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AP Statistics Unit 7: Inference for Quantitative Data: Means

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Unit 7: Inference for Quantitative Data: Means

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Question 1

A teacher wants to determine whether students improved from a pre-test to a post-test. Each of 15 students took both tests. What is the appropriate procedure?

A.Two-sample

t

-test comparing pre-test and post-test group meansB.Paired

t

-test on the differences (post − pre) for each studentC.One-sample

z

-test comparing the mean pre-test to the mean post-testD.Two-sample

z

-test for proportions

Question 2

A paired $t$ -test produces $\overset{ˉ}{d} = 5.2$ , $s_{d} = 7.1$ , $n = 18$ . Compute the $t$ statistic for testing $H_{0} : μ_{d} = 0$ vs. $H_{a} : μ_{d} > 0$ .

t \approx 0.73

t \approx 3.11

t \approx 3.56

t \approx 1.67

Question 3

A 95% confidence interval for the difference in mean scores $μ_{1} - μ_{2}$ between two schools is $(- 3, 11)$ . What can you conclude at $α = 0.05$ ?

A.There is convincing evidence that School 1 scores higher than School 2.B.We fail to reject

H_{0} : μ_{1} = μ_{2}

because 0 is contained in the confidence interval.C.The mean scores are definitely the same for both schools.D.School 1 scores are higher on average because the midpoint (4) is positive.

Question 4

A researcher collects a random sample of 20 measurements and wants to construct a confidence interval for the population mean. The population standard deviation is unknown. Which distribution should be used?

A.Standard normal (

z

) distributionB.

t

-distribution with 20 degrees of freedomC.

t

-distribution with 19 degrees of freedomD.

χ^{2}

distribution with 19 degrees of freedom

Question 5

A random sample of 16 light bulbs had a mean life of $\overset{x}{ˉ} = 1, 200$ hours with $s = 80$ hours. Construct a 95% confidence interval for the true mean life. (Use $t^{*} \approx 2.131$ for $df = 15$ .)

(1, 143, 1, 257)

(1, 157.4, 1, 242.6)

(1, 040, 1, 360)

(1, 182.6, 1, 217.4)

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Question 6

A two-sample $t$ -test compares mean test scores for two independent classes (Class A: $n_{1} = 25$ , Class B: $n_{2} = 30$ ). Assuming unequal variances, what are the degrees of freedom? (Use the conservative formula: $df = min (n_{1} - 1, n_{2} - 1)$ .)

A.55B.53C.24D.29

Question 7

Conditions for a one-sample $t$ -test include randomness, independence (10% condition), and normality. A sample of $n = 8$ shows a slightly skewed dotplot with no outliers. Is the normality condition met?

A.No;

n = 8 < 30

, so normality cannot be assumed by the CLT.B.Yes; slight skew with no outliers in a small sample is considered acceptable for the

t

-procedure.C.No; the

t

-procedure always requires

n \geq 30

.D.Yes; the

t

-procedure is valid for any sample distribution.

Question 8

A one-sample $t$ -test for the mean yields $t = 3.8$ with $df = 24$ and a two-sided $p$ -value of 0.0009. A student concludes that because $p = 0.0009$ , there is only a 0.09% chance the null hypothesis is true. Identify the error.

A.The student should use a one-sided

p

-value, which would be 0.00045.B.The

p

-value is the probability of obtaining data this extreme if

H_{0}

is true, not the probability that

H_{0}

is true.C.With

p = 0.0009

, the researcher should use

α = 0.001

instead of 0.05.D.A

t

-value of 3.8 with

df = 24

does not correspond to

p = 0.0009

Question 9

A researcher conducts a one-sample $t$ -test with $n = 12$ , $\overset{x}{ˉ} = 55$ , $s = 8$ , testing $H_{0} : μ = 50$ vs. $H_{a} : μ \neq = 50$ . Compute the $t$ test statistic.

t = 0.625

t \approx 2.17

t \approx 1.73

t = 5.0

Question 10

A study finds a statistically significant difference between two teaching methods at $α = 0.05$ . The mean score difference is 2 points on a 100-point exam. What is the most important follow-up consideration?

A.The study should be replicated because

n

must have been very small.B.The 2-point difference may be statistically significant but not practically meaningful for educational decisions.C.A one-sided test should have been used instead of a two-sided test.D.The confidence interval should be computed to replace the hypothesis test.

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Question 11

Two independent random samples are taken from two populations. Group 1: $n_{1} = 20$ , $\overset{x}{ˉ}_{1} = 45$ , $s_{1} = 6$ . Group 2: $n_{2} = 20$ , $\overset{x}{ˉ}_{2} = 41$ , $s_{2} = 7$ . What is the test statistic for testing $H_{0} : μ_{1} = μ_{2}$ ?

t \approx 0.615

t \approx 1.96

t \approx 4.0

t \approx 0.44

Question 12

A $t$ -test for the mean produces $t = 2.45$ with $df = 14$ . For $H_{a} : μ > 50$ , the $p$ -value is between 0.01 and 0.025. At $α = 0.05$ , what conclusion should be drawn?

A.Fail to reject

H_{0}

; the

p

-value is between 0.01 and 0.025.B.Reject

H_{0}

; there is convincing evidence that

μ > 50

.C.Reject

H_{0}

; the result is significant at

α = 0.01

.D.The conclusion cannot be determined without the exact

p

-value.

AP Statistics Unit 7: Inference for Quantitative Data: Means — Free with Answer Explanations | Test Practice Hub