samples make t-procedures more robust, but severe outliers still matter.
• For formula context, \(\bar{x}\pm t^*\frac{s}{\sqrt{n}}\).
  • Higher confidence increases \(t^*\) and the margin of error. Increasing \(n\) decreases the standard error and margin of error.
  • Interpret the interval as a plausible range for the population mean in the original units, not as a range containing a stated percentage of individual observations.
• Use confidence wording because the population mean is fixed while the interval varies from sample to sample.
  • In repeated sampling, a \(C\)% confidence-interval method captures the fixed population mean in about \(C\)% of the intervals it produces.
  • After one interval is calculated, the mean either is or is not inside it. Say, 'We are \(C\)% confident that the interval captures the population mean,' not that there is a \(C\)% probability the fixed mean is inside.
• For planning, estimate the sample size with \(n=(z^*\sigma/E)^2\).
  • Use a known or reasonable planning value for \(\sigma\); a standard deviation from an earlier study may be used when the population standard deviation is unavailable.
  • Use the z critical value because the t critical value depends on the sample size being found, and always round the result up.
• TI-84 TInterval workflow:
  • Choose Data when raw observations are in a list, or Stats when \(\bar{x}\), \(s\), and \(n\) are given. Enter the confidence level and calculate.
  • Read the interval and degrees of freedom, then state: 'We are \(C\)% confident that the population mean [context] is between [lower] and [upper] [units].'

Quick reference

• A hypothesis test evaluates a claim about a population parameter using sample evidence.
  • Write hypotheses with parameters such as \(p\), \(\mu\), or \(\sigma\), not sample statistics such as \(\hat{p}\), \(\bar{x}\), or \(s\).
  • The null hypothesis \(H_0\) contains equality and represents the starting status or claimed value. The alternative \(H_a\) states the change or difference the evidence is meant to support.
• The alternative hypothesis determines the test direction.
  • \(\lt\) gives a left-tailed test, \(\gt\) gives a right-tailed test, and \(\ne\) gives a two-tailed test. Left- and right-tailed tests are also called one-tailed tests.
  • Words such as decreased or less suggest left-tailed; increased, more, or greater suggest right-tailed; changed or different suggest two-tailed.
• The P-value measures evidence against \(H_0\).
  • It is the probability, assuming \(H_0\) is true, of obtaining a sample result at least as extreme in the direction of \(H_a\) as the observed result.
  • Smaller P-values give stronger evidence against \(H_0\). A P-value is not the probability that \(H_0\) is true.
• The P-value and critical-value approaches use the same evidence and make the same decision.
  • P-value approach: reject \(H_0\) when P-value \(\le\alpha\); otherwise, fail to reject \(H_0\).
  • Critical-value approach: use \(\alpha\) and the direction of \(H_a\) to mark the rejection region. Reject \(H_0\) when the test statistic falls beyond the critical value in that region.
  • When the test statistic is beyond the critical value, its P-value area is no larger than \(\alpha\), so both approaches reject. Otherwise, both approaches fail to reject.
  • Fail to reject does not mean accept or prove \(H_0\); the evidence may simply be insufficient.
  • A smaller \(\alpha\) makes rejection harder and lowers the chance of a Type I error.
• A testing decision can be wrong.
  • Type I error: reject a true \(H_0\). Its probability is \(\alpha\). Describe the false conclusion in the problem's context.
  • Type II error: fail to reject a false \(H_0\). Describe the real change or difference the test failed to detect.
• Write the conclusion in terms of the original claim.
  • When rejecting \(H_0\), say there is sufficient evidence for the alternative claim.
  • When failing to reject \(H_0\), say there is not sufficient evidence for the alternative claim. If the original claim is the null, say the data do not provide sufficient evidence against that claim; do not say the claim was proven true.

Quick reference

• Use a one-proportion z-test for a claim about one population proportion \(p\).
  • Write \(H_0:p=p_0\). Choose \(H_a:p\lt p_0\), \(p\gt p_0\), or \(p\ne p_0\) from the claim's wording.
  • Use the population parameter \(p\) in the hypotheses, not the observed sample proportion \(\hat{p}\).
• Check the one-proportion test conditions using the null value.
  • Use a random or representative sample and check \(n\le0.05N\) when sampling without replacement.
  • For this course's homework, check \(np_0(1-p_0)\ge10\). Test conditions use the null proportion \(p_0\), unlike interval conditions, which use the observed sample proportion.
• The one-proportion test statistic compares the observed proportion with the null proportion.
  • For formula context, \(z=\frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}\).
  • The P-value is the null-model tail area at least as extreme as the observed \(z\), using the direction in \(H_a\).
• Use either the P-value approach or the critical-value approach for a one-proportion z-test.
  • P-value approach: reject \(H_0\) when the calculator's P-value is no larger than \(\alpha\).
  • Critical-value approach: for a left-tailed test use invNorm(alpha,0,1); for a right-tailed test use invNorm(1-alpha,0,1); for a two-tailed test use the positive and negative values from invNorm(1-alpha/2,0,1).
  • Reject when the observed \(z\) falls beyond the critical value in the rejection region. Both approaches must give the same decision.
• TI-84 1-PropZTest workflow:
  • Open STAT → TESTS → 1-PropZTest. Enter \(p_0\), successes \(x\), sample size \(n\), and the alternative symbol.
  • Read \(z\), the P-value, and \(\hat{p}\). Compare the P-value with \(\alpha\); do not decide from whether \(\hat{p}\) is merely above or below \(p_0\).
• Use the Draw option in 1-PropZTest to see the P-value graph.
  • After entering the 1-PropZTest information and selecting the alternative symbol, highlight Draw instead of Calculate and press ENTER.
  • The calculator draws the standard normal curve, shades the P-value area, and displays the test statistic and P-value. A less-than alternative shades the left tail, a greater-than alternative shades the right tail, and a not-equal alternative shades both tails.
  • The Draw screen illustrates the P-value approach. It does not mark the critical z-value, so find that separately with invNorm when the critical-value approach is required.
• State the conclusion with evidence wording and context.
  • Reject \(H_0\): 'There is sufficient evidence to conclude that [alternative claim about the population proportion].'
  • Fail to reject \(H_0\): 'There is not sufficient evidence to conclude that [alternative claim].' Do not say the null is true or that no difference exists.
• Statistical significance does not guarantee practical importance.
  • A very large sample can make a small departure from \(p_0\) statistically significant. Report the observed size of the difference and its context when practical importance matters.

Quick reference

• Use a one-sample t-test for a claim about one population mean when \(\sigma\) is unknown.
  • Write \(H_0:\mu=\mu_0\). Choose a left-, right-, or two-tailed alternative from the claim.
  • Use the population mean \(\mu\) in the hypotheses, not the sample mean \(\bar{x}\). Degrees of freedom are \(df=n-1\).
• Check the one-sample t-test conditions.
  • Use a random or representative sample and check \(n\le0.05N\) when sampling without replacement.
  • For a small sample, verify approximate normality and no strong outliers with the data, histogram, boxplot, or normal probability plot. Larger samples are more robust, but extreme outliers can still distort the test.
• The t statistic measures the sample mean's distance from the null mean in standard-error units.
  • For formula context, \(t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}\).
  • The P-value is the t-distribution tail area at least as extreme as the observed statistic, using \(df=n-1\) and the direction in \(H_a\).
• Tail direction controls the critical value and the P-value area.
  • For a right-tailed test use \(t_{\alpha}\); for a left-tailed test use \(-t_{\alpha}\); for a two-tailed test use \(\pm t_{\alpha/2}\), all with \(df=n-1\).
  • Because invT uses left-tail area, enter invT(1-alpha,df) for a right critical value, invT(alpha,df) for a left critical value, or invT(1-alpha/2,df) for the positive two-tailed critical value.
  • Shade the P-value to the left for \(H_a:\mu\lt\mu_0\), to the right for \(H_a:\mu\gt\mu_0\), and in both tails for \(H_a:\mu\ne\mu_0\).
  • The P-value approach rejects when P-value \(\le\alpha\). The critical-value approach rejects when \(t\) falls beyond the critical value. Both approaches must agree.
• TI-84 T-Test workflow:
  • Choose Data when raw observations are stored in a list, or Stats when \(\bar{x}\), \(s\), and \(n\) are given. Enter \(\mu_0\) and select the correct alternative.
  • Read \(t\), the P-value, \(\bar{x}\), and \(n\). Compare the P-value with \(\alpha\).
• Use the Draw option in T-Test to see the P-value graph.
  • After entering the T-Test information and selecting the alternative symbol, highlight Draw instead of Calculate and press ENTER.
  • The calculator draws the t distribution, shades the P-value area, and displays the test statistic and P-value. A less-than alternative shades the left tail, a greater-than alternative shades the right tail, and a not-equal alternative shades both tails.
  • The Draw screen illustrates the P-value approach. It does not mark the critical t-value, so find that separately with invT when the critical-value approach is required.
• Finish with a decision and contextual conclusion.
  • Reject \(H_0\): there is sufficient evidence for the alternative claim about the population mean.
  • Fail to reject \(H_0\): there is not sufficient evidence for the alternative claim. Do not say the population mean equals \(\mu_0\) or that \(H_0\) was accepted.
• Separate statistical significance from practical significance.
  • A small P-value says the sample result would be unusual under \(H_0\); it does not say the difference \(\bar{x}-\mu_0\) is large enough to matter in practice.
  • Report the estimated difference and its units when judging real-world importance.