AP Statistics



Chapter 6 Practice Test Name____________________________

Part I: Multiple Choice (Questions 1-10) - Circle the answer of your choice.

1. The weights of cockroaches living in a typical college dormitory are approximately normally distributed with a mean of 80 grams and a standard deviation of 4 grams. The percentage of cockroaches weighing between 77 grams and 83 grams is about:

A) 99.7%

B) 95%

C) 68%

D) 55%

E) 34%

D;

[pic]

2. Scores on the American College Test (ACT) are normally distributed with a mean of 18 and a standard deviation of 6. The interquartile range of the scores is approximately:

A) 6

B) 7

C) 8.1

D) 10.3

E) 12

C;

[pic] [pic] [pic]

3. Some descriptive statistics for a set of test scores are shown in the Minitab printout below. For this test, a certain student has a standardized score of [pic]. What score did this student receive on the test?

|Variable N Mean Median TrMean StDev |

|Score 50 1045.7 1024.7 1041.9 221.9 |

| |

|Variable SE Mean Minimum Maximum Q1 Q3 |

|Score 31.4 628.9 1577.1 877.7 1219.5 |

A) 266.28

B) 779.42

C) 1008.02

D) 1083.38

E) 1311.98

B; [pic]

4. The test grades at a large school have an approximately normal distribution with a mean of 50. What is the standard deviation of the data so that 80% of the students are within 12 points (above or below) the mean?

A) 5.875

B) 9.375

C) 10.375

D) 14.5

E) Cannot be determined from the given information

B; [pic]

[pic]

5. The five-number summary of the distribution of scores on a statistics exam is

0 26 31 36 50

Suppose 316 students took the exam. The histogram of all 316 test scores was approximately normal. Thus the variance of test scores must be about

A) 5

B) 8

C) 19

D) 64

E) 55

E; recall that the standard deviation is the square root of the variance.

[pic]

6. You have a set of data that you suspect came from a normal distribution. In order to assess normality, you construct a normal probability plot. Which of the following would constitute evidence that the data actually came from a normal distribution?

A) A strongly linear relationship between the data and their standardized values.

B) A bell-shaped (normal) relationship between the data and their standardized values.

C) A random scattering of points when the standardized values are plotted against the data.

D) A strongly non-linear relationship (with no outliers) between the data and their percentiles.

E) A uniform relationship between the percentiles and the standardized values.

A;

7. Suppose that a Normal model describes fuel economy (miles per gallon) for automobiles

and that a Saturn has a standardized score (z-score) of +2.2. This means that Saturns . . .

A) get 2.2 miles per gallon.

B) get 2.2 times the gas mileage of the average car.

C) get 2.2 mpg more than the average car.

D) have a standard deviation of 2.2 mpg.

E) achieve fuel economy that is 2.2 standard deviations better than the average car.

E;

8. Jay Olshansky from the University of Chicago was quoted in Chance News as arguing that for the average life expectancy to reach 100, 18% of people would have to live to age 120. Assuming life expectancy is normally distributed, what standard deviation is he assuming for this statement to make sense?

A) 21.7

B) 24.4

C) 25.2

D) 35.0

E) 111.1

A;

[pic]

9. Which of the following are true statements?

I. In all normal distributions, the mean and median are equal.

II. All bell-shaped curves are normal distributions for some choice of [pic] and [pic].

III. Virtually all the area under a normal curve is within three standard deviations of the mean, no matter what the particular mean and standard deviation are.

A) I and II

B) I and III

C) II and III

D) I, II, and III

E) None of the above gives the complete set of true responses.

B; I and III are true; II is false, a bell-shape is not necessarily normal. You should construct a normal probability plot of the data and/or check the 68-95-99.7% rule.

10. A set of 5000 scores on a college readiness exam are known to be approximately normally distributed with mean 72 and standard deviation 6. To the nearest integer value, how many scores are there between 63 and 75?

A) 0.6247

B) 3123

C) 3227

D) 3650

E) 4115

B;

[pic]

Part II: Free Response (Questions 11-13) – Show your work and explain your results clearly.

11. The Graduate Record Examinations are widely used to help predict the performance of applicants to graduate schools. The range of possible scores on a GRE is 200 to 900. The psychology department at a university finds that the scores of its applicants on the quantitative GRE are approximately normal with mean of 544 and standard deviation of 103.

a) Determine the proportion of scores less than 500.

Using [pic],

[pic]

[pic]

b) Determine the proportion of scores between 500 and 700.

Using [pic], [pic].

[pic]

c) What minimum score would a student need in order to score better than 75% of those taking the test?

Using [pic], the 75th percentile is 613.5.

[pic]

[pic]

d) Determine the interquartile range of the scores.

[pic] [pic]

e) Determine the score that is the boundary for the upper 10% of scores.

[pic] [pic]

[pic]

12. A professional sports team evaluates potential players for a certain position based on two main characteristics, speed and strength.

a) Speed is measured by the time required to run a distance of 40 yards, with smaller times indicating

more desirable (faster) speeds. From previous speed data for all players in this position, the times to run 40 yards have a mean of 4.60 seconds and a standard deviation of 0.15 seconds, with a minimum time of 4.40 seconds, as shown in the table below.

[pic]

Based on the relationship between the mean, standard deviation, and minimum time, is it reasonable to believe that the distribution of 40-yard running times is approximately normal? Explain.

No, it is not reasonable to believe that the distribution of 40-yard running times is approximately normal, because the minimum time is only 1.33 standard deviations below the mean [pic]. In a normal distribution, approximately 9.2% of the z-scores are below -1.33. However, there are no running times less than 4.4 seconds, which indicates that there are no running times with a z-score less than -1.33. Therefore, the distribution of 40-yard running times is not approximately normal.

b) Strength is measured by the amount of weight lifted, with more weight indicating more desirable (greater) strength. From previous strength data for all players in this position, the amount of weight lifted has a mean of 310 pounds and a standard deviation of 25 pounds, as shown in the table below.

[pic]

Calculate and interpret the z-score for a player in this position who can lift a weight of 370 pounds.

The z-score for a player who can lift a weight of 370 pounds is [pic]. The z-score indicates that the amount of weight the player can lift is 2.4 standard deviations above the mean for all previous players in this position.

c) The characteristics of speed and strength are considered to be of equal importance to the team in selecting a player for the position. Based on the information about the means and standard deviations of the speed and strength data for all players and the measurements listed in the table below for Players A and B, which player should the team select if the team can only select one of the two players? Justify your answer.

[pic]

Because the two variables - time to run yards and amount of weight lifted – are recorded on different scales, it is important not only to compare the players’ values but also to take into account the standard deviations of the distributions of the variables. One reasonable way to do this is with z-scores.

The z-scores for 40-yard running times are:

[pic][pic]

The z-scores for the amount of weight lifted are:

[pic][pic]

The z-scores indicate that Player A has a much better running time (lower z-score) while Player B is better at weight lifting (higher z-score), however, the difference for weight lifting is very small. As a result, Player A is faster and only slightly less strong.

13. The amount of time a Woodward Academy AP Statistics student spends on homework each evening is approximately normally distributed. If 22% of the 75 students spend at least 47 minutes doing homework and Erin, who spends 65 minutes has a standardized score of 1.932, determine:

a) the mean and standard deviation of the distribution.

[pic] [pic]

Solve the system of equations.

[pic]

[pic]

b) the approximate number of students who spend less than 15 minutes on homework.

Using [pic], [pic].

[pic]

Approximately 7 students.

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