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AP Statistics Practice Exam 1: printable

▶ Section I: multiple choice ✎ Section II: free response Answers and scoring guide

AP Statistics Practice Exam 1

Name: ______________________   Date: __________

Multiple choice: 42 questions (two sets of 3 on one prompt), 90 minutes, graphing calculator. Free response: 4 questions, 10 points each, 90 minutes, graphing calculator. steplyclass.com/ap-statistics/practice-exam-1/

Section I: multiple choice

  1. Five students spent 12, 15, 9, 20 and 14 minutes on a homework task. What is the mean time, in minutes?
    1. A) 70
    2. B) 14
    3. C) 11
    4. D) 17.5
  2. Six bags of rice weigh 18, 25, 11, 30, 22 and 40 kilograms. What is the median weight, in kilograms?
    1. A) 23.5
    2. B) 24.33
    3. C) 29
    4. D) 20.5
  3. A sample of five plants grew 15, 9, 12, 18 and 6 cm. What is the sample standard deviation, to one decimal place?
    1. A) 4.2
    2. B) 22.5
    3. C) 12
    4. D) 4.7
  4. The data are 100, 104, 106, 108, 110, 111, 113, 150 and 60. Using the 1.5 × IQR rule, which values are outliers?
    1. A) none
    2. B) 150 only
    3. C) 60 and 150
    4. D) 60, 100 and 150
  5. The data are 12, 5, 8, 20, 15, 9, 11, 7 and 14. What is the third quartile, Q3?
    1. A) 14
    2. B) 14.5
    3. C) 11
    4. D) 7.5
  6. Scores on a test have mean 100 and standard deviation 15. What is the z-score of a score of 115?
    1. A) 15
    2. B) 7.67
    3. C) −1
    4. D) 1
  7. Times to finish a puzzle have mean 25 minutes and standard deviation 3 minutes. What is the z-score of a time of 18 minutes, to two decimal places?
    1. A) 2.33
    2. B) −7
    3. C) −2.33
    4. D) 6
  8. (Questions 8-10 use this.) X is the number of cars in a household in a town: P(X = 0) = 0.2, P(X = 1) = ?, P(X = 2) = 0.3, P(X = 3) = 0.1. What is P(X = 1)?
    1. A) 0.4
    2. B) 1
    3. C) 0.25
    4. D) 0.6
  9. (Same household data.) P(X = 0) = 0.2, P(X = 1) = 0.4, P(X = 2) = 0.3, P(X = 3) = 0.1. What is the mean of X?
    1. A) 1.5
    2. B) 1
    3. C) 0.25
    4. D) 1.3
  10. (Same household data.) What is the standard deviation of X?
    1. A) 0.81
    2. B) 0.9
    3. C) 1.3
    4. D) 2.5
  11. A team wins each game with probability 0.4, independently. In 6 games, what is the probability that it wins exactly 4?
    1. A) 0.0256
    2. B) 0.4
    3. C) 0.1382
    4. D) 0.6667
  12. X is normal with mean 50 and standard deviation 10. What is P(X < 60)?
    1. A) 0.8413
    2. B) 0.6826
    3. C) 0.1587
    4. D) 0.9772
  13. X is normal with mean 50 and standard deviation 10. What is P(45 < X < 60)?
    1. A) 1.1498
    2. B) 0.8413
    3. C) 0.3085
    4. D) 0.5328
  14. IQ-style scores are normal with mean 100 and standard deviation 15. What score marks the top 10%?
    1. A) 80.8
    2. B) 119.2
    3. C) 115
    4. D) 110
  15. In a club, boys: 12 like cats and 8 like dogs; girls: 10 like cats and 20 like dogs. A member who likes dogs is picked at random. What is the probability it is a girl?
    1. A) 23
    2. B) 25
    3. C) 57
    4. D) 12
  16. One card is drawn from a standard deck of 52. What is the probability that it is a face card or a heart?
    1. A) 1126
    2. B) 2552
    3. C) 352
    4. D) 14
  17. A population has mean 70 and standard deviation 8. For random samples of 16, what is the probability that the sample mean is more than 73?
    1. A) 0.3538
    2. B) 0.9332
    3. C) 0.0668
    4. D) 0.1587
  18. In a random sample of 400 adults, 120 said yes. What is a 95% confidence interval for the proportion who say yes?
    1. A) (0.2771, 0.3229)
    2. B) (0.2623, 0.3377)
    3. C) (0.25, 0.35)
    4. D) (0.2551, 0.3449)
  19. In a random sample of 90 students, 27 bike to school. What is a 90% confidence interval for the proportion who bike?
    1. A) (0.2053, 0.3947)
    2. B) (0.2205, 0.3795)
    3. C) (0.27, 0.33)
    4. D) (0.25, 0.35)
  20. In a random sample of 200 voters, 84 support a plan. What is a 95% confidence interval for the proportion who support it?
    1. A) (0.3516, 0.4884)
    2. B) (0.3626, 0.4774)
    3. C) (0.4176, 0.4224)
    4. D) (0.3851, 0.4549)
  21. A test of H0: p = 0.5 against Ha: p > 0.5 uses a random sample of 500 in which 270 say yes. At α = 0.05, what is the conclusion?
    1. A) Do not reject H0
    2. B) Accept Ha because 270 is more than 250
    3. C) Accept H0
    4. D) Reject H0
  22. A test of H0: p = 0.4 against Ha: p ≠ 0.4 uses a random sample of 200 in which 90 agree. At α = 0.05, what is the conclusion?
    1. A) Reject H0
    2. B) Do not reject H0
    3. C) Accept Ha
    4. D) Accept H0
  23. A test of H0: p = 0.5 against Ha: p < 0.5 uses a random sample of 1000 in which 470 say yes. At α = 0.05, what is the conclusion?
    1. A) Reject H0
    2. B) Do not reject H0
    3. C) Accept H0
    4. D) Accept Ha because the sample is large
  24. In two random samples of 100, 45 and 30 people say yes. A test of H0: p1 = p2 against Ha: p1 > p2 is done at α = 0.05. What is the conclusion?
    1. A) Do not reject H0
    2. B) Accept H0
    3. C) Reject H0
    4. D) Accept Ha because 45 is more than 30
  25. Boys: 20 yes, 30 no; girls: 30 yes, 20 no. A chi-square test of independence is done at α = 0.05. What is the conclusion?
    1. A) Accept Ha because the counts differ
    2. B) Do not reject H0
    3. C) Accept H0
    4. D) Reject H0
  26. Juniors: 30 bus, 20 car, 10 walk; seniors: 20 bus, 35 car, 15 walk. A chi-square test of independence is done at α = 0.05. What is the conclusion?
    1. A) Do not reject H0
    2. B) Reject H0
    3. C) Accept H0
    4. D) Accept Ha because more seniors drive
  27. A random sample of 16 has mean 50 and standard deviation 8. What is a 95% t-interval for the population mean?
    1. A) (46.08, 53.92)
    2. B) (32.952, 67.048)
    3. C) (45.738, 54.262)
    4. D) (48, 52)
  28. The population standard deviation is 8. A random sample of 64 has mean 50. What is a 95% z-interval for the mean?
    1. A) (48.04, 51.96)
    2. B) (48.355, 51.645)
    3. C) (34.32, 65.68)
    4. D) (49, 51)
  29. H0: μ = 30, Ha: μ < 30. A random sample of 16 has mean 28 and standard deviation 4. At α = 0.05, what is the conclusion?
    1. A) Do not reject H0
    2. B) Reject H0
    3. C) Accept H0
    4. D) Accept Ha because 28 is less than 30
  30. H0: μ = 40, Ha: μ ≠ 40. A random sample of 25 has mean 43 and standard deviation 5. At α = 0.05, what is the conclusion?
    1. A) Accept H0
    2. B) Do not reject H0
    3. C) Reject H0
    4. D) Accept Ha because 43 is not 40
  31. Five students took a test before and after a course. Before: 70, 65, 80, 75, 60. After: 74, 66, 85, 76, 64. A paired t-test of whether scores went up is done at α = 0.05. What is the conclusion?
    1. A) Reject H0
    2. B) Do not reject H0
    3. C) Accept H0
    4. D) Accept Ha because every score went up
  32. Group 1: mean 16.2, standard deviation 3, n = 50. Group 2: mean 15, standard deviation 4, n = 40. A test of H0: μ1 = μ2 against Ha: μ1 ≠ μ2 is done at α = 0.05. What is the conclusion?
    1. A) Reject H0
    2. B) Accept Ha because 16.2 is more than 15
    3. C) Accept H0
    4. D) Do not reject H0
  33. A researcher wants a 95% confidence interval for a mean with a margin of error of at most 3. The standard deviation is 12. What is the smallest sample size?
    1. A) 8
    2. B) 62
    3. C) 61
    4. D) 16
  34. A population has mean 70 and standard deviation 8. For random samples of 16, what are the mean and the standard deviation of the sample mean?
    1. A) mean 70, SD 8
    2. B) mean 70, SD 0.5
    3. C) mean 70, SD 2
    4. D) mean 17.5, SD 2
  35. (Questions 35-37 use this.) Five points (x, y): (1, 2), (2, 3), (3, 5), (4, 4), (5, 6). What is the correlation r?
    1. A) 1
    2. B) 0.81
    3. C) −0.9
    4. D) 0.9
  36. (Same five points.) Which is the least-squares regression line?
    1. A) y = 0.9x + 1.3
    2. B) y = x + 1
    3. C) y = 1.3x + 0.9
    4. D) y = 0.9x
  37. (Same five points.) Using the line y = 0.9x + 1.3, what is the residual of the point (4, 4)?
    1. A) 0.9
    2. B) 0
    3. C) 4.9
    4. D) −0.9
  38. A regression line is y = 2x + 1. For the point (3, 8), what is the residual?
    1. A) −1
    2. B) 7
    3. C) 1
    4. D) 8
  39. Four points (x, y): (1, 9), (3, 6), (5, 4), (7, 1). What is the correlation r, to three decimal places?
    1. A) 0.997
    2. B) −0.997
    3. C) −1.3
    4. D) −0.5
  40. Five points (x, y): (1, 6), (2, 3), (3, 1), (4, 1), (5, 0). Which is the least-squares regression line?
    1. A) y = −1.4x + 6.4
    2. B) y = 1.4x + 6.4
    3. C) y = −1.5x + 7.5
    4. D) y = −1.4x + 2.2
  41. Points (0, 2), (1, 4), (2, 8), (3, 16), (4, 32). Which model fits them best?
    1. A) linear
    2. B) quadratic
    3. C) neither
    4. D) exponential
  42. A game costs nothing to play. You win $10 with probability 15 and lose $2 otherwise. What is the expected value of one game, in dollars?
    1. A) 8
    2. B) 2
    3. C) 0.4
    4. D) −1.6

Section II, part A

Question 1: Does a reading app help? (10 points)

A school with 600 students wants to know whether a new reading app makes students read faster. 40 students volunteer. The school will measure each volunteer's reading speed (words per minute) after one month.

(a) Write an investigative question that this study could answer.

(b) Describe how to assign the 40 volunteers at random to two groups of 20: one uses the app, one does not.

(c) Why is it important to assign the students at random?

(d) The 40 students are volunteers, not a random sample of the school. How does this limit the conclusions? Describe one way the volunteers may be different.

Question 2: Hours of sleep (10 points)

A random sample of 8 students at a college reported their hours of sleep last night: 6, 7, 8, 5, 9, 7, 6, 8.

(a) Find the mean of the 8 values.

(b) Find the sample standard deviation of the 8 values, to one decimal place.

(c) Find the median and the quartiles. Describe the shape of the distribution.

(d) Another student slept 10 hours. Find the z-score of 10 hours using your answers to (a) and (b), and say what it means.

Question 3: Biking to work (10 points)

A town says that 30% of its working residents bike to work. In a random sample of 200 working residents, 75 bike to work. Is there convincing evidence, at the 5% level, that more than 30% of the town's working residents bike to work?

(a) State the hypotheses and say what the parameter is.

(b) Check the conditions for a one-sample z-test for a proportion.

(c) Find the test statistic and the p-value.

(d) State your conclusion in context.

Question 4: Checking items at a factory (10 points)

At a factory, each item from a machine needs a second check with probability 0.3, independently of other items. The weights of the items are normal with mean 70 grams and standard deviation 6 grams.

(a) 8 items are picked. Find the probability that exactly 2 of them need a second check.

(b) For 8 items, find the mean and the standard deviation of the number that need a second check.

(c) Find the probability that one item weighs more than 80 grams.

(d) Two items are picked. Find the probability that both weigh more than 80 grams. What do you have to assume?

Answer key

Section I: multiple choice

  1. B
  2. A
  3. D
  4. C
  5. B
  6. D
  7. C
  8. A
  9. D
  10. B
  11. C
  12. A
  13. D
  14. B
  15. C
  16. A
  17. C
  18. D
  19. B
  20. A
  21. D
  22. B
  23. A
  24. C
  25. D
  26. B
  27. C
  28. A
  29. B
  30. C
  31. A
  32. D
  33. B
  34. C
  35. D
  36. A
  37. D
  38. C
  39. B
  40. A
  41. D
  42. C

Section II

Question 1: Does a reading app help? (10 points)

(a) Write an investigative question that this study could answer.

Example: "For volunteer students at this school, is the mean reading speed after one month higher for students who use the app than for students who do not?"

A good question names the group, the variable (reading speed) and the comparison (app or no app).

Scoring guide (2 points):

  • 1 point: names the group and the variable, reading speed
  • 1 point: compares app and no app

(b) Describe how to assign the 40 volunteers at random to two groups of 20: one uses the app, one does not.

Give the volunteers the numbers 1 to 40.

Use a random number generator to pick 20 different numbers from 1 to 40 (skip repeats).

Those 20 students use the app; the other 20 do not. Both groups read the same texts and are measured the same way.

Scoring guide (3 points):

  • 1 point: numbers every volunteer
  • 1 point: a random method that picks 20 different students
  • 1 point: the chosen 20 use the app, the other 20 do not

(c) Why is it important to assign the students at random?

Random assignment spreads other things that affect reading speed (how much each student reads at home, age) evenly between the groups.

Then a clear difference in reading speed can be said to be caused by the app.

Scoring guide (2 points):

  • 1 point: other variables are balanced between the groups
  • 1 point: so a difference can be caused by the app

(d) The 40 students are volunteers, not a random sample of the school. How does this limit the conclusions? Describe one way the volunteers may be different.

The results apply only to students like these volunteers, not to all 600 students.

Volunteers may already like reading more, so they may gain more (or less) from the app than other students.

To say something about all 600 students, the 40 should be a random sample of the school.

Scoring guide (3 points):

  • 1 point: the conclusion is only for students like the volunteers
  • 1 point: a real way volunteers may differ, in context
  • 1 point: a random sample is needed to speak for the whole school

Question 2: Hours of sleep (10 points)

(a) Find the mean of the 8 values.

mean = 7

Scoring guide (2 points):

  • 1 point: working: one mark for each right line (Check my steps)
  • 1 point: the right answer, finished

(b) Find the sample standard deviation of the 8 values, to one decimal place.

SD ≈ 1.3

Scoring guide (3 points):

  • 2 points: working: one mark for each right line (Check my steps)
  • 1 point: the right answer, finished

(c) Find the median and the quartiles. Describe the shape of the distribution.

In order: 5, 6, 6, 7, 7, 8, 8, 9. Median = 7; Q1 = 6 (the median of 5, 6, 6, 7); Q3 = 8 (the median of 7, 8, 8, 9).

The data are about symmetric: the mean and the median are both 7.

Scoring guide (2 points):

  • 1 point: median 7, Q1 6, Q3 8
  • 1 point: about symmetric, with a reason

(d) Another student slept 10 hours. Find the z-score of 10 hours using your answers to (a) and (b), and say what it means.

z = (10 − 7)/1.3 ≈ 2.3.

10 hours is about 2.3 standard deviations above the mean sleep of the sample.

That is unusually long sleep for these students.

Scoring guide (3 points):

  • 1 point: z = (10 − mean)/SD
  • 1 point: z ≈ 2.3
  • 1 point: the meaning in context (2.3 SDs above the mean, unusual)

Question 3: Biking to work (10 points)

(a) State the hypotheses and say what the parameter is.

H0: p = 0.3 and Ha: p > 0.3.

p = the proportion of all working residents of the town who bike to work.

Scoring guide (2 points):

  • 1 point: H0: p = 0.3, Ha: p > 0.3
  • 1 point: p defined in context (all working residents of the town)

(b) Check the conditions for a one-sample z-test for a proportion.

Random: the sample is random. 10%: 200 is less than 10% of the town's working residents.

Large counts: 200 × 0.3 = 60 and 200 × 0.7 = 140, both at least 10.

Scoring guide (2 points):

  • 1 point: random and 10% conditions
  • 1 point: large counts 60 and 140

(c) Find the test statistic and the p-value.

p-value 0.0104 < α = 0.05 → reject H0

Scoring guide (4 points):

  • 1 point: pÌ‚ = 75200 = 0.375
  • 1 point: z = (0.375 − 0.3)/√(0.3 × 0.7200), with p0 = 0.3 in the SE
  • 1 point: z ≈ 2.31
  • 1 point: p-value ≈ 0.0104

(d) State your conclusion in context.

The p-value ≈ 0.0104 is less than α = 0.05, so we reject H0.

There is convincing evidence that more than 30% of the town's working residents bike to work.

Scoring guide (2 points):

  • 1 point: compares the p-value with 0.05 and rejects H0
  • 1 point: the conclusion about all working residents, in context

Question 4: Checking items at a factory (10 points)

(a) 8 items are picked. Find the probability that exactly 2 of them need a second check.

P(2 successes) ≈ 0.2965

Scoring guide (3 points):

  • 1 point: binomial with n = 8 and p = 0.3
  • 1 point: C(8, 2) × 0.32 × 0.76
  • 1 point: ≈ 0.2965

(b) For 8 items, find the mean and the standard deviation of the number that need a second check.

Mean = np = 8 × 0.3 = 2.4 items.

SD = √(np(1 − p)) = √(8 × 0.3 × 0.7) = √1.68 ≈ 1.296 items.

Scoring guide (2 points):

  • 1 point: mean 2.4
  • 1 point: SD ≈ 1.296

(c) Find the probability that one item weighs more than 80 grams.

P(X > 80) = 1 − 0.9525 = 0.0475

Scoring guide (3 points):

  • 1 point: z = (80 − 70)/6 ≈ 1.67
  • 1 point: the area to the right of z
  • 1 point: ≈ 0.0475

(d) Two items are picked. Find the probability that both weigh more than 80 grams. What do you have to assume?

P(both) = 0.0475 × 0.0475 ≈ 0.0023.

This assumes the two weights are independent of each other.

Scoring guide (2 points):

  • 1 point: 0.04752 ≈ 0.0023
  • 1 point: assumes independence

Every answer solved step by step: steplyclass.com/ap-statistics/practice-exam-1/answers/

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