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AP Statistics Unit 9: Inference for Quantitative Data: Slopes

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Unit 9: Inference for Quantitative Data: Slopes

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Question 1

A regression output for predicting salary from years of experience shows: slope $b = 3.2$ , $S E_{b} = 0.8$ . What is the $t$ statistic for testing $H_{0} : β = 0$ ?

t = 0.25

t = 4.0

t = 2.56

t = 3.2

Question 2

A researcher tests $H_{0} : β = 0$ vs. $H_{a} : β > 0$ with $t = 1.95$ , $df = 28$ . Using a $t$ -table, the one-sided $p$ -value is between 0.025 and 0.05. At $α = 0.05$ , what is the conclusion?

A.Fail to reject

H_{0}

; both bounds (0.025 and 0.05) need to be below

α

to reject.B.Reject

H_{0}

; the

p

-value is less than

α = 0.05

.C.Inconclusive; the range straddles

α = 0.05

.D.Fail to reject

H_{0}

; the two-sided

p

-value would be between 0.05 and 0.10, which exceeds

α

Question 3

Regression output for predicting fuel efficiency ( $y$ , mpg) from vehicle weight ( $x$ , pounds) gives $b = - 0.006$ , $S E_{b} = 0.001$ , $n = 30$ . A 95% confidence interval for the true slope is reported as $(- 0.008, - 0.004)$ . What does this interval mean?

A.We are 95% confident that each additional pound reduces fuel efficiency by between 0.004 and 0.008 mpg.B.There is a 95% probability the slope is negative.C.95% of vehicles lose between 0.004 and 0.008 mpg per pound.D.We are 95% confident the regression line has a negative

y

-intercept.

Question 4

A 95% confidence interval for the slope $β$ in a regression of test score on study hours is $(0.3, 1.9)$ . A student claims that because 0 is not in the interval, the null hypothesis $H_{0} : β = 0$ would be rejected at $α = 0.05$ . Is this correct?

A.No; the CI and hypothesis test are separate procedures and cannot be compared.B.Yes; a value outside a 95% CI corresponds to rejection at

α = 0.05

for a two-sided test.C.No; the CI is for

b

(sample slope), not

β

(population slope).D.No; rejecting

H_{0} : β = 0

requires

t > 2

regardless of the CI.

Question 5

Regression output for predicting blood pressure ( $y$ ) from sodium intake ( $x$ ) gives $b = 1.8$ , $S E_{b} = 0.45$ , $n = 22$ , $t^{*} = 2.086$ for a 95% CI. Compute the 95% confidence interval for the slope.

(0.918, 2.682)

(0.900, 2.700)

(1.350, 2.250)

(0.180, 3.420)

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Question 6

Computer output for a regression of plant height on water amount shows $p$ -value = 0.03 for the test $H_{0} : β = 0$ vs. $H_{a} : β \neq = 0$ . At $α = 0.05$ , what is the correct interpretation?

A.Fail to reject

H_{0}

; there is no linear relationship between water and height.B.Reject

H_{0}

; there is convincing evidence of a linear relationship between water amount and plant height.C.Reject

H_{0}

; water amount causes plant height to increase.D.Reject

H_{0}

; the regression model explains 97% of variation in height.

Question 7

A residual plot for a regression of price on square footage shows a fan-shaped pattern (residuals get larger as $x$ increases). What condition is violated, and what does this mean for inference?

A.Linearity is violated; the relationship is not linear.B.Equal variance is violated; inference on the slope may be unreliable.C.Normality is violated; the residuals are skewed.D.Independence is violated; the data have a time-trend.

Question 8

A study fits a regression of reading score on age for 30 children. The ANOVA table for regression shows $F = 16.4$ , $p < 0.001$ . A researcher notes that the $t$ statistic for the slope also gives $p < 0.001$ . What is the relationship between these two tests?

A.They test different things;

F

tests model fit and

t

tests the slope.B.In simple linear regression,

F = t^{2}

; both tests are equivalent tests of

H_{0} : β = 0

.C.The

F

test requires more assumptions than the

t

test.D.The

F

test is always more powerful than the

t

test.

Question 9

Which conditions must be satisfied to perform inference on the slope of a regression line? Select all that apply:

I. Linearity (the relationship between $x$ and $y$ is linear)
II. Independence (observations are independent)
III. Normality (residuals are approximately normal)
IV. Equal variance (spread of residuals is roughly constant for all $x$ )

A.I and III onlyB.I, II, and IV onlyC.I, II, III, and IVD.None; regression inference has no conditions beyond having data.

Question 10

Regression of hours of TV watched per day on sleep hours gives the following computer output:

	Coef	SE Coef	T	P
Constant	9.1	0.8	11.4	0.000
TV hours	−0.4	0.15	−2.67	0.014

$n = 25$ , $r^{2} = 0.24$

Which of the following best describes the result of the test of the slope?

A.The slope is not significant because

r^{2} = 0.24

is low.B.At

α = 0.05

, reject

H_{0} : β = 0

; there is evidence of a negative linear relationship between TV hours and sleep.C.The slope is significant because

t = - 2.67

is greater than 2.D.Each hour of TV causes sleep to decrease by 0.4 hours.

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Question 11

A regression of test score on hours of study is based on $n = 25$ observations. What are the degrees of freedom for the $t$ -test of the slope?

A.25B.24C.23D.1

Question 12

A researcher fits a regression of crop yield ( $y$ ) on rainfall ( $x$ ) and gets $b = 0.85$ , $S E_{b} = 0.52$ , $n = 18$ . The two-sided $p$ -value is 0.12. A colleague suggests increasing $n$ to 100 would likely make the slope statistically significant. Why might this be true?

A.A larger

n

changes the slope

b

, making it more likely to be significant.B.A larger

n

reduces

S E_{b}

, increasing the

t

statistic and making it easier to detect a non-zero slope.C.A larger

n

reduces the critical

t^{*}

value, making rejection easier.D.A larger

n

increases

r^{2}

, which directly reduces the

p

-value.

AP Statistics Unit 9: Inference for Quantitative Data: Slopes — Free with Answer Explanations | Test Practice Hub