AP Statistics Unit 9 Practice Test: Inference for Quantitative Data: Slopes

The correct answer is (A). The appropriate confidence interval for the slope of a regression model is a t-interval for the slope.

The correct answer is (D). The best critical value you can get from the t-table provided on the AP Statistics exam is:

$1.962 \,$ for $\, df = n − 2 = 1002 - 2 = 1000$

1.960 is very close , but 1.962 is better in this case. The confidence interval will be:

$3 \pm 1.962*2.37 $ $ = (-1.650, 7.650)$

This includes zero. Which means that our 95% confidence interval contains zero, which means the $p$-value would not be below .05, so we wouldn’t reject the null hypothesis that the slope is zero.

Answer choice (B) is incorrect because we do need a linear relationship (or we at least need it not to look like a nonlinear relationship) between $x$ and $y$.

Answer choice (C) is incorrect — it is there to confuse you.

The correct answer is (C). The margin of error for the slope is $t∗SE$, the standard error, which can be calculated by:

$ \dfrac{s}{s_x\sqrt{n-1}} = \dfrac{0.3}{0.25\sqrt{29}} ≈ 0.223$

$2.467*0.223≈0.550$

The correct answer is (A). The interval estimate is:

$b \pm t*(SE_b)$

In this case, $b=0.386, SE_b=0.082$, and $\, t* = 2.074 \,$ $ (p=0.025 \,$ with $\, df \; n − 2= 24 − 2 = 22)$.

The correct set-up for the calculation is:

$0.386 ± 2.074(0.082)$

The correct answer is (B). All other factors remaining the same, the width of the confidence interval will decrease as the sample size increases. This is because the standard error has $n$ in the denominator, whether it’s a proportion, mean, difference in means, etc.

The correct answer is (D). Appropriate alternative hypotheses can use <, >, or ≠ but cannot use ≤ or ≥ as this would overlap with the null hypothesis.

The correct answer is (A). The true relationship between $x$ and $y$ needs to be LINEAR (or at least look linear). All of the others are conditions for using linear regression to get the slope, and for performing the significance test of the slope.

The correct answer is (C). The distribution of the slope of a regression model, assuming all conditions are satisfied and the null hypothesis is true, is a t-distribution. That is why we use the t-table to get the confidence intervals for slopes.

The correct answer is (C). If the $p$-value ≤ $\alpha$, then reject the null hypothesis; if the $p$-value > $\alpha$, accept the null hypothesis. $\alpha$ is the significance level, the threshold for rejecting the null hypothesis.

The correct answer is (D). The null hypothesis says there is no relationship, that the slope of the regression line = 0. Because the confidence interval does not contain zero, the student can reject the null hypothesis and conclude that there is a linear relationship between library use and ACT scores.