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AP Statistics Full-length practice exam 3

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Full-length practice exam 3

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Question 1

A call center randomly selects 50 of its 500 daily calls to monitor for quality. This is an example of what type of sample, and what is its main advantage over a census?

A.Convenience sample; easier to obtainB.Simple random sample; it reduces cost and time while providing unbiased estimatesC.Cluster sample; entire groups of calls are selectedD.Systematic sample; it eliminates variability

Question 2

Computer output for a regression of weight ( $y$ , in lbs) on height ( $x$ , in inches) gives: $\overset{y}{^} = - 130 + 4.5 x$ , $r^{2} = 0.71$ . Interpret the slope.

A.For each additional pound, height is predicted to increase by 4.5 inches.B.For each additional inch of height, weight is predicted to increase by 4.5 lbs.C.The average weight is 4.5 lbs.D.71% of the increase in weight is caused by height.

Question 3

A chi-square goodness-of-fit test is performed on a spinner with 3 sections claimed to be equally likely. The test statistic is $χ^{2} = 5.4$ with $df = 2$ . The $p$ -value is approximately 0.067. At $α = 0.05$ , is there evidence the spinner is unfair?

A.Yes;

χ^{2} = 5.4

is large, so the spinner is definitely unfair.B.No;

p

-value = 0.067 >

α = 0.05

, so we fail to reject the null hypothesis that the spinner is fair.C.Yes; any

χ^{2} > 5

with small

df

indicates a significant result.D.No; the

p

-value should be doubled for a two-sided test.

Question 4

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 5

A large population has proportion $p = 0.60$ with a certain characteristic. A random sample of $n = 100$ is taken. What is the mean of the sampling distribution of $\overset{p}{^}$ ?

A.0.060B.60C.0.60D.0.049

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Question 6

The regression equation for predicting a city's high temperature in July ( $y$ , °F) from its latitude ( $x$ , degrees north) is $\overset{y}{^} = 115 - 1.1 x$ . Interpret the $y$ -intercept in context.

A.At 0° latitude (equator), the predicted July high is 115°F.B.For each degree increase in latitude, temperature decreases by 115°F.C.The average July high temperature across all cities is 115°F.D.The

y

-intercept has no meaningful interpretation because latitude cannot equal zero.

Question 7

Which statement best describes what the Central Limit Theorem (CLT) tells us about the sampling distribution of $\overset{x}{ˉ}$ ?

A.The population must be normally distributed for

\overset{x}{ˉ}

to be approximately normal.B.For large enough sample sizes, the sampling distribution of

\overset{x}{ˉ}

is approximately normal regardless of the population shape.C.For large samples, the sample mean equals the population mean.D.As sample size increases,

\overset{x}{ˉ}

converges to

σ

Question 8

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 9

A survey asks: "Do you agree that the city council wastes taxpayer money on unnecessary projects?" What problem does this question have?

A.The question does not use a random sampling method.B.The question is too short to collect meaningful data.C.The question is leading and will likely produce response bias toward agreement.D.The question is subject to nonresponse bias.

Question 10

Researchers conduct a two-sample $z$ -test for proportions comparing recovery rates between two treatments. They obtain $p$ -value = 0.08 with $α = 0.05$ . A skeptic says the treatment difference is statistically significant. Is the skeptic correct?

A.Yes;

p

-value = 0.08 is close to

α = 0.05

, which is nearly significant.B.No; since

p

-value = 0.08 >

α = 0.05

, we fail to reject

H_{0}

and cannot conclude a significant difference.C.Yes; any

p

-value below 0.10 indicates significance.D.No; a two-sided test should be used, which would double the

p

-value to 0.16.

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Question 11

A population is normal with $μ = 100$ and $σ = 15$ . Random samples of size $n = 9$ are repeatedly drawn. What is the shape of the sampling distribution of $\overset{x}{ˉ}$ ?

A.Approximately normal, by the CLT since

n = 9 \geq 5

.B.Exactly normal, because the population is normal.C.Unknown shape, since

n < 30

.D.Approximately normal with mean 100/9.

Question 12

A company claims that 80% of its customers are satisfied. A consumer group surveys 100 customers and finds 72 are satisfied. Which hypothesis should be used to test whether the satisfaction rate is lower than claimed?

H_{0} : p = 0.72

H_{a} : p < 0.72

H_{0} : p = 0.80

H_{a} : p \neq = 0.80

H_{0} : p = 0.80

H_{a} : p < 0.80

H_{0} : p < 0.80

H_{a} : p = 0.80

Question 13

A researcher wants to estimate a population proportion to within ±0.04 with 95% confidence, but has no prior estimate of $p$ . What minimum sample size is needed?

A.385B.601C.25D.2401

Question 14

A residual plot for a linear regression model shows a clear curved (U-shaped) pattern. What does this indicate?

A.The regression line fits the data well.B.There is likely an outlier in the data.C.A linear model is not appropriate; the relationship is non-linear.D.The standard deviation of the residuals is too large.

Question 15

Researchers studying the effect of sleep on memory randomly assign 30 college students to either 6 hours or 8 hours of sleep, then test their recall the next morning. What is the explanatory variable?

A.Memory recall scoreB.Hours of sleepC.The college studentsD.Whether the study was blinded

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Question 16

A chi-square test of independence produces $χ^{2} = 9.2$ with $df = 3$ . The critical value at $α = 0.05$ is 7.815. What is the correct conclusion?

A.Fail to reject

H_{0}

; the variables are independent.B.Reject

H_{0}

; there is convincing evidence of an association between the two variables.C.Reject

H_{0}

; we have proven the variables are dependent.D.The test is inconclusive because

χ^{2} \approx 9

is not far above the critical value.

Question 17

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 18

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 19

A school wants to estimate the proportion of students who bike to school. They survey every 10th student on the enrollment list. What type of sampling method is this?

A.Simple random sample (SRS)B.Systematic random sampleC.Stratified random sampleD.Cluster sample

Question 20

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 21

A student claims that because $P (A) = 0.5$ and $P (B) = 0.4$ , events $A$ and $B$ cannot both occur since $P (A) + P (B) = 0.9 < 1$ . Is this reasoning valid?

A.Yes; if

P (A) + P (B) < 1

, they must be mutually exclusive.B.No;

P (A) + P (B) < 1

does not prevent both events from occurring simultaneously.C.Yes; the sum of individual probabilities must equal 1 for events to overlap.D.No; events can co-occur only if

P (A) + P (B) > 1

Question 22

A population of SAT scores is strongly right-skewed. Samples of size $n = 5$ are taken repeatedly. Which of the following best describes the sampling distribution of $\overset{x}{ˉ}$ for $n = 5$ ?

A.Approximately normal, because the CLT applies for any sample size.B.Exactly normal, because all sampling distributions of

\overset{x}{ˉ}

are normal.C.Still right-skewed, but less skewed than the population.D.Identical in shape to the population distribution.

Question 23

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 24

A chi-square test for independence is conducted on a $3 \times 4$ contingency table. What are the degrees of freedom?

A.12B.6C.11D.2

Question 25

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

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Question 26

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 27

A bag contains 4 red marbles and 6 blue marbles. One marble is drawn at random. What is the probability of drawing a red marble?

A.0.40B.0.60C.0.44D.0.67

Question 28

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 29

An experiment randomly assigns participants to one of three diets. After 8 weeks, mean weight loss is compared across groups. Which principle of experimental design ensures that unmeasured variables (like initial metabolism) are evenly distributed among the groups?

A.ReplicationB.ControlC.Random assignmentD.Blinding

Question 30

A researcher surveys 150 randomly selected students about their preferred study method (solo, group, or tutoring) and tests whether preference is related to major (STEM vs. non-STEM). Which chi-square test is appropriate?

A.Chi-square goodness-of-fit testB.Chi-square test of independenceC.Chi-square test of homogeneityD.Two-sample

t

-test

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Question 31

An experiment randomly assigns 40 patients to receive a new drug or a placebo. Neither the patients nor the doctors evaluating outcomes know which treatment each patient received. What design feature does this describe?

A.ReplicationB.BlockingC.Double-blind designD.Matched pairs design

Question 32

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 33

The regression of $ln (y)$ on $x$ produces a better fit than the regression of $y$ on $x$ . What does this suggest about the relationship between $x$ and $y$ ?

A.The relationship between

x

and

y

is exponential.B.The relationship between

x

and

y

is linear.C.The relationship between

x

and

y

is a power function.D.

x

must be transformed, not

y

Question 34

For a regression of final exam score on midterm score, the regression equation is $\overset{y}{^} = 10 + 0.80 x$ , $r = 0.90$ . A student scored 70 on the midterm. What is the student's predicted final exam score?

A.63B.66C.80D.70

Question 35

For the test $H_{0} : p = 0.80$ vs. $H_{a} : p < 0.80$ using the data from the previous question ( $n = 100$ , $\overset{p}{^} = 0.72$ ), the $z$ -test statistic is approximately $- 2.0$ and the $p$ -value is 0.023. At $α = 0.05$ , what is the correct conclusion?

A.Fail to reject

H_{0}

; there is insufficient evidence that fewer than 80% of customers are satisfied.B.Reject

H_{0}

; there is convincing evidence that fewer than 80% of customers are satisfied.C.Reject

H_{0}

; we have proved the true satisfaction rate is exactly 0.72.D.Accept

H_{0}

; the satisfaction rate is 80%.

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Question 36

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 37

A student scored 85 on a test where the class mean was 78 and the standard deviation was 7. What is the student's standardized score ( $z$ -score)?

z = 0.88

z = 1.00

z = - 1.00

z = 7.00

Question 38

Events $A$ and $B$ are mutually exclusive with $P (A) = 0.30$ and $P (B) = 0.45$ . What is $P (A or B)$ ?

A.0.135B.0.75C.0.775D.1.00

Question 39

A dot plot shows the ages (in years) of 9 trees: 3, 5, 5, 7, 8, 9, 10, 12, 40. Which measure of center best represents the typical age, and why?

A.Mean, because it uses all values in the data set.B.Median, because it is resistant to the extreme value of 40.C.Mean, because it equals the median in symmetric distributions.D.Mode, because 5 appears twice.

Question 40

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

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