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AP Statistics Hard practice

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Hard practice

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Question 1

A researcher observes that students who eat breakfast score higher on standardized tests than those who skip breakfast, using data from a voluntary survey. She concludes that eating breakfast improves test scores. Why is this an invalid causal conclusion?

A.The sample size from a voluntary survey is always too small.B.The data are observational; students were not randomly assigned to eat or skip breakfast, so confounding variables may explain the difference.C.Test score differences between groups cannot be measured accurately.D.Voluntary surveys always overrepresent students who skip breakfast.

Question 2

A large school district conducts a two-sample $z$ -test and finds a statistically significant difference in graduation rates between two schools ( $p$ -value = 0.002). The school with the lower rate has $\overset{p}{^}_{1} = 0.91$ vs. $\overset{p}{^}_{2} = 0.93$ . What is the most important caveat when reporting these results?

A.The test should not have been performed because both rates are above 0.90.B.Statistical significance does not imply practical significance; a 2-percentage-point difference may not be educationally meaningful.C.A

p

-value of 0.002 proves the schools are different in graduation policy.D.With a large enough sample, any small difference will become statistically significant, so no conclusion is possible.

Question 3

A factory produces bolts with a mean diameter of $μ = 10$ mm and $σ = 0.5$ mm. Quality control selects random samples of 16 bolts. In repeated sampling, 95% of sample means will fall within what range?

A.9.00 mm to 11.00 mmB.9.75 mm to 10.25 mmC.9.50 mm to 10.50 mmD.9.875 mm to 10.125 mm

Question 4

Regression output for predicting plant height ( $y$ , cm) from weekly fertilizer dose ( $x$ , grams) is shown below:

Predictor	Coef	SE Coef	T	P
Constant	5.2	1.1	4.73	0.000
Dose	3.8	0.6	6.33	0.000

$r^{2} = 0.77$

A plant received 4 grams of fertilizer per week. What is its predicted height, and what does $r^{2}$ tell us?

A.Predicted height = 20.4 cm; 77% of the variation in height is explained by fertilizer dose.B.Predicted height = 20.4 cm;

r = 0.77

indicates a moderately strong positive association.C.Predicted height = 20.4 cm; 23% of the variation in height is unexplained.D.Predicted height = 17.4 cm; 77% of plants are predicted correctly.

Question 5

A statistician notes that cities with more churches tend to have more crime. Before concluding that religion causes crime, what is the most important alternative explanation?

A.The correlation between churches and crime is not strong enough to draw any conclusion.B.Population size is a confounding variable — larger cities have both more churches and more crime.C.The data should be analyzed with a

χ^{2}

test, not correlation.D.Crime causes people to build more churches for moral guidance.

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Question 6

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 7

A researcher says a score at the 90th percentile in a normal distribution with mean 50 and SD 6 corresponds to a score of about 57.7. Another student says the 90th percentile score is about 42.3. Who is correct, and why?

A.The student, because 42.3 is 1.28 standard deviations below the mean.B.Both are wrong; the 90th percentile requires the exact normal table value.C.The researcher, because

z^{*} \approx 1.28

for the 90th percentile, giving

x = 50 + 1.28 (6) \approx 57.7

.D.Neither; the correct value is 56.0, using

z^{*} = 1.00

Question 8

A sample proportion $\overset{p}{^} = 0.55$ is calculated from a sample of $n = 200$ . The population proportion is claimed to be $p = 0.50$ . What is the approximate $z$ -score for $\overset{p}{^}$ ?

z \approx 1.41

z \approx 2.00

z \approx 0.14

z \approx 3.16

Question 9

In a large population, 1% of people have a certain disease. A test has a sensitivity of 95% (true positive rate) and a specificity of 90% (true negative rate). A randomly selected person tests positive. What is the approximate probability they actually have the disease?

A.About 95%B.About 87%C.About 8.7%D.About 1%

Question 10

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.

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Question 11

A researcher conducts an experiment on pain relief using a matched pairs design. Each of 25 participants tries both Drug A and Drug B on different days, with order randomly assigned. What is the primary advantage of this design over a completely randomized design?

A.It eliminates the placebo effect entirely.B.It reduces variability by controlling for individual differences in pain sensitivity.C.It doubles the effective sample size by using each person twice.D.It allows the researcher to test more than two treatments simultaneously.

Question 12

A set of exam scores has mean 70 and standard deviation 10. Each score is multiplied by 1.05 (a 5% curve), then 3 points are added. What are the new mean and standard deviation?

A.Mean = 73, SD = 10B.Mean = 76.5, SD = 10.5C.Mean = 76.5, SD = 10D.Mean = 73, SD = 10.5

Question 13

A fair coin is flipped 8 times. Using the binomial distribution, what is the probability of getting exactly 5 heads?

\frac{5}{8} = 0.625

(5 8) (0.5)^{5} (0.5)^{3} = \frac{56}{256} \approx 0.219

(0.5)^{8} \approx 0.004

(5 8) (0.5)^{8} \approx 0.219

is wrong; the correct answer is 0.273.

Question 14

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.

Question 15

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.

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