AP Statistics Unit 4: Inference for Quantitative Data: Means

The sampling distribution of a sample mean and the Central Limit Theorem, t distributions, confidence intervals and significance tests for one mean and for a mean difference from paired data, and for the difference between two independent means.

Topics: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10 · free response

Key formulas and facts

• Sampling distribution of $\bar x$: $\mu_{\bar x}=\mu$, $\sigma_{\bar x}=\dfrac{\sigma}{\sqrt n}$; normal if the population is normal, approximately normal when $n\ge30$ (Central Limit Theorem)
• One-sample t interval: $\bar x\pm t^*\dfrac{s_x}{\sqrt n}$ with df $=n-1$; paired data: apply it to the differences, $\bar x_d\pm t^*\dfrac{s_d}{\sqrt n}$
• One-sample t test: $t=\dfrac{\bar x-\mu_0}{s_x/\sqrt n}$, df $=n-1$
• Difference of means: $\mu_{\bar x_1-\bar x_2}=\mu_1-\mu_2$, $\sigma_{\bar x_1-\bar x_2}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}$
• Two-sample t interval: $(\bar x_1-\bar x_2)\pm t^*\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}$; test: $t=\dfrac{\bar x_1-\bar x_2}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$
• Two-sample df: use technology (Welch–Satterthwaite) or, conservatively, the smaller of $n_1-1$ and $n_2-1$

Inference procedures and conditions

• One-sample t interval/test for μ (or $\mu_d$ with paired data). Random: random sample or random assignment. 10%: $n\le0.10N$ when sampling without replacement. Normal/Large Sample: the population is normal, or $n\ge30$, or a graph of the sample (of the differences, for paired data) shows no strong skew or outliers.
• Two-sample t interval/test for $\mu_1-\mu_2$: two independent random samples or two randomly assigned treatment groups; 10% for each sample; Normal/Large Sample for each group.
• State: name the parameter(s) in context (μ, $\mu_d$ or $\mu_1-\mu_2$), the hypotheses or confidence level and α. Plan: name the procedure and check the conditions. Do: statistic, df, interval or test statistic and p-value. Conclude in context.
• Paired or two-sample? Paired when the same individual is measured twice, or individuals are matched in pairs; two-sample when the groups are separate and independent.

Worked examples

• CLT: if waiting times have $\mu=12$ and $\sigma=5$ minutes, then for samples of 50, $\sigma_{\bar x}=\frac{5}{\sqrt{50}}\approx0.707$ and $P(\bar x>13)\approx P(Z>1.414)\approx0.0786$.
• One-sample interval: $\bar x=64.3$, $s_x=5.1$, $n=18$; df = 17, $t^*\approx2.110$, so a 95% interval is $64.3\pm2.110\frac{5.1}{\sqrt{18}}=(61.76,\ 66.84)$.
• One-sample test: $H_0:\mu=50$, $H_a:\mu>50$, $\bar x=52.4$, $s_x=6$, $n=25$ gives $t=2.0$, df = 24 and p-value $\approx0.0285$.
• Paired data: differences 3, 5, −1, 4, 2, 6 have $\bar x_d=3.1\overline{6}$ and $s_d\approx2.483$; $t=\frac{3.167}{2.483/\sqrt6}\approx3.124$ with df = 5.
• Two-sample interval: $\bar x_1=88$, $s_1=10$, $n_1=40$ and $\bar x_2=82$, $s_2=12$, $n_2=35$: $SE=\sqrt{2.5+4.114}\approx2.572$, Welch df $\approx66.5$, and a 95% interval is about $(0.87,\ 11.13)$.

Common traps

• Using $z^*$ or the normal distribution when $\sigma$ is estimated by $s_x$.
• Treating paired data as two independent samples (or the reverse): ask whether each value in one group is linked to one value in the other.
• Checking the Normal/Large Sample condition with the original two columns instead of the differences for paired data.
• Saying the Central Limit Theorem makes the population (or the sample) normal: it is about the sampling distribution of $\bar x$.
• Using $\sigma/n$ instead of $\sigma/\sqrt n$, or adding standard deviations instead of variances for two independent samples.
• Interpreting a confidence interval for a mean as a range for individual values.

Practice questions

80 multiple-choice questions (4 options each, as on the new exam) and 3 free-response questions worth 10 points each. A calculator is allowed throughout. Try each one before opening the answer. Each line of a scoring guide is one point, scored 0 or 1 and earned independently of the other lines, the style of the new course's 10-point free-response scoring (no E/P/I part scores). A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Topic 4.1: The sampling distribution of a sample mean

8 multiple-choice questions.

Question 1 · Multiple choice

The amount of coffee dispensed by a machine has mean 240 mL and standard deviation 12 mL. For random samples of 16 cups, what is the standard deviation of the sample mean $\bar x$?

1. 0.75
2. 3.00
3. 12.00
4. 48.00

Question 2 · Multiple choice

The weights of a breed of adult cat are approximately normal with mean 4.5 kg and standard deviation 0.8 kg. What is the probability that the mean weight of a random sample of 10 cats is more than 4.8 kg?

1. 0.0300
2. 0.1178
3. 0.3538
4. 0.8822

Question 3 · Multiple choice

The number of minutes customers spend in a store is strongly skewed to the right, with mean 18 and standard deviation 11. Which statement about the sampling distribution of $\bar x$ for random samples of 50 customers is correct?

1. It is strongly skewed to the right, like the population.
2. It cannot be described because the population is not normal.
3. It is approximately normal by the Central Limit Theorem, with mean 18 and standard deviation about 1.56.
4. It is approximately normal with mean 18 and standard deviation 11.

Question 4 · Multiple choice

Daily step counts for adults in a city have mean 7,400 and standard deviation 2,600. For random samples of 100 adults, what is the probability that the sample mean is below 7,000 steps?

1. 0.0400
2. 0.0620
3. 0.4389
4. 0.9380

Question 5 · Multiple choice

For random samples from a population with standard deviation 20, what sample size makes the standard deviation of $\bar x$ equal to 2.5?

1. 8
2. 50
3. 64
4. 160

Question 6 · Multiple choice

Battery lifetimes have mean 52 hours. For random samples of 25 batteries, what is the mean of the sampling distribution of $\bar x$?

1. 2.1
2. 10.4
3. 52.0
4. 1300.0

Question 7 · Multiple choice

Scores on a standardized reading test are approximately normal with mean 68 and standard deviation 8. For a random sample of 36 students, what is the probability that the sample mean is within 2 points of 68?

1. 0.1974
2. 0.6800
3. 0.8664
4. 0.9500

Question 8 · Multiple choice

Why does the sampling distribution of $\bar x$ for samples of size 5 from a strongly skewed population NOT look normal?

1. The Central Limit Theorem needs a large sample (a common guideline is $n\ge30$) when the population is not normal.
2. The standard deviation formula only works for $n\ge30$.
3. The mean of the sampling distribution is not $\mu$ for small samples.
4. Samples of size 5 are biased.

Topic 4.2: Confidence interval for a population mean or a population mean difference (paired data)

8 multiple-choice questions.

Question 9 · Multiple choice

A random sample of 25 commuters in a city had a mean commute time of 31.4 minutes with standard deviation 6.2 minutes. The sample data show no strong skew or outliers. Which is a 95% confidence interval for the mean commute time of all commuters in the city?

1. $(18.60,\ 44.20)$
2. $(28.84,\ 33.96)$
3. $(28.97,\ 33.83)$
4. $(30.89,\ 31.91)$

Question 10 · Multiple choice

What is the critical value $t^*$ for a 90% confidence interval for a mean based on a sample of 15?

1. 1.345
2. 1.645
3. 1.761
4. 2.145

Question 11 · Multiple choice

A quality inspector weighs a random sample of 10 bags of trail mix (table). A dotplot of the weights shows no strong skew or outliers. Which is a 95% confidence interval for the mean weight of all bags?

1. $(11.09,\ 13.35)$
2. $(11.56,\ 12.28)$
3. $(11.86,\ 12.58)$
4. $(11.91,\ 12.53)$

Question 12 · Multiple choice

Eight randomly selected patients had their systolic blood pressure measured before and after a month on a new exercise plan (table). The differences (before − after) show no strong skew or outliers. Which is a 90% confidence interval for the mean difference (before − after) for all such patients?

1. $(0.26,\ 9.74)$
2. $(1.07,\ 8.93)$
3. $(3.07,\ 6.93)$
4. $(6.07,\ 9.93)$

Question 13 · Multiple choice

A random sample of 40 tomatoes from a farm had mean weight 142 g and standard deviation 18 g. What is the margin of error for a 99% confidence interval for the mean weight of all the farm's tomatoes?

1. 1.22
2. 5.58
3. 7.33
4. 7.71

Question 14 · Multiple choice

Why do we use $t^*$ instead of $z^*$ for a confidence interval for a population mean?

1. The t distribution is only used for small populations.
2. We estimate $\sigma$ with $s_x$, which adds variability, and t distributions have heavier tails to account for it.
3. The population is never normal.
4. The sample mean is biased.

Question 15 · Multiple choice

A random sample of 12 house prices in a town includes one price far above the others, and a boxplot shows strong right skew. Is it appropriate to construct a one-sample t interval for the mean price?

1. No: with only 12 values, strong skew and an outlier, the Normal/Large Sample condition is not met.
2. No: the sample must have at least 100 values.
3. Yes: t intervals work for any sample size.
4. Yes: the sample is random.

Question 16 · Multiple choice

A random sample of 60 college students slept a mean of 5.8 hours the night before an exam, with standard deviation 1.9 hours. Which is a 95% confidence interval for the mean for all students at the college?

1. $(3.90,\ 7.70)$
2. $(5.00,\ 6.60)$
3. $(5.31,\ 6.29)$
4. $(5.55,\ 6.05)$

Topic 4.3: Using an interval for a population mean or mean difference to justify a claim

8 multiple-choice questions.

Question 17 · Multiple choice

A 95% confidence interval for the mean amount of soda in a brand's 355 mL cans is (352.1, 354.6) mL. Is there convincing evidence that the mean fill differs from 355 mL?

1. Yes: 355 is not in the interval, so it is not a plausible value for the mean.
2. Yes: the sample mean is exactly 353.35.
3. No: individual cans vary.
4. No: 355 is close to 354.6.

Question 18 · Multiple choice

A 90% confidence interval for the mean time to complete a puzzle is (8.4, 11.2) minutes. The manufacturer claims the mean is 10 minutes. What does the interval suggest?

1. The claim is proven true.
2. The claim is false because the interval is wide.
3. 10 is in the interval, so the claim is plausible.
4. The claim is false because the center is 9.8.

Question 19 · Multiple choice

A 95% confidence interval for the mean weekly hours of exercise for adults in a town is (3.2, 4.6). Which interpretation is correct?

1. 95% of adults in the town exercise between 3.2 and 4.6 hours a week.
2. There is a 95% chance the sample mean is between 3.2 and 4.6.
3. We are 95% confident that the interval from 3.2 to 4.6 hours captures the true mean weekly exercise time of all adults in the town.
4. The mean of all adults is 3.9 hours.

Question 20 · Multiple choice

A confidence interval for a population mean is (46.2, 53.8). What is the sample mean?

1. 3.8
2. 7.6
3. 46.2
4. 50.0

Question 21 · Multiple choice

Using the interval (46.2, 53.8), what is the margin of error?

1. 1.9
2. 3.8
3. 7.6
4. 50.0

Question 22 · Multiple choice

A 99% confidence interval for the mean difference in reaction time (after caffeine − before) for a random sample of adults is (−0.042, −0.008) seconds. What does the interval suggest?

1. All plausible mean differences are negative, so there is convincing evidence that reaction times are shorter after caffeine.
2. There is no evidence of a change because the values are small.
3. Reaction times are longer after caffeine.
4. The interval is invalid because it is negative.

Question 23 · Multiple choice

A researcher computes a 95% confidence interval for a mean and then a 90% interval from the same data. Which is true?

1. The 90% interval cannot contain the sample mean.
2. The 90% interval is narrower and contained in the 95% interval.
3. The 90% interval is wider.
4. The intervals have different centers.

Question 24 · Multiple choice

A 95% confidence interval for the mean height of a plant species is (41.2, 47.6) cm. A student says, "95% of these plants are between 41.2 and 47.6 cm tall." What is wrong?

1. The interval should be about the sample mean.
2. Nothing is wrong.
3. The interval is about the mean height, not individual plants; individual heights vary much more.
4. It should say 5% of plants are outside the interval.

Topic 4.4: Setting up a test for a population mean or a population mean difference

8 multiple-choice questions.

Question 25 · Multiple choice

A bakery says its muffins weigh 120 g on average. A customer suspects they weigh less. Which hypotheses should be tested?

1. $H_0: \bar x=120$, $H_a: \bar x<120$
2. $H_0: \mu=120$, $H_a: \mu<120$, where μ is the mean weight of all the bakery's muffins
3. $H_0: \mu=120$, $H_a: \mu\ne120$, where μ is the mean of the customer's sample
4. $H_0: \mu<120$, $H_a: \mu=120$

Question 26 · Multiple choice

Each of 20 students took a practice test before and after a study workshop. Which procedure tests whether scores improved on average?

1. A paired t test on the differences (after − before)
2. A one-sample z test for a proportion
3. A chi-square test for independence
4. A two-sample t test comparing before and after scores

Question 27 · Multiple choice

Which study calls for a two-sample t test rather than a paired t test?

1. Comparing each twin's IQ with the other twin's IQ
2. Comparing each runner's time with old shoes and with new shoes
3. Comparing the mean commute times of a random sample of 40 drivers and a separate random sample of 35 bus riders
4. Comparing the left-hand and right-hand grip strength of each of 30 people

Question 28 · Multiple choice

A random sample of 8 delivery times (minutes) is 14, 16, 15, 18, 17, 15, 16, 41. Is the Normal/Large Sample condition met for a one-sample t test?

1. No: the population standard deviation is unknown.
2. Yes: the sample is random.
3. Yes: most values are between 14 and 18.
4. No: the sample is small (n < 30) and 41 is an outlier.

Question 29 · Multiple choice

A random sample of 10 rainfall totals (mm) is 22, 25, 19, 24, 27, 21, 23, 26, 20, 24. Is the Normal/Large Sample condition met?

1. Yes: a dotplot of the data shows no strong skew or outliers.
2. Yes: the mean and median are both 24.
3. No: the data are not exactly normal.
4. No: the sample size is less than 30.

Question 30 · Multiple choice

A nutritionist wants to test whether the mean sodium content of a brand's soup differs from the 480 mg on the label. Which alternative hypothesis is appropriate?

1. $H_a: \bar x\ne480$
2. $H_a: \mu\ne480$
3. $H_a: \mu=480$
4. $H_a: \mu>480$

Question 31 · Multiple choice

For a paired t test, the parameter of interest is $\mu_d$. What does $\mu_d$ represent?

1. The mean of the sample differences
2. The mean of the differences for all individuals in the population (or the true mean difference for the treatment)
3. The difference between the two sample means
4. The difference between two population standard deviations

Question 32 · Multiple choice

A random sample of 45 apartments in a city is used to test a claim about mean rent. The rents are moderately skewed to the right. Is the Normal/Large Sample condition met?

1. Yes: rents are quantitative.
2. No: rents are not normally distributed.
3. No: the rents are skewed.
4. Yes: $n=45\ge30$, so the Central Limit Theorem applies.

Topic 4.5: Carrying out a test for a population mean or a population mean difference

8 multiple-choice questions.

Question 33 · Multiple choice

For the muffin test ($H_0: \mu=120$, $H_a: \mu<120$), a random sample of 20 muffins had $\bar x=117.2$ g and $s_x=6.4$ g. What is the test statistic?

1. -2.800
2. -1.957
3. -0.437
4. 1.957

Question 34 · Multiple choice

Using the muffin sample ($\bar x=117.2$, $s_x=6.4$, n = 20; $H_a: \mu<120$), what is the p-value?

1. 0.0252
2. 0.0326
3. 0.0653
4. 0.9674

Question 35 · Multiple choice

For the muffin test ($H_0: \mu=120$, $H_a: \mu<120$), a random sample of 20 muffins had $\bar x=117.2$ g and $s_x=6.4$ g, with no strong skew or outliers. What is the conclusion at $\alpha=0.05$?

1. Reject $H_0$: every muffin weighs less than 120 g.
2. Reject $H_0$: there is convincing evidence that the mean weight of the bakery's muffins is less than 120 g.
3. Accept $H_0$: the mean weight is 120 g.
4. Fail to reject $H_0$: there is not convincing evidence that the mean weight is less than 120 g.

Question 36 · Multiple choice

For the blood-pressure data (before and after a month of exercise; 8 patients), test $H_0: \mu_d=0$ against $H_a: \mu_d>0$ where d = before − after. What is the p-value?

1. 0.0009
2. 0.0017
3. 0.0086
4. 0.0500

Question 37 · Multiple choice

A random sample of 36 phone batteries lasted a mean of 4.62 hours on a test with $s_x=0.9$ hours. For $H_0: \mu=4.3$ and $H_a: \mu\ne4.3$, what is the p-value?

1. 0.0200
2. 0.0400
3. 0.3200
4. 0.9600

Question 38 · Multiple choice

A company claims the mean assembly time of its desk is at most 50 minutes. A random sample of 15 customers had $\bar x=51.3$ minutes and $s_x=4.8$ minutes (no strong skew or outliers). At $\alpha=0.05$, is there convincing evidence that the mean time is more than 50 minutes?

1. Yes: 51.3 is more than 50.
2. Yes: reject $H_0$; the mean assembly time is more than 50 minutes.
3. No: accept $H_0$; the mean is exactly 50 minutes.
4. No: fail to reject $H_0$; there is not convincing evidence that the mean assembly time is more than 50 minutes.

Question 39 · Multiple choice

A one-sample t test with n = 12 gives t = 2.31 for $H_a: \mu>\mu_0$. What is the p-value?

1. 0.0104
2. 0.0206
3. 0.0231
4. 0.0413

Question 40 · Multiple choice

A paired t test of $H_0: \mu_d=0$ gives a p-value of 0.18. Which conclusion is correct?

1. Reject $H_0$ because 0.18 > 0.05.
2. There is an 18% chance that $H_0$ is true.
3. Fail to reject $H_0$; the data do not give convincing evidence of a nonzero mean difference.
4. Accept $H_0$; the mean difference is 0.

Topic 4.6: The sampling distribution of a difference between two sample means

8 multiple-choice questions.

Question 41 · Multiple choice

Male and female adult penguins in a colony have mean masses 5.4 kg and 4.8 kg with standard deviations 0.6 kg and 0.5 kg. For independent random samples of 20 males and 25 females, what is the mean of $\bar x_M-\bar x_F$?

1. -0.60
2. 0.10
3. 0.60
4. 10.20

Question 42 · Multiple choice

For the penguins (standard deviations 0.6 kg and 0.5 kg; samples of 20 and 25), what is the standard deviation of $\bar x_M-\bar x_F$?

1. 0.0280
2. 0.1000
3. 0.1673
4. 0.2342

Question 43 · Multiple choice

Penguin masses are approximately normal: males mean 5.4 kg (SD 0.6), females mean 4.8 kg (SD 0.5). For independent random samples of 20 males and 25 females, what is the probability that $\bar x_M-\bar x_F$ is greater than 0.8 kg?

1. 0.1160
2. 0.2000
3. 0.3446
4. 0.8840

Question 44 · Multiple choice

Two machines fill bottles with standard deviations 9 mL and 12 mL. For independent random samples of 30 and 40 bottles, what is the standard deviation of $\bar x_1-\bar x_2$?

1. 0.159
2. 2.510
3. 3.000
4. 3.541

Question 45 · Multiple choice

Machine 1 fills bottles with mean 503 mL (SD 9 mL) and Machine 2 with mean 500 mL (SD 12 mL). For independent random samples of 30 and 40 bottles, what is the probability that the Machine 2 sample mean is larger than the Machine 1 sample mean ($\bar x_1-\bar x_2<0$)?

1. 0.1160
2. 0.4013
3. 0.5000
4. 0.8840

Question 46 · Multiple choice

When is the sampling distribution of $\bar x_1-\bar x_2$ approximately normal?

1. Only when the two population means are equal
2. Only when the two sample sizes are equal
3. Whenever the samples are random, regardless of size or shape
4. When both populations are normal, or both sample sizes are large (each at least 30)

Question 47 · Multiple choice

City A's daily high temperatures in July have mean 31°C and City B's have mean 27°C. For random samples of days from each city, what is the mean of the sampling distribution of $\bar x_B-\bar x_A$?

1. -4.0
2. 2.0
3. 4.0
4. 58.0

Question 48 · Multiple choice

Why can't the formula $\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}$ be used for before-and-after measurements on the same people?

1. The sample sizes are equal.
2. The populations are not normal.
3. The means are the same.
4. The two sets of measurements are not independent, so their variances do not simply add.

Topic 4.7: Confidence interval for the difference between two population means

8 multiple-choice questions.

Question 49 · Multiple choice

Independent random samples of students from two high schools took the same math test: School A ($n=32$, $\bar x=78.4$, $s=8.2$) and School B ($n=28$, $\bar x=73.1$, $s=9.6$). Which is a 95% confidence interval for $\mu_A-\mu_B$ (technology, Welch df)?

1. $(-9.96,\ -0.64)$
2. $(-0.56,\ 11.16)$
3. $(0.64,\ 9.96)$
4. $(2.36,\ 8.24)$

Question 50 · Multiple choice

Using the two schools ($s_A=8.2$, $n_A=32$; $s_B=9.6$, $n_B=28$), what is the standard error of $\bar x_A-\bar x_B$?

1. 0.415
2. 2.322
3. 3.264
4. 5.300

Question 51 · Multiple choice

For the two schools (A: $s=8.2$, $n=32$; B: $s=9.6$, $n=28$), what are the Welch degrees of freedom that a calculator uses for the two-sample t interval?

1. 27.0
2. 31.0
3. 53.5
4. 58.0

Question 52 · Multiple choice

In an experiment, 16 seedlings were randomly assigned to a new fertilizer or the standard one, 8 each. Heights after 4 weeks are in the table (neither group shows strong skew or outliers). Which is a 90% confidence interval for $\mu_{new}-\mu_{standard}$?

1. $(-6.36,\ -2.39)$
2. $(1.39,\ 7.36)$
3. $(2.39,\ 6.36)$
4. $(3.89,\ 7.86)$

Question 53 · Multiple choice

Random samples of customers at two gyms reported weekly visit minutes: Gym 1 ($n=50$, $\bar x=42.5$, $s=11.3$) and Gym 2 ($n=45$, $\bar x=47.9$, $s=12.8$). Which is a 99% confidence interval for $\mu_1-\mu_2$?

1. $(-14.95,\ 4.15)$
2. $(-11.95,\ 1.15)$
3. $(-10.26,\ -0.54)$
4. $(-1.15,\ 11.95)$

Question 54 · Multiple choice

Using the two gyms (Gym 1: n = 50, s = 11.3; Gym 2: n = 45, s = 12.8), what is the margin of error of a 95% confidence interval for $\mu_1-\mu_2$?

1. 2.49
2. 3.75
3. 4.95
4. 6.55

Question 55 · Multiple choice

Which condition is needed for a two-sample t interval for $\mu_1-\mu_2$?

1. The two sample standard deviations must be equal.
2. Two independent random samples (or random assignment) with each sample size at least 30 or each sample showing no strong skew or outliers
3. Each pair of observations must come from the same individual.
4. The two samples must be the same size.

Question 56 · Multiple choice

Using the fertilizer experiment (New: 24, 27, 22, 26, 29, 25, 23, 28; Standard: 20, 23, 19, 22, 21, 24, 18, 22), what is the point estimate of $\mu_{new}-\mu_{standard}$?

1. 3.500
2. 4.375
3. 21.125
4. 25.500

Topic 4.8: Using an interval for mu1 - mu2 to justify a claim

8 multiple-choice questions.

Question 57 · Multiple choice

A 95% confidence interval for $\mu_A-\mu_B$, the difference in mean test scores at Schools A and B, is (0.6, 10.0). What does it suggest?

1. School A's mean is exactly 5.3 points higher.
2. 0 is not in the interval, so there is convincing evidence that School A's mean score is higher.
3. 0 is plausible, so the schools may have equal means.
4. Every student at A scores higher than every student at B.

Question 58 · Multiple choice

A 99% confidence interval for $\mu_1-\mu_2$ (Gym 1 − Gym 2 weekly minutes) is (−12.1, 1.3). Is a difference of 0 plausible, and what does the interval suggest?

1. 0 is in the interval, so there is not convincing evidence that the mean visit times differ.
2. Gym 1's members exercise more.
3. Gym 2's members exercise more because most of the interval is negative.
4. The gyms definitely have the same mean.

Question 59 · Multiple choice

A 95% confidence interval for $\mu_{new}-\mu_{old}$ in a randomized experiment on battery designs is (2.1, 5.9) hours. Which interpretation is correct?

1. We are 95% confident that the interval from 2.1 to 5.9 hours captures the true difference in mean battery life (new − old) for batteries like these.
2. There is a 95% chance that $\bar x_{new}-\bar x_{old}$ is between 2.1 and 5.9.
3. The new design lasts exactly 4 hours longer.
4. 95% of new batteries last 2.1 to 5.9 hours longer than old ones.

Question 60 · Multiple choice

A 90% interval for $\mu_{city}-\mu_{rural}$ (mean annual rainfall, cm) is (−18.4, −3.2). Which statement is supported?

1. The city mean rainfall is greater.
2. There is no evidence of a difference.
3. The interval is invalid.
4. The rural mean rainfall is greater than the city mean.

Question 61 · Multiple choice

A confidence interval for $\mu_1-\mu_2$ from an observational study is entirely above 0. What conclusion is justified?

1. The population means differ, but we cannot conclude the grouping variable causes the difference.
2. Every individual in group 1 scores higher.
3. Group 1 membership causes higher values.
4. The sample means are equal.

Question 62 · Multiple choice

A 95% confidence interval for $\mu_1-\mu_2$ is (−2.4, 6.8). What is the point estimate of the difference?

1. 2.2
2. 4.4
3. 4.6
4. 9.2

Question 63 · Multiple choice

Two researchers estimate the same $\mu_1-\mu_2$ with the same data, one at 90% and one at 99% confidence. The 90% interval is (0.4, 3.0). Which could be the 99% interval?

1. (−0.4, 3.8)
2. (0.4, 3.0)
3. (1.2, 4.8)
4. (0.8, 2.6)

Question 64 · Multiple choice

A study finds a 95% interval for the difference in mean daily screen time (teens − adults) of (1.2, 1.8) hours. A newspaper writes, "Teens spend 1.5 hours more on screens than adults." What is missing?

1. The fact that 0 is in the interval.
2. Nothing; the statement is exact.
3. The uncertainty: plausible values for the difference run from 1.2 to 1.8 hours.
4. The fact that the difference is not significant.

Topic 4.9: Setting up a test for the difference between two population means

8 multiple-choice questions.

Question 65 · Multiple choice

A researcher wants to know whether students who sleep at least 8 hours have a higher mean GPA than those who sleep less. Which hypotheses are appropriate?

1. $H_0: \mu_{8+}-\mu_{<8}>0$, $H_a: \mu_{8+}-\mu_{<8}=0$
2. $H_0: \mu_{8+}-\mu_{<8}=0$, $H_a: \mu_{8+}-\mu_{<8}>0$
3. $H_0: \mu_d=0$, $H_a: \mu_d>0$
4. $H_0: \bar x_{8+}-\bar x_{<8}=0$, $H_a: \bar x_{8+}-\bar x_{<8}>0$

Question 66 · Multiple choice

An experiment randomly assigns 60 volunteers to a meditation app or a control app and measures stress scores after 4 weeks. Which procedure tests whether the meditation app lowers the mean stress score?

1. A two-sample t test for $\mu_1-\mu_2$
2. A chi-square test for homogeneity
3. A two-sample z test for proportions
4. A paired t test

Question 67 · Multiple choice

Random samples of 18 and 22 tutoring sessions at two centers have session lengths with no strong skew or outliers in either sample. Is the Normal/Large Sample condition met for a two-sample t test?

1. Yes: neither sample shows strong skew or outliers, so it is reasonable to treat both population distributions as roughly normal.
2. No: both samples must have at least 30 observations.
3. No: the samples have different sizes.
4. Yes: the means are close.

Question 68 · Multiple choice

Which situation calls for a paired t test instead of a two-sample t test?

1. Measuring each of 25 cars' fuel economy with regular gas and with premium gas
2. Comparing fuel economy of 25 sedans and 25 trucks
3. Comparing heights of a random sample of 30 boys and 30 girls
4. Comparing test scores of students in two different classes

Question 69 · Multiple choice

Why does the Random condition for a two-sample t test in an observational study require the two samples to be independent?

1. So that the sample sizes are equal
2. So that the populations are normal
3. So that the sample means are equal
4. So that the variance of $\bar x_1-\bar x_2$ is the sum of the two variances

Question 70 · Multiple choice

A company tests whether two shipping methods differ in mean delivery time. Which alternative hypothesis is appropriate?

1. $H_a: \mu_1-\mu_2=0$
2. $H_a: \mu_1-\mu_2>0$
3. $H_a: \bar x_1-\bar x_2\ne0$
4. $H_a: \mu_1-\mu_2\ne0$

Question 71 · Multiple choice

A two-sample t test compares random samples of 12 and 40 commute times. The sample of 12 has a strong outlier. Is the Normal/Large Sample condition met?

1. No: the sample sizes differ.
2. Yes: the combined sample size is 52.
3. Yes: the sample of 40 is large.
4. No: the sample of 12 is small and has a strong outlier.

Question 72 · Multiple choice

In a two-sample t test, what does the null hypothesis $H_0: \mu_1=\mu_2$ say?

1. The two populations have the same standard deviation.
2. Every individual in the two populations has the same value.
3. The two population (true) means are equal.
4. The two sample means are equal.

Topic 4.10: Carrying out a test for the difference between two population means

8 multiple-choice questions.

Question 73 · Multiple choice

For the fertilizer experiment (New: 24, 27, 22, 26, 29, 25, 23, 28; Standard: 20, 23, 19, 22, 21, 24, 18, 22), what is the two-sample t statistic for $H_0: \mu_{new}-\mu_{standard}=0$?

1. 1.944
2. 3.889
3. 4.375
4. 5.389

Question 74 · Multiple choice

Using the fertilizer experiment and $H_a: \mu_{new}-\mu_{standard}>0$, what is the p-value (Welch df)?

1. 0.0009
2. 0.0017
3. 0.0109
4. 0.0500

Question 75 · Multiple choice

Using the two schools (A: n = 32, $\bar x=78.4$, s = 8.2; B: n = 28, $\bar x=73.1$, s = 9.6), test $H_0: \mu_A=\mu_B$ against $H_a: \mu_A\ne\mu_B$ at $\alpha=0.05$. What is the conclusion?

1. Reject $H_0$: School A's teachers are better.
2. Fail to reject $H_0$: not convincing evidence of a difference.
3. Reject $H_0$: there is convincing evidence that the mean scores at the two schools differ.
4. Accept $H_0$: the means are equal.

Question 76 · Multiple choice

Using the two gyms (Gym 1: n = 50, $\bar x=42.5$, s = 11.3; Gym 2: n = 45, $\bar x=47.9$, s = 12.8) and $H_a: \mu_1<\mu_2$, what is the p-value?

1. 0.0164
2. 0.0327
3. 0.0500
4. 0.9836

Question 77 · Multiple choice

Students were randomly assigned to study with flashcards (n = 25, mean recall 14.2 words, s = 3.1) or by rereading (n = 25, mean 13.1, s = 2.8). Neither group's data show strong skew or outliers. At $\alpha=0.05$, is there convincing evidence that flashcards give a higher mean recall?

1. No: accept $H_0$; the methods are equally good.
2. No: fail to reject $H_0$; there is not convincing evidence that flashcards give a higher mean recall.
3. Yes: reject $H_0$; flashcards give higher mean recall.
4. Yes: 14.2 is greater than 13.1.

Question 78 · Multiple choice

Using the flashcard experiment (14.2, s = 3.1, n = 25 versus 13.1, s = 2.8, n = 25), what is the two-sample t statistic?

1. 0.188
2. 1.100
3. 1.317
4. 2.620

Question 79 · Multiple choice

In the flashcard experiment, which error could the researchers have made with their conclusion, and what would it mean?

1. A Type I error: flashcards are not better, but the test said they were.
2. A Type I error: the samples were not random.
3. No error is possible because the sample sizes were equal.
4. A Type II error: flashcards really do give higher mean recall, but the test did not detect it.

Question 80 · Multiple choice

Random samples of 40 adults from Region 1 ($\bar x=68.2$ in., s = 5.4) and 38 from Region 2 ($\bar x=64.9$ in., s = 6.1) are compared. For $H_a: \mu_1>\mu_2$, what is the p-value?

1. 0.0069
2. 0.0137
3. 0.0330
4. 0.9931

Free response

Write a complete answer, in context, as you would type it on the exam. The scoring guides are ours, written in the new course's style: 10 points per question, each point scored 0 or 1 and earned independently (no E/P/I part scores). They are not College Board scoring guidelines.

Question 81 · Free response (10 points)

A shoe company had 10 randomly selected members of a running club each run a 300-meter sprint in its old shoe model and in its new model, in random order on different days. The times (seconds) are in the table.

1. (a) Explain why a paired procedure is appropriate. [1 point]
2. (b) Construct and interpret a 95% confidence interval for the mean difference (old − new). Use the four steps. [7 points]
3. (c) Does the interval give convincing evidence that the new shoe reduces mean sprint time? Explain. [2 points]

Question 82 · Free response (10 points)

A school randomly assigned 68 volunteer students to two review formats for a unit test: a game-based review (n = 35, mean score 81.6, s = 7.9) or a lecture review (n = 33, mean score 77.2, s = 9.4). Neither group's scores show strong skew or outliers.

1. (a) State the hypotheses for testing whether the game-based review leads to a higher mean score, defining the parameters. [2 points]
2. (b) Name the procedure and check the conditions. [2 points]
3. (c) Find the test statistic, degrees of freedom and p-value. [3 points]
4. (d) State a conclusion at $\alpha=0.05$, and say which type of error could have been made. [3 points]

Question 83 · Free response (10 points)

The time a customer spends at an ATM is strongly skewed to the right with mean 6.8 minutes and standard deviation 2.4 minutes.

1. (a) Can you find the probability that one randomly chosen customer spends more than 7.5 minutes? Explain. [2 points]
2. (b) For random samples of 40 customers, describe the sampling distribution of $\bar x$ (shape, center, variability). [3 points]
3. (c) Find the probability that the mean time for a random sample of 40 customers is more than 7.5 minutes. [2 points]
4. (d) How large a sample is needed for the standard deviation of $\bar x$ to be 0.2 minutes? [3 points]