AP Statistics Unit 7 Practice Test: Inference for Quantitative Data: Means

Below is our free AP Statistics unit 7 practice test. These questions are about means — population means and sample means. This includes using the T and Z distributions to test null hypotheses about population means (including the differences between the population means of 2 groups) or to get confidence intervals for population means (or differences).

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Question 1

When is it appropriate to use the T distribution to find a confidence interval for a mean? Choose the best answer.

A
When the population mean is unknown.
B
When the population mean is known.
C
When the population standard deviation is unknown.
D
When the population standard deviation is known.
Question 1 Explanation: 
The correct answer is (C). When population standard deviation is not known, the appropriate confidence interval procedure for estimating the population mean of one quantitative variable for one sample is to use a T table to find the appropriate T critical value and use that to find the margin of error and then the interval. Generally this is done when the population mean is unknown, but that’s not the specific reason that it’s a T value.
Question 2

Which statements are true of t-distributions?

  1. More of the area of the distribution is allocated to the tails than in a normal distribution.

  2. Less of the area of the distribution is allocated to the tails than in a normal distribution.

  3. As degrees of freedom increase, the area in the tails of a t-distribution increases.

  4. As degrees of freedom increase, the area in the tails of a t-distribution decreases.

A
I and III
B
I and IV
C
II and III
D
II and IV
Question 2 Explanation: 
The correct answer is (B). A t-distribution differs from a normal distribution with more of the area allocated to the tails. The area in the tails decreases as degrees of freedom increase because the T becomes closer and closer to the standard normal distribution.
Question 3

What is the margin of error for a one-sample t-interval with sample size 20 and sample standard deviation 1.2 with a confidence level of 95%? Use the condensed table of values below.

df

95%

19

2.093

20

2.086

21

2.080
A
0.5581
B
0.5597
C
0.5616
D
0.5762
Question 3 Explanation: 
The correct answer is (C). The formula is: $ t*\left(\dfrac{s}{\sqrt{n}}\right)$

Where $t$ is the value obtained using degrees of freedom $(df=n-1)$ on a chart, $s$ is the sample standard deviation, and $n$ is the sample size.

$t \ast \dfrac{s}{\sqrt{n}} = 2.093 \ast \dfrac{1.2}{\sqrt{20}} $ $ = 0.5616$
Question 4

Which of the following statements is NOT true about the interpretation of a confidence interval?

A
We can be C% confident that the confidence interval for a population captures the population mean.
B
An interpretation of the confidence interval includes a reference to the sample and details about the population it represents.
C
Each interval is based on data from a random sample.
D
The confidence interval will be the same for every random sample.
Question 4 Explanation: 
The correct answer is (D). Because the data (and specifically the sample standard deviation of the data) will vary from sample to sample, the confidence interval will not be the same for every random sample.
Question 5

Which could be an appropriate null and alternative hypothesis for a one-sample t-test for a population mean?

A
$H_0:µ=µ_0,$ $\, H_a:µ≠µ_0$
B
$H_0:µ<µ_0,$ $\, H_a:µ>µ_0$
C
$H_0:µ≠µ_0,$ $\, H_a:µ=µ_0$
D
$H_0:µ=µ_0,$ $\, H_a:µ≥µ_0$
Question 5 Explanation: 
The correct answer is (A). For a one-sample t-test, $H_0$ is $µ=µ_0$ where $µ_0$ is the hypothesized value.

$H_a$ may be $µ<µ_0 , µ>µ_0$, or $µ≠µ_0$.
Question 6

What should an interpretation of the p-value of a significance test for a population mean recognize?

A
The p-value is computed by assuming the true population mean is equal to the value stated in the null hypothesis.
B
The p-value is computed by assuming the true population mean is equal to the value stated in the alternative hypothesis.
C
The p-value is computed by assuming the true population mean is not equal to the value stated in the null hypothesis.
D
The p-value is computed without input from the null or alternative hypothesis.
Question 6 Explanation: 
The correct answer is (A). The p-value is computed by assuming the true population mean is equal to the value stated in the null hypothesis. The p-value is the probability that we would see a statistic as extreme as or more extreme than the statistic we see now, given that the value of the mean specified in the null hypothesis.
Question 7

Which of the following is a necessary condition to calculate confidence intervals for the difference of two population means?

A
The sampling distributions of $x_1$ and $x_2$ should both be approximately normal; if skewed either $n_1$ or $n_2$ must be greater than 30.
B
The sampling distributions of $x_1$ and $x_2$ should both be approximately normal; if skewed both $n_1$ or $n_2$ must be greater than or equal to 30.
C
The sampling distributions of $x_1$ and $x_2$ should both be approximately normal; if skewed either $n_1$ or $n_2$ must be equal to 30.
D
None of the above are necessary conditions.
Question 7 Explanation: 
The correct answer is (B). To calculate confidence intervals for the difference of two population means, the sampling distribution should be approximately normal; if the observations are skewed, both $n_1$ and $n_2$ must be greater than or equal to 30.
Question 8

What effect does the sample size have on the width of a confidence interval for the difference of two means?

A
The width of the confidence interval for the difference of two means does not change as the sample sizes increase.
B
The width of the confidence interval for the difference of two means tends to increase as the sample sizes increase.
C
The width of the confidence interval for the difference of two means tends to increase exponentially as the sample sizes increase.
D
The width of the confidence interval for the difference of two means tends to decrease as the sample sizes increase.
Question 8 Explanation: 
The correct answer is (D). The width of the confidence interval for the difference of two means tends to decrease as the sample sizes increase, all other factors remaining the same. This is because the standard error has the sample sizes in the denominator. This is generally true when doing a one-sample test and/or a test for proportion as well, as the n is in the denominators of the standard errors in those cases as well.
Question 9

What could not be appropriate null and alternative hypotheses for a two-sample t-test for a difference of two population means?

A
$H_0:µ_1=µ_2,$ $ \, H_a:µ_1≤µ_2$
B
$H_0:µ_1=µ_2,$ $ \, H_a:µ_1>µ_2$
C
$H_0:µ_1-µ_2=0,$ $ \, H_a:µ_1<µ_2$
D
$H_0:µ_1-µ_2=0,$ $ \, H_a:µ_1≠µ_2$
Question 9 Explanation: 
The correct answer is (A). An appropriate null hypothesis is a statement of equality, either $µ_1=µ_2 \,$ or $\, µ_1-µ_2=0$.

An appropriate alternative hypothesis can not contain an overlapping statement, so the form $µ_1≥µ_2 \,$ or $\, µ_1≤µ_2$ cannot be appropriate for the alternative hypothesis.
Question 10

When conducting a test for the difference of two population means, both with unknown variances, what is an appropriate number for degrees of freedom for sample sizes of 43 and 37?

A
80
B
43
C
78
D
79
Question 10 Explanation: 
The correct answer is (C). The value should be:

$n_1+n_2-2$

$43+37-2=78$
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