Statistics

TI-84-focused notes, quick references, and representative examples for an introductory statistics course.

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Module 1: Introduction, Data Collection, Sampling, and Experimental Design

Students learn the language of statistics, study design, sampling methods, bias, and experiment design.

Section 1.1

Introduction to the Practice of Statistics

Use the basic vocabulary of statistics and classify variables correctly.

Quick reference

  • Statistics is the science of collecting, organizing, summarizing, and analyzing information to answer questions.
  • Descriptive statistics summarizes data. Inferential statistics uses sample data to draw conclusions about a population.
  • A population is the entire group being studied. A sample is a subset of that population.
  • A parameter describes a population. A statistic describes a sample.
  • Qualitative variables describe categories. Quantitative variables are numerical and may be discrete or continuous.
Example 1. A researcher surveys \(420\) adults from a city to estimate the percent of all adults in the city who support a new park. Identify the population and sample.
Example 2. A report says \(61\%\) of the \(420\) surveyed adults support the park. Is \(61\%\) a statistic or a parameter?
Example 3. Classify each variable as qualitative or quantitative: eye color, number of siblings, commute time, and phone brand.
Example 4. A graph summarizes the ages of students in one class. Is this descriptive or inferential statistics?
Section 1.2

Observational Studies versus Designed Experiments

Distinguish observational studies from experiments and avoid unsupported causal claims.

Quick reference

  • An observational study measures characteristics without trying to change the subjects.
  • A designed experiment imposes treatments and observes the response.
  • The explanatory variable may help explain changes in the response variable.
  • A lurking or confounding variable can make it hard to know what caused an observed result.
  • Observational studies can show association, but they do not prove cause and effect by themselves.
Example 1. A researcher records coffee intake and sleep hours for \(200\) adults without assigning coffee amounts. Is this observational or experimental?
Example 2. Students are randomly assigned to either use a new study app or not use it, then their exam scores are compared. Is this observational or experimental?
Example 3. In a study of study time and exam score, identify the explanatory variable and response variable.
Example 4. A study finds that people who exercise more also have lower stress. What wording is safest?
Section 1.3

Simple Random Sampling

Recognize and create simple random samples.

Quick reference

  • A census collects data from every individual in the population.
  • A simple random sample of size \(n\) gives every group of \(n\) individuals the same chance of being selected.
  • A sampling frame is the list of individuals from which the sample is chosen.
  • Random sampling helps reduce selection bias.
  • TI-84 note: randInt( can generate random integers when individuals are numbered.
Example 1. A school numbers all \(1200\) students from \(1\) to \(1200\), then uses random integers to choose \(80\) student numbers. What sampling method is being used?
Example 2. A teacher surveys the first \(30\) students who enter the cafeteria. Is this a simple random sample?
Example 3. A city wants a simple random sample of \(50\) addresses from a list of \(18{,}000\) addresses. What should be numbered?
Example 4. Why is random selection preferred over letting volunteers respond?
Section 1.4

Other Effective Sampling Methods

Classify stratified, cluster, and systematic sampling.

Quick reference

  • Stratified sampling separates the population into groups, then samples from every group.
  • Cluster sampling separates the population into groups, randomly selects some groups, then surveys everyone in those selected groups.
  • Systematic sampling selects every \(k\)th individual after a random starting point.
  • Convenience sampling uses individuals who are easy to reach and is not an effective random method.
  • Stratified samples preserve representation from each group; cluster samples are often cheaper to collect.
Example 1. A college samples \(40\) freshmen, \(40\) sophomores, \(40\) juniors, and \(40\) seniors. What method is this?
Example 2. A researcher randomly selects \(8\) classrooms and surveys every student in those classrooms. What method is this?
Example 3. A quality checker inspects every \(25\)th item coming off an assembly line after a random start. What method is this?
Example 4. A website posts a poll and lets anyone answer. What sampling concern is present?
Section 1.5

Bias in Sampling

Identify common sources of bias and improve sampling plans.

Quick reference

  • Sampling bias occurs when the sampling method tends to favor certain outcomes.
  • Undercoverage happens when some groups in the population are left out or underrepresented.
  • Nonresponse bias happens when selected individuals do not respond.
  • Response bias happens when answers are inaccurate because of wording, interviewer effects, or other pressure.
  • Wording bias occurs when the question wording influences the response.
Example 1. A phone survey only calls landlines to estimate opinions of all adults. What type of bias is likely?
Example 2. A survey asks, "Do you support the wasteful new tax increase?" What type of bias is present?
Example 3. A mail survey is sent to \(2000\) people, but only \(180\) respond. What bias is a concern?
Example 4. How could a mall-intercept survey about city transportation be improved?
Section 1.6

The Design of Experiments

Identify experimental design features and explain their purpose.

Quick reference

  • Experimental units are the individuals or objects receiving treatments.
  • A treatment is a condition applied in an experiment.
  • Random assignment helps create comparable treatment groups.
  • A control group provides a comparison, and a placebo can help measure placebo effect.
  • Blinding hides treatment information from subjects or evaluators; double-blinding hides it from both.
Example 1. A study randomly assigns \(90\) patients to a new medication, old medication, or placebo. Identify the experimental units and treatments.
Example 2. Why should patients be randomly assigned to treatments?
Example 3. A researcher groups subjects by age group first, then randomly assigns treatments within each age group. What design is being used?
Example 4. In a drug study, neither patients nor doctors evaluating outcomes know who received the real drug. What is this called?

Module 2: Organizing and Displaying Data

Students organize qualitative and quantitative data using tables and graphs, then identify misleading displays.

Section 2.1

Organizing Qualitative Data

Build and interpret frequency tables and categorical graphs.

Quick reference

  • Frequency is the count in a category.
  • Relative frequency is \(\dfrac{\text{category frequency}}{\text{total frequency}}\).
  • Bar graphs and Pareto charts are common displays for qualitative data.
  • A Pareto chart orders bars from largest frequency to smallest frequency.
  • Pie charts show parts of a whole, usually as percentages.
Example 1. Use the table and bar graph to find the relative frequency for each favorite subject.
Subject Math English Science History
Frequency 18 12 15 5

Favorite Subject Bar Graph

The tallest bar is Math at 18 students, followed by Science, English, and History.

Bar graph of favorite subjects with Math 18, English 12, Science 15, and History 5 Favorite Subject Subject Frequency 18 12 15 5 Math English Science History
Example 2. Use the Pareto chart. Which club has the greatest frequency, and why is this graph a Pareto chart?

Club Signups Pareto Chart

The categories are ordered from largest frequency to smallest frequency.

Pareto chart of club signups ordered from Drama 24 to Music 20 to Robotics 11 to Chess 5 After-School Club Signups Club Frequency 24 20 11 5 Drama Music Robotics Chess
Example 3. Use the pie chart. Which format is most popular, and what percent chose either online or hybrid?

Preferred Course Format

Online is the largest slice at 40 percent, and Weekend is the smallest slice at 15 percent.

Pie chart of preferred course formats with online 40 percent, hybrid 25 percent, evening 20 percent, and weekend 15 percent Preferred Course Format Online 40% Hybrid 25% Evening 20% Weekend 15%
Example 4. Should a histogram or bar graph be used for eye-color categories?
Section 2.2

Organizing Quantitative Data: The Popular Displays

Read and create grouped frequency displays for quantitative data.

Quick reference

  • A frequency distribution groups quantitative data into classes.
  • Class width is the distance from one lower class limit to the next lower class limit.
  • A histogram uses touching bars because the data are quantitative.
  • Distribution shapes may be symmetric, skewed left, skewed right, uniform, or bimodal.
  • Relative-frequency histograms show proportions instead of counts.
Example 1. Complete the grouped-frequency idea: the lower class limits are \(10,20,30,40\). What is the class width?
Class 10-19 20-29 30-39 40-49
Frequency 6 11 9 4
Example 2. Use the histogram. Describe the shape of the distribution.

Clinic Wait Times

Most wait times are in the lower classes, and the bars taper to the right.

Histogram of wait times with high frequencies in lower classes and a right tail Clinic Wait Times Wait time in minutes Frequency 0-9 10-19 20-29 30-39 40-49
Example 3. In the wait-time histogram, which class has the greatest frequency?

Clinic Wait Times

Most wait times are in the lower classes, and the bars taper to the right.

Histogram of wait times with high frequencies in lower classes and a right tail Clinic Wait Times Wait time in minutes Frequency 0-9 10-19 20-29 30-39 40-49
Example 4. A class has frequency \(12\) out of \(80\). What is the relative frequency?
Section 2.3

Additional Displays of Quantitative Data

Use dot plots, stem-and-leaf plots, time-series plots, and ogives when appropriate.

Quick reference

  • A dot plot stacks or places dots to show each data value.
  • A stem-and-leaf plot keeps the original data values visible while organizing them.
  • A time-series plot shows data values over time.
  • An ogive displays cumulative frequency or cumulative relative frequency.
  • Choose a display based on the data type and what pattern you need to show.
Example 1. Use the dot plot. What commute time occurs most often?

Commute Time Dot Plot

Each dot represents one student. The most common commute time is 15 minutes.

Dot plot of commute times with most dots around 15 minutes Student Commute Times Minutes 5 10 15 20 25
Example 2. Use the stem-and-leaf plot. What original data value is represented by stem \(6\) with leaf \(2\)?

Key: 4|7 represents 47.

Stem Leaves
4 2 7 9
5 0 3 8
6 2 2 5
7 1
Example 3. Use the time-series plot. What trend is shown from January through May?

Tutoring Center Visits

The line rises over time, showing an upward trend from January to May.

Time-series plot of tutoring visits increasing from January to May Tutoring Center Visits Month Visits JanFebMarAprMay
Example 4. Use the ogive. What does the point at upper class boundary \(79.5\) represent?

Reading Score Ogive

The cumulative frequency increases from left to right because each point includes all earlier classes.

Ogive showing cumulative frequency increasing as score classes increase Cumulative Frequency for Reading Scores Upper class boundary Cumulative frequency 59.569.579.589.599.5
Section 2.4

Graphical Misrepresentations of Data

Identify graphs that distort or exaggerate data.

Quick reference

  • A graph can mislead if the vertical axis is truncated or scaled oddly.
  • Pictographs can mislead when pictures change area rather than just height.
  • Different class widths can distort histograms.
  • Missing labels, units, or context make graphs harder to interpret correctly.
  • A good graph should show the data honestly and clearly.
Example 1. Compare the two bar graphs. Why is the left graph misleading?

Same Data, Different Vertical Scales

The left graph starts at 48 and exaggerates the change. The right graph starts at 0 and shows the change more honestly.

Two bar graphs comparing a truncated vertical axis to a full zero-based vertical axis Median Wait Time, Same Data Misleading: starts at 48 Better: starts at 0 48 52 56 MonTueWedThu 0 30 60 MonTueWedThu
Example 2. A pictograph doubles both height and width of a symbol. Why is that misleading?

Pictograph Area Distortion

Doubling both height and width makes the area four times as large.

Pictograph comparison showing a small square and a square with double height and double width Same Symbol Scaled Two Ways Original 2x height and width Area is 4 times as large
Example 3. A graph has no axis labels or units. What is the problem?
Example 4. How can a misleading graph usually be improved?

Module 3: Descriptive Statistics

Students compute and interpret center, spread, position, outliers, five-number summaries, and boxplots.

Section 3.1

Measures of Central Tendency

Find and interpret mean, median, and mode.

Quick reference

  • The mean is the arithmetic average.
  • The median is the middle value after data are ordered.
  • The mode is the value that occurs most often.
  • The median is resistant to outliers; the mean is not.
  • TI-84 note: enter data in STAT lists and use 1-Var Stats for summary statistics.
Example 1. Find the mean, median, and mode of \(4,7,7,10,12\).
Example 2. Which measure of center is more resistant to the outlier in \(8,9,10,11,52\)?
Example 3. A data set has no repeated values. What is the mode?
Example 4. A TI-84 1-Var Stats output is shown in the answer. Which entries give the mean and median?
Section 3.2

Measures of Dispersion

Use range, variance, and standard deviation to describe spread.

Quick reference

  • Range is maximum minus minimum.
  • Standard deviation measures a typical distance from the mean.
  • Sample standard deviation uses \(s\); population standard deviation uses \(\sigma\).
  • The long standard-deviation formulas explain the idea, but calculator output is usually used for computation.
  • TI-84 warning: Sx is sample standard deviation and sigma x is population standard deviation.
Example 1. Find the range of \(12,18,21,25,30\).
Example 2. A teacher records scores from one class and treats them as a sample of possible students. Should the TI-84 output use \(Sx\) or \(\sigma x\)?
Example 3. A data set includes every employee at a small company. Should the standard deviation be \(s\) or \(\sigma\)?
Example 4. Two classes have the same mean score. Class A has standard deviation \(4\), and Class B has standard deviation \(12\). Which class has more spread?
Section 3.3

Measures of Central Tendency and Dispersion from Tables

Use frequency tables and grouped data to estimate summary measures.

Quick reference

  • A frequency table can be used like repeated data values.
  • For a frequency table, multiply each value by its frequency before adding.
  • Grouped-data summaries using class midpoints are approximations.
  • TI-84 can use one list for values and another list for frequencies.
  • Always interpret summaries in the units of the original variable.
Example 1. Use the frequency table to find the mean.
Score 6 8 10
Frequency 2 3 5
Example 2. For grouped data, why do we use class midpoints?
Example 3. On the TI-84, what does the frequency list represent in 1-Var Stats?
Section 3.4

Measures of Position and Outliers

Use z-scores, percentiles, quartiles, and outlier fences.

Quick reference

  • A z-score tells how many standard deviations a value is from the mean: \(z=\dfrac{x-\mu}{\sigma}\) or \(z=\dfrac{x-\bar{x}}{s}\).
  • Positive z-scores are above the mean; negative z-scores are below the mean.
  • A percentile describes the percent of data at or below a value.
  • IQR is \(Q_3-Q_1\).
  • Outlier fences are \(Q_1-1.5(IQR)\) and \(Q_3+1.5(IQR)\).
Example 1. A score is \(84\), the mean is \(70\), and the standard deviation is \(7\). Find the z-score.
Example 2. A student's score is at the \(80\)th percentile. What does that mean?
Example 3. If \(Q_1=12\) and \(Q_3=28\), find the IQR.
Example 4. Using \(Q_1=12\), \(Q_3=28\), and \(IQR=16\), find the outlier fences.
Section 3.5

The Five-Number Summary and Boxplots

Summarize data with quartiles, IQR, outliers, and boxplots.

Quick reference

  • The five-number summary is minimum, \(Q_1\), median, \(Q_3\), and maximum.
  • A boxplot displays the five-number summary visually.
  • The box runs from \(Q_1\) to \(Q_3\), and the median is marked inside the box.
  • Whiskers extend to non-outlier values.
  • Outliers should be plotted separately in a modified boxplot.
Example 1. Find the five-number summary for \(2,4,5,8,10,12,15\).
Example 2. Use the modified boxplot. What does the separate point at \(52\) represent?

Modified Boxplot

The box runs from Q1 to Q3, the median is marked inside the box, and the outlier is plotted separately.

Modified boxplot with minimum 12, first quartile 18, median 24, third quartile 31, maximum non-outlier 38, and outlier 52 Quiz Scores 1020304050 min 12 Q1 18 med 24 Q3 31 38 outlier 52
Example 3. If \(Q_1=20\) and \(Q_3=35\), what is the IQR?
Example 4. In the modified boxplot, what does the line inside the box represent?

Modified Boxplot

The box runs from Q1 to Q3, the median is marked inside the box, and the outlier is plotted separately.

Modified boxplot with minimum 12, first quartile 18, median 24, third quartile 31, maximum non-outlier 38, and outlier 52 Quiz Scores 1020304050 min 12 Q1 18 med 24 Q3 31 38 outlier 52

Module 4: Correlation and Regression

Students describe relationships between two quantitative variables and use linear regression carefully.

Section 4.1

Scatter Diagrams and Correlation

Describe scatterplots and interpret the correlation coefficient.

Quick reference

  • A scatterplot displays paired quantitative data.
  • Describe direction, form, and strength when discussing a scatterplot.
  • The correlation coefficient \(r\) measures strength and direction of a linear relationship.
  • The value of \(r\) is between \(-1\) and \(1\).
  • Correlation does not imply causation.
Example 1. Use the scatterplot. Describe the direction, form, and strength of the association.

Scatterplot: Study Time and Score

The points rise from left to right and stay close to a line.

Scatterplot with points rising from left to right in a strong positive linear pattern Study Time and Exam Score Study time (hours) Exam score
Example 2. Which correlation is stronger: \(r=-0.91\) or \(r=0.45\)?
Example 3. Can \(r=1.24\) be a correlation coefficient?
Example 4. A study finds a strong correlation between ice cream sales and drowning incidents. What causal wording should be avoided?
Section 4.2

Least-Squares Regression

Use regression equations for prediction and interpret slope, intercept, residuals, and extrapolation.

Quick reference

  • A least-squares regression line predicts a response variable from an explanatory variable.
  • The slope gives the predicted change in \(y\) for a one-unit increase in \(x\).
  • A residual is observed value minus predicted value.
  • The coefficient of determination \(r^2\) describes the percent of variation in \(y\) explained by the linear model.
  • Interpolation predicts inside the data range; extrapolation predicts outside the data range.
  • TI-84 note: use LinReg for regression and turn diagnostics on when \(r\) and \(r^2\) are needed.
Example 1. Use the regression graph. Interpret the slope in context.

Least-Squares Regression Line

The line models the upward linear trend in the paired data.

Scatterplot with a least-squares regression line through positively associated points Predicted Score from Study Time y-hat = 52.4 + 4.8x
Example 2. A TI-84 LinReg output gives \(a=52.4\), \(b=4.8\), \(r=0.94\), and \(r^2=0.884\). Write the regression equation and interpret \(r^2\).
Example 3. Use the residual plot. Does the residual plot show an obvious curved pattern?

Residual Plot

Residuals are scattered around zero without a clear curve, which supports a linear model.

Residual plot with points scattered around the zero residual line Residuals from a Linear Model Explanatory variable Residual 0
Example 4. Data were collected for \(x\)-values from \(2\) to \(15\). Is predicting at \(x=40\) interpolation or extrapolation?

Module 5: Probability

Students compute probabilities using complements, addition rules, multiplication rules, conditional probability, and counting techniques.

Section 5.1

Probability Rules

Use basic probability language and simple probability rules.

Quick reference

  • A probability is always between \(0\) and \(1\), inclusive.
  • The sample space contains all possible outcomes.
  • An event is a collection of outcomes from the sample space.
  • The complement of event \(A\) is the event that \(A\) does not occur.
  • The complement rule is \(P(A^c)=1-P(A)\).
Example 1. If \(P(A)=0.37\), find \(P(A^c)\).
Example 2. Can a probability equal \(1.08\)?
Example 3. A fair die is rolled. What is the sample space?
Example 4. A fair die is rolled. Find the probability of rolling an even number.
Section 5.2

The Addition Rule and Complements

Compute probabilities involving "or" statements.

Quick reference

  • Mutually exclusive events cannot happen at the same time.
  • For mutually exclusive events, \(P(A\text{ or }B)=P(A)+P(B)\).
  • General addition rule: \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\).
  • Use complements when it is easier to find the probability that something does not happen.
  • In probability, "or" usually means one event, the other event, or both.
Example 1. If \(P(A)=0.30\), \(P(B)=0.25\), and \(A\) and \(B\) are mutually exclusive, find \(P(A\text{ or }B)\).
Example 2. If \(P(A)=0.40\), \(P(B)=0.35\), and \(P(A\text{ and }B)=0.10\), find \(P(A\text{ or }B)\).
Example 3. A card is drawn from a deck. Are "heart" and "club" mutually exclusive?
Example 4. If the probability of at least one success is needed, what complement is often easier?
Section 5.3

Independence and the Multiplication Rule

Use multiplication rules for independent and dependent events.

Quick reference

  • Independent events do not affect each other's probabilities.
  • For independent events, \(P(A\text{ and }B)=P(A)P(B)\).
  • Without replacement usually creates dependent events.
  • With replacement often keeps probabilities the same.
  • For dependent events, update the probability after the first event.
Example 1. A coin is flipped twice. Find the probability of two heads.
Example 2. A bag has \(5\) red and \(7\) blue marbles. Two marbles are drawn without replacement. Find the probability both are red.
Example 3. Are two draws from a deck independent if the first card is not replaced?
Example 4. Are two rolls of a fair die independent?
Section 5.4

Conditional Probability and the General Multiplication Rule

Compute and interpret conditional probabilities.

Quick reference

  • Conditional probability \(P(A\mid B)\) means the probability of \(A\) given that \(B\) has occurred.
  • \(P(A\mid B)=\dfrac{P(A\text{ and }B)}{P(B)}\).
  • \(P(A\mid B)\) and \(P(B\mid A)\) are usually different.
  • Two-way tables are common sources for conditional probabilities.
  • General multiplication rule: \(P(A\text{ and }B)=P(B)P(A\mid B)\).
Example 1. If \(P(A\text{ and }B)=0.18\) and \(P(B)=0.30\), find \(P(A\mid B)\).
Example 2. Out of \(80\) students, \(30\) take statistics and \(18\) of those also take biology. Find the probability a student takes biology given they take statistics.
Example 3. Does \(P(A\mid B)\) always equal \(P(B\mid A)\)?
Example 4. If \(P(B)=0.40\) and \(P(A\mid B)=0.25\), find \(P(A\text{ and }B)\).
Section 5.5

Counting Techniques

Use the fundamental counting principle, permutations, and combinations.

Quick reference

  • The fundamental counting principle multiplies choices across stages.
  • A factorial \(n!\) multiplies \(n(n-1)(n-2)\cdots1\).
  • Use permutations when order matters.
  • Use combinations when order does not matter.
  • TI-84 note: use \(nPr\) for permutations and \(nCr\) for combinations.
Example 1. A code uses \(3\) letters followed by \(2\) digits. If repetition is allowed, how many codes are possible?
Example 2. How many ways can \(4\) students be arranged in a line from a group of \(10\)?
Example 3. How many committees of \(4\) can be chosen from \(10\) students?
Example 4. Should order matter when choosing a committee?

Module 6: Discrete Random Variables and Binomial Distributions

Students work with discrete probability distributions, expected value, and binomial probabilities.

Section 6.1

Discrete Random Variables

Determine whether a distribution is valid and compute expected values.

Quick reference

  • A random variable assigns a numerical value to outcomes of a probability experiment.
  • A discrete random variable has countable possible values.
  • A probability distribution must have probabilities between \(0\) and \(1\), and the probabilities must sum to \(1\).
  • Expected value is the long-run average: \(\mu=\sum xP(x)\).
  • The standard deviation of a random variable describes typical distance from its expected value.
  • TI-84 can compute weighted statistics using values in one list and probabilities or frequencies in another list.
Example 1. Is this a valid probability distribution?
\(x\) 0 1 2 3
\(P(x)\) 0.20 0.35 0.25 0.20
Example 2. Find the expected value for \(x=0,1,2\) with probabilities \(0.2,0.5,0.3\).
Example 3. A TI-84 weighted 1-Var Stats output gives \(\bar{x}=1.1\) and \(\sigma x=0.7\) for a probability distribution. Interpret the standard deviation.
Example 4. A probability table sums to \(1.08\). Is it valid?
Section 6.2

The Binomial Probability Distribution

Recognize binomial settings and compute binomial probabilities with technology.

Quick reference

  • A binomial experiment has fixed trials, independent trials, two outcomes, and constant success probability.
  • Use \(n\) for number of trials, \(p\) for success probability, \(q=1-p\), and \(x\) for number of successes.
  • The binomial formula \(P(X=x)=\binom{n}{x}p^xq^{n-x}\) shows the structure.
  • For a binomial distribution, \(\mu=np\) and \(\sigma=\sqrt{npq}\).
  • TI-84 binompdf(n,p,x) finds \(P(X=x)\).
  • TI-84 binomcdf(n,p,x) finds \(P(X\le x)\).
Example 1. A basketball player makes \(70\%\) of free throws and shoots \(8\) free throws. Identify \(n\) and \(p\).
Example 2. Use binomial notation to compute the probability of exactly \(3\) successes in \(10\) trials when \(p=0.40\).
Example 3. For \(X\sim Binomial(12,0.25)\), what TI-84 command gives \(P(X\le 4)\)?
Example 4. For \(X\sim Binomial(12,0.25)\), how can \(P(X\ge 5)\) be found using the complement?
Example 5. For \(X\sim Binomial(12,0.25)\), find the mean and standard deviation.

Module 7: Normal Distributions

Students use normal distribution properties, normal probabilities, inverse normal values, and normality checks.

Section 7.1

Properties of the Normal Distribution

Understand the normal curve, z-scores, and the empirical rule.

Quick reference

  • A normal distribution is bell-shaped and symmetric.
  • For a normal distribution, mean, median, and mode are equal.
  • The total area under a normal curve is \(1\).
  • The z-score formula is \(z=\dfrac{x-\mu}{\sigma}\).
  • The empirical rule gives about \(68\%\), \(95\%\), and \(99.7\%\) within \(1\), \(2\), and \(3\) standard deviations of the mean.
Example 1. A normal distribution has mean \(50\) and standard deviation \(6\). Find the z-score for \(x=62\).
Example 2. Use the normal curve. Using the empirical rule, about what percent of values lie within \(2\) standard deviations of the mean?

Empirical Rule on a Normal Curve

About 68 percent is within one standard deviation, 95 percent within two, and 99.7 percent within three.

Normal curve marked with mean and one, two, and three standard deviations Empirical Rule mu - 3sdmu - 2sdmu - 1sdmumu + 1sdmu + 2sdmu + 3sd 68% near the center
Example 3. A normal distribution has mean \(100\) and standard deviation \(15\). What interval is within \(1\) standard deviation?
Example 4. What is the total area under a normal curve?
Section 7.2

Applications of the Normal Distribution

Use TI-84 normalcdf and invNorm to compute probabilities and cutoff values.

Quick reference

  • Use normalcdf(lower, upper, mean, standard deviation) for normal probabilities.
  • Use a very large negative lower bound for "less than" probabilities.
  • Use a very large positive upper bound for "greater than" probabilities.
  • Use invNorm(area, mean, standard deviation) to find a value from a left-tail area.
  • Always sketch or name the shaded region before choosing the calculator command.
Example 1. Let \(X\) be normal with \(\mu=70\), \(\sigma=8\). What TI-84 command finds \(P(X<82)\)?
Example 2. Use the shaded normal curve. What TI-84 command finds \(P(65

Shaded Normal Probability

The shaded area represents P(65 less than X less than 80).

Normal curve with the area between 65 and 80 shaded Normal Model: mean 70, standard deviation 8 657080 shade between
Example 3. Let \(X\) be normal with \(\mu=70\), \(\sigma=8\). What TI-84 command finds the 90th percentile?
Example 4. If the calculator gives \(P(X>90)=0.0228\), interpret the answer.
Section 7.3

Assessing Normality

Use graphs and normal probability plots to decide whether normal methods are reasonable.

Quick reference

  • A histogram that is roughly bell-shaped supports normality.
  • Strong skewness or outliers are warnings against normality.
  • A normal probability plot that is roughly linear supports normality.
  • Normality checks are about whether normal-based methods are reasonable, not whether data are perfectly normal.
  • StatCrunch is useful for normal probability plots and larger data sets.
Example 1. Use the normal probability plot. What does this suggest about normality?

Normal Probability Plot

The points are close to a straight line, so a normal model appears reasonable.

Normal probability plot with points close to a straight line Normal Probability Plot Data value
Example 2. A histogram is strongly skewed right with several outliers. Are normal methods automatically reasonable?
Example 3. What graph is specifically designed to assess whether data are approximately normal?
Example 4. Does a data set have to be perfectly normal for normal methods to be useful?

Module 8: Sampling Distributions

Students study sampling distributions for sample means and sample proportions.

Section 8.1

Distribution of the Sample Mean

Use the sampling distribution of \(\bar{x}\) and the Central Limit Theorem.

Quick reference

  • The sampling distribution of \(\bar{x}\) describes sample means from repeated samples of the same size.
  • The mean of \(\bar{x}\) is \(\mu_{\bar{x}}=\mu\).
  • The standard deviation of \(\bar{x}\) is \(\sigma_{\bar{x}}=\dfrac{\sigma}{\sqrt{n}}\), also called standard error.
  • The Central Limit Theorem says \(\bar{x}\) is approximately normal for large samples under the usual conditions.
  • Use TI-84 normalcdf with the standard error for probabilities involving sample means.
Example 1. A population has \(\mu=80\), \(\sigma=12\), and samples of size \(36\). Find \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\).
Example 2. For the same setting, what TI-84 command finds \(P(\bar{x}<83)\)?
Example 3. What happens to the standard error as sample size increases?
Example 4. Why is the Central Limit Theorem useful?
Section 8.2

Distribution of the Sample Proportion

Use the sampling distribution of \(\hat{p}\) and check normal approximation conditions.

Quick reference

  • The sampling distribution of \(\hat{p}\) describes sample proportions from repeated samples of the same size.
  • The mean is \(\mu_{\hat{p}}=p\).
  • The standard deviation is \(\sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{n}}\).
  • The normal approximation usually requires \(np\ge10\) and \(n(1-p)\ge10\).
  • Use TI-84 normalcdf with \(p\) and the standard error for probabilities involving \(\hat{p}\).
Example 1. Let \(p=0.40\) and \(n=100\). Find the mean and standard deviation of \(\hat{p}\).
Example 2. Check the normal approximation conditions for \(p=0.40\), \(n=100\).
Example 3. What TI-84 command finds \(P(\hat{p}<0.45)\) when \(p=0.40\), \(n=100\), and standard error is \(0.049\)?
Example 4. If \(np=7\), what concern should be noted?

Module 9: Confidence Intervals

Students estimate population proportions and means using confidence intervals.

Section 9.1

Estimating a Population Proportion

Use one-proportion confidence intervals and interpret them correctly.

Quick reference

  • A point estimate for a population proportion is \(\hat{p}=\dfrac{x}{n}\).
  • A confidence interval estimates a population parameter with a margin of error.
  • For a one-proportion interval, check large-count conditions before using normal methods.
  • TI-84 note: use 1-PropZInt for one-proportion confidence intervals.
  • Correct wording: we are confident the interval captures the population proportion.
Example 1. In a sample of \(250\) adults, \(142\) support a proposal. Find \(\hat{p}\).
Example 2. A sample has \(x=142\) supporters out of \(n=250\). Use TI-84 1-PropZInt for a \(95\%\) confidence interval and interpret the output.
Example 3. A \(95\%\) confidence interval is \((0.51,0.63)\). Interpret it in context.
Example 4. Is it correct to say there is a \(95\%\) probability the fixed population proportion is in this computed interval?
Section 9.2

Estimating a Population Mean

Use one-sample t-intervals and interpret them correctly.

Quick reference

  • A point estimate for a population mean is \(\bar{x}\).
  • Use a t-interval for a population mean when \(\sigma\) is unknown.
  • Degrees of freedom for one sample are \(df=n-1\).
  • TI-84 note: use TInterval with data or summary statistics.
  • Check normality/large-sample conditions before using the interval.
Example 1. A sample has \(n=18\). What are the degrees of freedom for a one-sample t-interval?
Example 2. Use TI-84 TInterval with summary statistics \(n=18\), \(\bar{x}=14.6\), \(s=3.8\), and confidence level \(90\%\). Interpret the interval.
Example 3. A \(90\%\) confidence interval for mean wait time is \((12.4,16.8)\) minutes. Interpret it.
Example 4. Why should outliers be checked for a small-sample t-interval?

Module 10: Hypothesis Testing

Students set up and run hypothesis tests for population proportions and means using correct statistical wording.

Section 10.1

The Language of Hypothesis Testing

Use hypothesis-test vocabulary and write correct conclusions.

Quick reference

  • The null hypothesis \(H_0\) is the starting claim tested by the procedure.
  • The alternative hypothesis \(H_a\) describes what the evidence is trying to support.
  • The significance level \(\alpha\) is the cutoff for deciding whether the evidence is statistically significant.
  • The P-value is the probability of observing results as extreme or more extreme, assuming \(H_0\) is true.
  • A Type I error rejects a true \(H_0\); a Type II error fails to reject a false \(H_0\).
  • Decisions are reject \(H_0\) or fail to reject \(H_0\). Do not say accept \(H_0\).
Example 1. A claim says the population proportion is greater than \(0.40\). Write the alternative hypothesis.
Example 2. If \(H_a:\mu<25\), what type of test is this?
Example 3. If \(P\text{-value}=0.032\) and \(\alpha=0.05\), what is the decision?
Example 4. If the P-value is large, should we say we accept \(H_0\)?
Example 5. In a medical screening study, \(H_0\) says the test has the advertised accuracy. What kind of error is rejecting \(H_0\) when the advertised accuracy is actually true?
Section 10.2

Hypothesis Tests for a Population Proportion

Use one-proportion z-tests with TI-84 and state conclusions correctly.

Quick reference

  • Use a one-proportion z-test when testing a claim about one population proportion.
  • Set hypotheses using the population proportion \(p\), not the sample proportion \(\hat{p}\).
  • Check large-count conditions using the null value \(p_0\).
  • TI-84 note: use 1-PropZTest for a one-proportion hypothesis test.
  • Conclusion should include the decision and context tied to the original claim.
Example 1. A claim says fewer than \(30\%\) of students work full time. Write \(H_0\) and \(H_a\).
Example 2. For \(H_0:p=0.30\), \(n=120\), check the large-count conditions.
Example 3. A sample of \(120\) students has \(27\) who work full time. Test the claim \(p<0.30\) at \(\alpha=0.05\) using TI-84 1-PropZTest.
Example 4. A test gives \(P\text{-value}=0.081\) with \(\alpha=0.05\). What is the decision?
Example 5. State the conclusion for a failed rejection when the claim was \(p<0.30\).
Section 10.3

Hypothesis Tests for a Population Mean

Use one-sample t-tests with TI-84 and state conclusions correctly.

Quick reference

  • Use a one-sample t-test when testing a claim about one population mean and \(\sigma\) is unknown.
  • Set hypotheses using the population mean \(\mu\), not the sample mean \(\bar{x}\).
  • Check normality or large-sample conditions before trusting the test.
  • TI-84 note: use T-Test with data or summary statistics.
  • Reject \(H_0\) or fail to reject \(H_0\), then state the conclusion in context.
Example 1. A claim says the population mean wait time is different from \(15\) minutes. Write \(H_0\) and \(H_a\).
Example 2. If \(H_a:\mu>42\), what type of test is this?
Example 3. A sample has \(n=16\), \(\bar{x}=43.8\), and \(s=3.2\). Test the claim \(\mu>42\) at \(\alpha=0.05\) using TI-84 T-Test.
Example 4. A t-test gives \(P\text{-value}=0.014\) with \(\alpha=0.05\). What is the decision?
Example 5. State the conclusion for rejecting \(H_0\) when the claim was \(\mu>42\).
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