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Unit 1: Exploring One-Variable Data and Collecting Data

Topic 1.1 · Say it in contextA researcher surveys 400 of a district's 12,000 students about sleep. Name the population, the sample, $N$ and $n$ .

Population: all 12,000 students in the district, so $N = 12,000$ . Sample: the 400 students surveyed, so $n = 400$ . Data from the sample are used to answer a question about the population.
Review: 1.1 Introducing Statistics: What Can We Learn from Data?
Topic 1.1 · Spot the mistakeAfter looking at the data, a team rewrites its investigative question to match a pattern it noticed. What is wrong with that?

An investigative question has a defined purpose that is set before the analysis. Changing it to fit the results means the study no longer answers the question it was designed to answer.
Review: 1.1 Introducing Statistics: What Can We Learn from Data?
Topic 1.2 · Which one fitsClassify each variable: number of siblings, eye color, time to run a mile.

Number of siblings: quantitative and discrete, since the values can be counted. Eye color: categorical. Mile time: quantitative and continuous, since it can take any value in an interval.
Review: 1.2 Variables
Topic 1.2 · Which one fitsParameter or statistic: the mean height of 50 randomly chosen players, used to estimate the mean height of every player in a league.

A statistic, because it summarizes a sample. The mean height of all players in the league is the parameter, and it is usually unknown.
Review: 1.2 Variables
Topic 1.3 · Know the termHow does a relative frequency table differ from a frequency table?

A frequency table gives the count in each category. A relative frequency table gives the proportion in each category, so its entries add to 1, or 100%.
Review: 1.3 Tabular Representation and Summary Statistics for One Categorical Variable
Topic 1.4 · Which one fitsYou want to compare which lunch options students choose at two schools of very different sizes. Should the bar charts show counts or relative frequencies?

Relative frequencies. With groups of different sizes, counts mostly show which school is bigger. Proportions put both schools on the same scale.
Review: 1.4 Graphical Representations for One Categorical Variable
Topic 1.5 · Say it in contextTwo students make histograms of the same data and they look different. Neither made a mistake. How is that possible?

They chose different bin widths. Wider bins can hide gaps and clusters, and narrower bins can make the shape look jagged, even though the data are the same.
Review: 1.5 Graphical Representations for One Quantitative Variable
Topic 1.6 · Know the termA distribution has its peak on the left and a long tail stretching to the right. What is its shape called?

Skewed to the right, also called positively skewed. Skew is named for the side with the longer tail, not the side where the peak is.
Review: 1.6 Descriptions for One Quantitative Variable Distributions
Topic 1.6 · Spot the mistakeA student describes a histogram of test scores as "centered around 40 with a range of 20." What is missing?

The shape and any unusual features such as outliers, gaps or clusters, all in context. A full description covers shape, center, variability (spread) and unusual features.
Review: 1.6 Descriptions for One Quantitative Variable Distributions
Topic 1.7 · Which one fitsA data set of home prices includes a few extremely expensive homes. Which measures of center and variability should you report?

The median and the IQR. They are resistant: a few extreme values barely move them. The mean is pulled toward the outliers, and the range and standard deviation are greatly affected by them.
Review: 1.7 Summary Statistics for One Quantitative Variable
Topic 1.7 · Work it out $Q_{1} = 20$ and $Q_{3} = 32$ . Using the $1.5 \times IQR$ rule, which values count as outliers?

$IQR = 12$ and $1.5 \times 12 = 18$ . Values below $20 - 18 = 2$ or above $32 + 18 = 50$ are outliers.
Review: 1.7 Summary Statistics for One Quantitative Variable
Topic 1.7 · Say it in contextCommute times for a group of workers have a standard deviation of $s = 4$ minutes. Interpret that value.

A typical commute time for these workers differs from their mean commute time by about 4 minutes.
Review: 1.7 Summary Statistics for One Quantitative Variable
Topic 1.8 · Say it in contextOn a boxplot, the median sits close to $Q_{1}$ and the right whisker is much longer than the left. How do you expect the mean to compare with the median?

The distribution is skewed right, so the mean is usually larger than the median. The long right tail pulls the mean up.
Review: 1.8 Graphical Representations of Summary Statistics for One Quantitative Variable
Topic 1.8 · Know the termWhich five values make up a five-number summary, and what does the box of a boxplot show?

Minimum, $Q_{1}$ , median, $Q_{3}$ and maximum. The box runs from $Q_{1}$ to $Q_{3}$ , the middle 50% of the data, with a line at the median.
Review: 1.8 Graphical Representations of Summary Statistics for One Quantitative Variable
Topic 1.9 · Work it outAna scored 82 on a test with mean 74 and standard deviation 4. Ben scored 88 on a test with mean 80 and standard deviation 8. Who did better relative to their class?

Ana: $z = (82 - 74) / 4 = 2$ . Ben: $z = (88 - 80) / 8 = 1$ . Ana's score is 2 standard deviations above her class mean and Ben's is 1, so Ana did better relative to her class.
Review: 1.9 Comparisons of the Distributions for One Quantitative Variable
Topic 1.9 · Spot the mistakeYou compare two distributions with side-by-side boxplots. What can the boxplots not show you?

Gaps, clusters, shape details such as two peaks, the sample size and the individual values. Boxplots compare center, variability, outliers and skewness. Use dotplots or histograms to see the rest.
Review: 1.9 Comparisons of the Distributions for One Quantitative Variable
Topic 1.10 · Which one fitsResearchers record the diets and heart health of 2,000 adults who chose their own diets. Experiment or observational study? Can they conclude that diet causes better heart health?

Observational study, because no treatments were imposed. They cannot conclude cause and effect: a confounding variable such as exercise could explain the association.
Review: 1.10 The Investigative Question Revisited and Data Collection
Topic 1.10 · Know the termWhat makes a variable a confounding variable in an observational study?

It is associated with both the explanatory variable and the response variable, so it gives another explanation for the relationship the study found.
Review: 1.10 The Investigative Question Revisited and Data Collection
Topic 1.10 · Say it in contextA fitness study uses volunteers from one gym. To whom can the results be generalized?

Only to people similar to those volunteers. Without random selection from a larger population, generalizing to that whole population is not appropriate.
Review: 1.10 The Investigative Question Revisited and Data Collection
Topic 1.11 · Which one fitsA principal divides students by grade level and takes a simple random sample from each grade. Which sampling method is this?

A stratified random sample. The grades are the strata, groups of similar individuals, and a simple random sample is taken within each one.
Review: 1.11 Random Sampling
Topic 1.11 · Spot the mistakeHow is a cluster random sample different from a stratified random sample?

A stratified sample takes a random sample from every group. A cluster sample randomly selects whole groups and collects data from everyone in the chosen groups. Ideally, each cluster looks like the population.
Review: 1.11 Random Sampling
Topic 1.12 · Which one fitsA radio host asks listeners to call in with their opinion of a new law. Which bias is most likely?

Voluntary response bias. People with strong opinions are the most likely to call, so the callers may not represent all listeners.
Review: 1.12 Potential Problems with Sampling
Topic 1.12 · Which one fitsOnly 30% of the people randomly selected for a phone survey answer it. What kind of bias could this cause?

Nonresponse bias. The people who did not answer may differ from those who did in ways that matter to the question.
Review: 1.12 Potential Problems with Sampling
Topic 1.12 · Which one fitsA survey asks, "Don't you agree that the overcrowded park needs a new entrance?" Which bias does the wording create?

Response bias from question wording. A leading question pushes answers in one direction.
Review: 1.12 Potential Problems with Sampling
Topic 1.13 · Know the termName the four features of a well-designed experiment.

Comparison of at least two treatment groups (one can be a control group), random assignment of treatments, replication (more than one experimental unit per treatment), and direct control of extraneous sources of variation.
Review: 1.13 Experimental Design
Topic 1.13 · Which one fitsYou are testing two fertilizers on garden plots. Half the plots are in full sun and half are in shade, and sunlight affects growth. Which design handles this, and how?

A randomized block design that blocks on sunlight. Form a sunny block and a shady block, then randomly assign the fertilizers within each block. Comparing within blocks separates out the variation caused by sunlight.
Review: 1.13 Experimental Design
Topic 1.13 · Say it in contextWhy does random assignment of treatments allow a cause-and-effect conclusion?

It tends to make the treatment groups similar on extraneous variables, which reduces the potential for confounding. A difference in the response too large to be explained by chance can then be attributed to the treatments.
Review: 1.13 Experimental Design
Topic 1.13 · Know the termIn a double-blind experiment, who does not know which treatment each participant receives?

Neither the participants nor the members of the research team who interact with them. In a single-blind experiment, only one of those two groups is kept unaware.
Review: 1.13 Experimental Design

Unit 2: Probability, Random Variables, and Probability Distributions

Topic 2.1 · Which one fitsWhich graphs display the relationship between two categorical variables?

Side-by-side bar charts, segmented bar charts and mosaic plots. A two-way table shows the same data in numbers.
Review: 2.1 Tabular and Graphical Representations for the Distributions of Two Categorical Variables
Topic 2.2 · Work it out60 of 150 seniors and 40 of 200 juniors drive to school. Find the conditional relative frequencies. Do they suggest an association between grade and driving?

Seniors: $60 / 150 = 0.40$ . Juniors: $40 / 200 = 0.20$ . The proportions differ, which suggests an association between grade level and driving to school.
Review: 2.2 Summary Statistics for Two Categorical Variables
Topic 2.2 · Know the termJoint, marginal or conditional: a cell count divided by the total for its row.

Conditional relative frequency. A cell count divided by the table total is joint, and a row or column total divided by the table total is marginal.
Review: 2.2 Summary Statistics for Two Categorical Variables
Topic 2.3 · Say it in contextWhat does the law of large numbers say about flipping a fair coin many times?

As the number of independent flips grows, the relative frequency of heads gets closer and closer to a single value, 0.5. It says nothing about the next flip making up for a streak.
Review: 2.3 Estimating Probabilities Using Simulation
Topic 2.3 · Which one fitsEach day has a 30% chance of rain, independently. How could you use random digits to estimate the probability of rain on at least 2 of the next 3 days?

Let 0, 1 and 2 mean rain and 3 through 9 mean no rain. One trial is three random digits; record whether at least 2 are rain. Run many trials and use the relative frequency of those trials as the estimate.
Review: 2.3 Estimating Probabilities Using Simulation
Topic 2.4 · Work it outThe probability that a shipment has no defective items is 0.83. What is the probability it has at least one?

Use the complement: $1 - 0.83 = 0.17$ . "At least one" is the complement of "none."
Review: 2.4 Introduction to Probability
Topic 2.5 · Spot the mistakeMutually exclusive events and independent events are the same thing. True or false?

False. Mutually exclusive means $P (A \cap B) = 0$ , so the events cannot happen together. If each has a positive probability, knowing one happened makes the other impossible, so they are not independent.
Review: 2.5 Mutually Exclusive Events
Topic 2.6 · Work it out $P (A) = 0.5$ and $P (A \cap B) = 0.2$ . Find $P (B | A)$ .

$P (B | A) = P (A \cap B) / P (A) = 0.2 / 0.5 = 0.4$ .
Review: 2.6 Conditional Probability
Topic 2.7 · Check the conditionsHow can you check whether events $A$ and $B$ are independent?

Check whether $P (A | B) = P (A)$ , or whether $P (A \cap B) = P (A) \cdot P (B)$ . If the values are equal, the events are independent.
Review: 2.7 Independent Events and Unions of Events
Topic 2.7 · Work it out $P (A) = 0.4$ , $P (B) = 0.3$ and $P (A \cap B) = 0.1$ . Find $P (A \cup B)$ .

$P (A \cup B) = 0.4 + 0.3 - 0.1 = 0.6$ . Subtract the overlap so it is not counted twice.
Review: 2.7 Independent Events and Unions of Events
Topic 2.8 · Check the conditionsWhat must be true of a table for it to be a valid probability distribution of a discrete random variable?

Every probability is between 0 and 1, and the probabilities over all possible values add to exactly 1.
Review: 2.8 Introduction to Random Variables and Probability Distributions
Topic 2.9 · Say it in contextFor a raffle ticket, the expected net gain is $E (X) = - 0.40$ dollars. Interpret this value.

Over many, many tickets, a buyer loses about 40 cents per ticket on average. It is a long-run average, not the result of any one ticket.
Review: 2.9 Parameters of Random Variables
Topic 2.9 · Say it in contextThe number of customers per hour at a shop has $σ_{X} = 3$ . Interpret this value.

Over the long run, the number of customers in an hour typically differs from the mean number of customers per hour by about 3.
Review: 2.9 Parameters of Random Variables
Topic 2.10 · Check the conditionsWhat makes a random variable binomial?

It counts the number of successes in a fixed number $n$ of independent trials, where each trial has two outcomes, success or failure, and the same probability of success $p$ .
Review: 2.10 The Binomial Distribution
Topic 2.10 · Work it out $X$ is binomial with $n = 50$ and $p = 0.2$ . Find its mean and standard deviation.

$μ_{X} = n p = 10$ and $σ_{X} = \sqrt{n p (1 - p)} = \sqrt{8} \approx 2.83$ .
Review: 2.10 The Binomial Distribution
Topic 2.11 · Work it outHeights are approximately normal with mean 170 cm and standard deviation 8 cm. About what percent of heights fall between 154 cm and 186 cm?

About 95%. Both values are 2 standard deviations from the mean, and the empirical rule puts about 95% of values within 2 standard deviations.
Review: 2.11 The Normal Distribution
Topic 2.11 · Which one fitsTo find the boundaries of the middle 90% of a normal distribution, how much area goes in each tail?

5% in each tail. The middle 90% leaves 10%, split equally between the two tails. For the standard normal, the boundaries are about $z = - 1.645$ and $z = 1.645$ .
Review: 2.11 The Normal Distribution
Topic 2.12 · Spot the mistakeWhat is the difference between the distribution of one sample and the sampling distribution of $\overline{x}$ ?

The distribution of a sample shows the individual values in that one sample. The sampling distribution shows the values of $\overline{x}$ across all possible samples of the same size from the population.
Review: 2.12 Sampling Distributions and the Central Limit Theorem
Topic 2.12 · Know the termWhat does the central limit theorem say?

The sampling distribution of the mean of a random sample has a shape that can be approximated by a normal distribution, and the approximation gets better as the sample size grows.
Review: 2.12 Sampling Distributions and the Central Limit Theorem

Unit 3: Inference for Categorical Data: Proportions

Topic 3.1 · Know the termWhat does it mean for an estimator to be unbiased?

On average, it does not overestimate or underestimate the population parameter. The sample proportion $\hat{p}$ is an unbiased estimator of $p$ .
Review: 3.1 Estimators
Topic 3.2 · Work it outA population has $p = 0.3$ . For random samples of size $n = 100$ , find the mean and standard deviation of the sampling distribution of $\hat{p}$ .

Mean: $p = 0.3$ . Standard deviation: $\sqrt{0.3 (0.7) / 100} \approx 0.046$ .
Review: 3.2 Sampling Distributions for Sample Proportions
Topic 3.2 · Check the conditionsWhen is the sampling distribution of $\hat{p}$ approximately normal?

When the sample is large enough that $n p \geq 10$ and $n (1 - p) \geq 10$ : the expected numbers of successes and failures are both at least 10.
Review: 3.2 Sampling Distributions for Sample Proportions
Topic 3.3 · Which one fitsA poll of 1,000 randomly selected voters finds that 54% support a ballot measure. Which procedure estimates the proportion of all voters who support it?

A one-sample z-interval for a population proportion.
Review: 3.3 Constructing a Confidence Interval for a Population Proportion
Topic 3.3 · Check the conditionsName the three conditions for a one-sample z-interval for a population proportion.

Randomization condition: the data come from a random sample. 10% condition: when sampling without replacement, the sample is no more than 10% of the population. Normality condition: the observed successes $n \hat{p}$ and failures $n (1 - \hat{p})$ are both at least 10.
Review: 3.3 Constructing a Confidence Interval for a Population Proportion
Topic 3.3 · Work it outWhat sample size gives a margin of error of 0.03 at 95% confidence, with no earlier estimate of $p$ ?

Use $\hat{p} = 0.5$ and $z^{*} = 1.96$ : $n = (1.96)^{2} (0.5) (0.5) / (0.03)^{2} \approx 1067.1$ . Round up, so at least 1,068.
Review: 3.3 Constructing a Confidence Interval for a Population Proportion
Topic 3.4 · Say it in contextInterpret a 95% confidence interval of (0.51, 0.57) for the proportion of a city's voters who support a ballot measure.

We are 95% confident that the interval from 0.51 to 0.57 contains the true proportion of all voters in this city who support the measure.
Review: 3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion
Topic 3.4 · Say it in contextWhat does a 95% confidence level mean?

In repeated random sampling with the same sample size, about 95% of the intervals built this way would capture the true value of the population parameter. It describes the method, not any one interval.
Review: 3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion
Topic 3.4 · Work it outTo cut the width of a confidence interval in half, by what factor must the sample size grow?

By a factor of 4. Width is roughly proportional to $1 / \sqrt{n}$ , so halving it requires $\sqrt{n}$ to double.
Review: 3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion
Topic 3.5 · Which one fitsA company claims 90% of its orders ship on time. You suspect fewer do. Write the hypotheses.

$H_{0} : p = 0.90$ and $H_{a} : p < 0.90$ , where $p$ is the true proportion of all the company's orders that ship on time.
Review: 3.5 Setting Up a Test for a Population Proportion
Topic 3.5 · Spot the mistakeFor the normality condition in a one-sample z-test for a proportion, do you use $\hat{p}$ or $p_{0}$ ?

$p_{0}$ . The test assumes $H_{0}$ is true, so check that the expected successes $n p_{0}$ and expected failures $n (1 - p_{0})$ are both at least 10. A confidence interval uses the observed $\hat{p}$ instead.
Review: 3.5 Setting Up a Test for a Population Proportion
Topic 3.6 · Say it in contextTesting $H_{0} : p = 0.5$ against $H_{a} : p > 0.5$ , where $p$ is the proportion of all customers who prefer a new design, gives a p-value of 0.03. Interpret it.

Assuming the true proportion of customers who prefer the new design is 0.5, there is a 0.03 probability of getting a sample proportion as large as or larger than the one observed, by chance alone.
Review: 3.6 p-Values
Topic 3.6 · Spot the mistake"A p-value of 0.40 means there is a 40% chance the null hypothesis is true." What is wrong?

The p-value is computed assuming $H_{0}$ is true, so it cannot be the probability that $H_{0}$ is true. A large p-value means the result would not be unusual if $H_{0}$ were true. It is not evidence that $H_{0}$ is true.
Review: 3.6 p-Values
Topic 3.7 · Which one fitsThe p-value is 0.062 and $α = 0.05$ . What is the decision, and what do you conclude?

Fail to reject $H_{0}$ , because $0.062 > 0.05$ . There is not convincing statistical evidence for the alternative hypothesis, stated in context. Never say you accept or prove $H_{0}$ .
Review: 3.7 Carrying Out a Test for a Population Proportion
Topic 3.7 · Spot the mistakeWhy does the standard error in a one-sample z-test for a proportion use $p_{0}$ and not $\hat{p}$ ?

The test statistic is computed assuming $H_{0}$ is true, so the standard error uses the null value: $z = \frac{\hat{p} - p_{0}}{\sqrt{p_{0} (1 - p_{0}) / n}}$ .
Review: 3.7 Carrying Out a Test for a Population Proportion
Topic 3.8 · Say it in context $H_{0}$ : a new drug works no better than the current drug. $H_{a}$ : the new drug works better. Describe a Type I error in context.

Concluding that the new drug works better when it really does not. Patients could switch to a drug that gives them no added benefit.
Review: 3.8 Potential Errors When Performing Tests
Topic 3.8 · Which one fitsName two ways a researcher can increase the power of a test before collecting data.

Increase the sample size, or use a larger significance level $α$ . Power also rises when the true parameter is farther from the null value, but the researcher does not control that.
Review: 3.8 Potential Errors When Performing Tests
Topic 3.8 · Work it outA test has power 0.80. What is the probability of a Type II error?

$1 - 0.80 = 0.20$ . The probability of a Type II error is 1 minus the power.
Review: 3.8 Potential Errors When Performing Tests
Topic 3.9 · Check the conditionsFor the sampling distribution of ${\hat{p}}_{1} - {\hat{p}}_{2}$ , what does the randomization condition require when the data come from an experiment rather than from samples?

The treatments must be randomly assigned to the experimental units. When the data come from samples, they must come from two independent random samples.
Review: 3.9 Sampling Distributions for the Difference Between Sample Proportions
Topic 3.10 · Which one fitsStudents are randomly assigned to a tutoring program or no program, and pass rates are recorded. Which procedure estimates the difference in pass rates?

A two-sample z-interval for the difference between two population proportions.
Review: 3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions
Topic 3.11 · Say it in contextA 95% confidence interval for $p_{1} - p_{2}$ is $(- 0.04, 0.10)$ . Is there convincing evidence that the two proportions differ?

No. The interval contains 0, so no difference is a plausible value. There is not convincing evidence that the two population proportions differ.
Review: 3.11 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions
Topic 3.12 · Check the conditionsIn a two-sample z-test for proportions, why does the normality check use the combined proportion ${\hat{p}}_{c}$ ?

The null hypothesis says $p_{1} = p_{2}$ , so under $H_{0}$ both groups share one proportion, and ${\hat{p}}_{c}$ estimates it. Check that $n_{1} {\hat{p}}_{c}$ , $n_{1} (1 - {\hat{p}}_{c})$ , $n_{2} {\hat{p}}_{c}$ and $n_{2} (1 - {\hat{p}}_{c})$ are all at least 10.
Review: 3.12 Setting Up a Test for the Difference Between Two Population Proportions
Topic 3.13 · Say it in contextTesting $H_{a} : p_{1} > p_{2}$ , where $p_{1}$ and $p_{2}$ are the proportions of students at School 1 and School 2 who recycle, gives a p-value of 0.004 with $α = 0.05$ . State the conclusion.

Reject $H_{0}$ , because $0.004 \leq 0.05$ . There is convincing statistical evidence that the proportion of students who recycle is greater at School 1 than at School 2.
Review: 3.13 Carrying Out a Test for the Difference Between Two Population Proportions
Topic 3.14 · Which one fitsSeparate random samples from three cities are asked which way they usually commute. Which chi-square test compares the cities?

A chi-square test for homogeneity: independent samples from several populations, comparing the distribution of one categorical variable.
Review: 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence
Topic 3.14 · Which one fitsOne random sample of 500 adults is classified by age group and by preferred news source. Which chi-square test asks whether these are associated?

A chi-square test for independence: one sample, two categorical variables, asking whether they are associated in that population.
Review: 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence
Topic 3.14 · Check the conditionsWhat is the expected counts condition for a chi-square test?

All expected counts are greater than 5. Check the expected counts, not the observed counts.
Review: 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence
Topic 3.15 · Work it outA cell's row total is 80, its column total is 50, and the table total is 200. What is the expected count for that cell?

$(80) (50) / 200 = 20$ . Expected count = (row total)(column total)/(table total).
Review: 3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence
Topic 3.15 · Work it outHow many degrees of freedom does a chi-square test on a table with 3 rows and 4 columns have?

$(3 - 1) (4 - 1) = 6$ .
Review: 3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence

Unit 4: Inference for Quantitative Data: Means

Topic 4.1 · Work it outA population has mean 50 and standard deviation 12. For random samples of size 36, find the mean and standard deviation of the sampling distribution of $\overline{x}$ .

Mean: 50. Standard deviation: $12 / \sqrt{36} = 2$ .
Review: 4.1 Sampling Distributions for Sample Means
Topic 4.1 · Check the conditionsWhen can the sampling distribution of $\overline{x}$ be modeled with a normal distribution?

When the population is normal, for any sample size, or approximately when $n \geq 30$ for a population that is not normal. An extremely skewed population may need a sample much larger than 30.
Review: 4.1 Sampling Distributions for Sample Means
Topic 4.2 · Which one fitsWhy does inference for a population mean use a t-distribution instead of the standard normal?

The population standard deviation $σ$ is usually unknown, so the sample standard deviation $s$ is used instead. The t-distributions, with their wider tails, account for that.
Review: 4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference
Topic 4.2 · Which one fitsTwenty runners are timed before and after a training plan. Which procedure estimates the mean improvement?

A one-sample t-interval for a population mean difference. Find each runner's difference, with the order of subtraction stated, and analyze that one sample of differences.
Review: 4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference
Topic 4.2 · Check the conditionsA random sample of $n = 18$ is strongly skewed with an outlier, and nothing says the population is normal. Can you use a one-sample t-interval?

No. When $n < 30$ , the sample data must be free from strong skewness and outliers, unless the population is known to be approximately normal. This sample fails the sample data condition.
Review: 4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference
Topic 4.3 · Say it in contextInterpret a 90% confidence interval of (12.1, 14.7) hours for the mean weekly screen time of a school's students.

We are 90% confident that the interval from 12.1 to 14.7 hours contains the true mean weekly screen time for all students at this school.
Review: 4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference
Topic 4.3 · Spot the mistakeDoes that 90% interval of (12.1, 14.7) hours mean 90% of students have screen times in that range?

No. The interval estimates the population mean, not where individual values fall. Individual students' screen times vary much more widely than the mean does.
Review: 4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference
Topic 4.4 · Which one fitsDoes a training plan lower 5K times? Using differences $d = before - after$ , write the hypotheses.

$H_{0} : μ_{d} = 0$ and $H_{a} : μ_{d} > 0$ , where $μ_{d}$ is the true mean of the before-minus-after differences in 5K times for runners like these. A positive difference means a faster time after the plan.
Review: 4.4 Setting Up a Test for a Population Mean or Population Mean Difference
Topic 4.5 · Work it out $\overline{x} = 52$ , $μ_{0} = 50$ , $s = 6$ and $n = 16$ . Find the test statistic and its degrees of freedom.

$t = (52 - 50) / (6 / \sqrt{16}) = 2 / 1.5 \approx 1.33$ , with $df = 15$ .
Review: 4.5 Carrying Out a Test for a Population Mean or Population Mean Difference
Topic 4.6 · Check the conditionsWhen can the sampling distribution of ${\overline{x}}_{1} - {\overline{x}}_{2}$ be modeled as approximately normal?

When both population distributions can be modeled as normal, or when both samples are large: $n_{1} \geq 30$ and $n_{2} \geq 30$ .
Review: 4.6 Sampling Distributions for the Difference Between Two Sample Means
Topic 4.7 · Which one fitsIndependent random samples of commuters in two cities record commute times. Which procedure estimates the difference in mean commute times?

A two-sample t-interval for the difference between two population means. Technology gives the degrees of freedom.
Review: 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means
Topic 4.7 · Spot the mistakeIn each of 25 pairs of twins, one twin is randomly chosen for a new reading program. Two-sample t procedure or paired t procedure?

Paired. Each twin is matched to the other, so the groups are not independent. Analyze the differences within each pair with one-sample t procedures for a mean difference.
Review: 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means
Topic 4.8 · Say it in contextA 95% confidence interval for $μ_{A} - μ_{B}$ , the difference in mean commute times in minutes for City A minus City B, is (2.3, 6.8). What does it suggest?

The interval does not contain 0, so there is convincing evidence that the two population means differ. Every plausible value is positive, which suggests the mean commute time is longer in City A, by about 2.3 to 6.8 minutes.
Review: 4.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means
Topic 4.9 · Which one fitsWrite the hypotheses for testing whether two teaching methods produce different mean test scores.

$H_{0} : μ_{1} = μ_{2}$ and $H_{a} : μ_{1} \neq μ_{2}$ , where $μ_{1}$ and $μ_{2}$ are the true mean scores for students taught with method 1 and with method 2.
Review: 4.9 Setting Up a Test for the Difference Between Two Population Means
Topic 4.10 · Say it in contextA two-sample t-test for $H_{a} : μ_{1} \neq μ_{2}$ gives a p-value of 0.21. Interpret it.

Assuming the two population means are equal, there is a 0.21 probability of getting a difference in sample means at least as extreme as the one observed, by chance alone.
Review: 4.10 Carrying Out a Test for the Difference Between Two Population Means

Unit 5: Regression Analysis

Topic 5.1 · Know the termWhat four features belong in a description of a scatterplot?

Form (linear or non-linear), direction (positive or negative), strength (strong, moderate or weak) and unusual features, all in context.
Review: 5.1 Graphical Representations Between Two Quantitative Variables
Topic 5.2 · Spot the mistake"The correlation is $r = 0.95$ , so a straight line is a good model." What is wrong?

A correlation close to 1 or -1 does not by itself mean a linear model is appropriate. Check the scatterplot and the residual plot for curvature.
Review: 5.2 Correlation
Topic 5.2 · Spot the mistakeTowns with higher ice cream sales have more drownings, with $r = 0.8$ . Does ice cream cause drowning?

No. Correlation does not imply causation. Hot weather plausibly increases both swimming and ice cream sales.
Review: 5.2 Correlation
Topic 5.3 · Spot the mistakeA regression line fit to children aged 5 to 15 is used to predict height at age 40. What is the problem?

Extrapolation. Age 40 is far beyond the ages used to fit the line, so the prediction is unreliable.
Review: 5.3 Linear Regression Models
Topic 5.4 · Work it outA model predicts $\hat{y} = 72$ , and the observed value is $y = 68$ . Find the residual and say what it means.

Residual $= y - \hat{y} = 68 - 72 = - 4$ . It is negative, so the model overpredicted this value by 4.
Review: 5.4 Residuals
Topic 5.4 · Say it in contextA residual plot for a linear model shows a clear curved pattern. What does that tell you?

A linear model is not the most appropriate model for these data. Random scatter with no pattern would support the linear model.
Review: 5.4 Residuals
Topic 5.5 · Say it in contextFor used cars, $\hat{price} = 15,000 - 1,200 (age)$ , with price in dollars and age in years. Interpret the slope.

For each additional year of age, the predicted price of these used cars decreases by 1,200 dollars.
Review: 5.5 Least-Squares Regression
Topic 5.5 · Say it in contextPredicting exam score from hours studied gives $r^{2} = 0.64$ . Interpret this value.

64% of the variation in exam scores is explained by the linear relationship with hours studied.
Review: 5.5 Least-Squares Regression
Topic 5.5 · Spot the mistakeWhen does the y-intercept of a regression line not have a reasonable interpretation?

When $x = 0$ is outside the range of x-values in the data, which makes it an extrapolation, or when the predicted value at $x = 0$ is impossible in context, such as a negative height.
Review: 5.5 Least-Squares Regression
Topic 5.5 · Know the termWhy is it called the least-squares regression line, and which point does it always pass through?

It is the line that minimizes the sum of the squared residuals. It always passes through $(\overline{x}, \overline{y})$ .
Review: 5.5 Least-Squares Regression