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AP Statistics Full-length practice exam 1

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Full-length practice exam 1

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Question 1

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 2

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.

Question 3

A researcher increases the confidence level from 90% to 99% while keeping the sample size constant. What happens to the confidence interval?

A.The interval becomes narrower because higher confidence means more precision.B.The interval becomes wider because a higher

z^{*}

is used.C.The interval stays the same because the sample data do not change.D.The interval becomes wider and the point estimate changes.

Question 4

A chi-square test of homogeneity compares the distribution of preferred music genre across three age groups (teenagers, adults, seniors). The null hypothesis is:

H_{0}

: Music genre and age group are independent.B.

H_{0}

: The distribution of music genre preference is the same for all three age groups.C.

H_{0}

: All genres are equally preferred in each age group.D.

H_{0}

: There is no difference in mean age between genre groups.

Question 5

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.

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Question 6

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.

Question 7

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 8

A 3×2 contingency table (3 education levels × 2 voting outcomes) shows the following row totals and column totals:

	Voted	Did not vote	Row total
High school			80
Some college			60
College grad			60
Col total	120	80	200

The expected count for the High school/Voted cell is $E_{11}$ . Compute $χ^{2}$ for this cell if the observed count is 40.

E_{11} = 48

; cell contribution

= 1.33

E_{11} = 40

; cell contribution

= 0

E_{11} = 60

; cell contribution

= 6.67

E_{11} = 48

; cell contribution

= 0.133

Question 9

A geometric random variable $X$ counts the number of trials until the first success, where $p = 0.25$ . What is $P (X = 3)$ ?

0.2 5^{3} \approx 0.016

(0.75)^{2} (0.25) \approx 0.141

(0.75)^{3} \approx 0.422

(1 3) (0.25)^{1} (0.75)^{2} \approx 0.422

Question 10

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

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Question 11

Two sampling distributions are described: (I) $n = 25$ samples from a symmetric, bell-shaped population; (II) $n = 100$ samples from a heavily skewed population. Which is more likely to be approximately normal?

A.Distribution I, because the population is symmetric.B.Distribution II, because

n = 100

is large enough by the CLT even for skewed populations.C.Both are equally normal, because the CLT applies in both cases.D.Neither is approximately normal, because neither comes from a perfectly normal population.

Question 12

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 13

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 14

Two independent random variables $X$ and $Y$ have means $μ_{X} = 10$ , $μ_{Y} = 6$ , and standard deviations $σ_{X} = 3$ , $σ_{Y} = 4$ . What are the mean and standard deviation of $X + Y$ ?

μ = 16

σ = 7

μ = 16

σ = 5

μ = 4

σ = 5

μ = 16

σ = 25

Question 15

A die is rolled 60 times and the observed frequencies for each face are recorded. A chi-square goodness-of-fit test is used to check if the die is fair. What are the degrees of freedom?

A.6B.5C.59D.1

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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 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 18

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 19

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 20

A scatterplot of hours studied ( $x$ ) versus exam score ( $y$ ) for 20 students shows a positive linear association with no obvious outliers. The correlation coefficient is $r = 0.83$ . What does this value tell us?

A.83% of the variation in exam scores is explained by hours studied.B.There is a strong positive linear association between hours studied and exam score.C.Studying more hours causes higher exam scores.D.Each additional hour of studying increases the score by 0.83 points.

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Question 21

A random sample of 150 teenagers found 90 who own a smartphone. A random sample of 120 adults found 60 who own a smartphone. Researchers test $H_{0} : p_{teen} = p_{adult}$ vs. $H_{a} : p_{teen} \neq = p_{adult}$ . What is the pooled proportion $\overset{p}{^}_{c}$ ?

\overset{p}{^}_{c} = 0.575

\overset{p}{^}_{c} = \frac{90 + 60}{150 + 120} = \frac{150}{270} \approx 0.556

\overset{p}{^}_{c} = \frac{90}{150} = 0.60

\overset{p}{^}_{c} = \frac{0.60 + 0.50}{2} = 0.55

Question 22

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 23

Which condition must be met for the sampling distribution of $\overset{p}{^}$ to be approximately normal?

n \geq 30

n p \geq 10

and

n (1 - p) \geq 10

p \geq 0.5

D.The population must be normally distributed.

Question 24

A candy machine has a 30% chance of dispensing a rare flavor. You try the machine 10 times. Using the binomial model, what is the probability of getting the rare flavor at least once?

1 - (0.30)^{10}

10 \times 0.30 = 0.30

1 - (0.70)^{10} \approx 0.972

(0.30)^{1} (0.70)^{9} \approx 0.121

Question 25

The expected cell count for a cell in a $3 \times 3$ contingency table with total $n = 200$ is computed as follows: the row total is 80 and the column total is 60. What is the expected count?

A.24B.140C.40D.0.024

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Question 26

A discrete random variable $X$ has the probability distribution shown. What is $E (X)$ ?

$x$	1	2	3	4
$P (X = x)$	0.10	0.30	0.40	0.20

A.2.50B.2.70C.3.00D.1.00

Question 27

A stemplot of hourly wages (in dollars) for 10 part-time workers is shown below. What is the median wage?

Stem	Leaves
8	2 5
9	1 3 7
10	0 4 4 8
11	2 6

A.$9.70B.$10.00C.$10.02D.$9.85

Question 28

A researcher wants to estimate the proportion of adults who exercise regularly. She samples 400 adults and finds 160 exercise regularly. What is the 95% confidence interval for the true proportion?

(0.35, 0.45)

(0.352, 0.448)

(0.36, 0.44)

(0.40, 0.40)

Question 29

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 30

A least-squares regression line is fit to data on advertising spending ( $x$ , thousands of dollars) and sales ( $y$ , thousands of units). The residual for one observation is $+ 12$ . What does this mean?

A.The actual sales were 12 thousand units less than predicted.B.The actual sales were 12 thousand units more than predicted.C.Advertising spending was overestimated by 12 thousand dollars.D.The regression line passes 12 units above the data point.

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Question 31

A population proportion is $p = 0.40$ . A random sample of $n = 100$ is drawn. What is the standard deviation of the sampling distribution of $\overset{p}{^}$ ?

A.0.40B.0.0024C.0.049D.0.490

Question 32

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 33

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 34

In a chi-square test of independence, which of the following conditions must be met to proceed?

I. The data come from a random sample.
II. Expected cell counts are at least 5.
III. All observed counts are at least 5.

A.I onlyB.I and III onlyC.I and II onlyD.I, II, and III

Question 35

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 36

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 37

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

Question 38

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)

Question 39

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 40

A 90% confidence interval for the proportion of defective items is $(0.04, 0.12)$ . A quality engineer claims the defect rate is 15%. Is the claim consistent with the interval at the 10% significance level?

A.Yes; since 15% is close to 12%, it is consistent.B.No; since 15% falls outside the 90% CI, a two-sided test at

α = 0.10

would reject

H_{0} : p = 0.15

.C.Cannot determine without computing the

p

-value.D.Yes; a 90% CI at

α = 0.10

can only reject proportions below 0.04.

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