Section S.6 – Practice Problems

1. What does a z-score measure?

2. Consider the standard normal distribution. The mean is always _____ and the standard deviation is always ______.

3. What is the z-score of x = 12, if it is two standard deviations to the right of the mean?

4. What is the z-score of x = 9, if it is 1.5 standard deviations to the left of the mean?

5. What is the z-score of x = –2, if it is 2.78 standard deviations to the right of the mean?

6. What is the z-score of x = 7, if it is 0.133 standard deviations to the left of the mean?

7. Suppose X is normally distributed with mean -1 and standard deviation 2. What is the z-score of x = 2?

8. Suppose a variable is approximately normally distributed with mean 12 and standard deviation 6. What is the z-score of x = 2?

9. Suppose a variable is normally distributed with mean 9 and standard deviation 3. What is the z-score of x = 9?

10. Suppose a normal distribution has a mean of six and a standard deviation of 1.5. What is the z-score of x = 5.5?

11. In a normal distribution, x = 5 and z = –1.25. This tells you that x = 5 is ____ standard deviations to the ____ (right or left) of the mean.

12. In a normal distribution, x = 3 and z = 0.67. This tells you that x = 3 is ____ standard deviations to the ____ (right or left) of the mean.

13. In a normal distribution, x = –2 and z = 6. This tells you that x = –2 is ____ standard deviations to the ____ (right or left) of the mean.

14. In a normal distribution, x = –5 and z = –3.14. This tells you that x = –5 is ____ standard deviations to the ____ (right or left) of the mean.

15. In a normal distribution, x = 6 and z = –1.7. This tells you that x = 6 is ____ standard deviations to the ____ (right or left) of the mean.

16. The life of Sunshine DVD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A DVD player is guaranteed for three years. We are interested in the length of time a DVD player lasts. Find the z-score corresponding to the guaranteed life of 3 years.

17. The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005–2006 season. The heights of basketball players have an approximately normal distribution with mean 79 inches and a standard deviation 3.89 inches. For each of the following heights, calculate the z-score and interpret it using complete sentences.

a. 77 inches
b. 85 inches

18. The systolic blood pressure (given in millimeters) of males has an approximately normal distribution with mean 125 and standard deviation 14. Systolic blood pressure for males follows a normal distribution.

a. Calculate the z-scores for the male systolic blood pressures 100 and 150 millimeters.

b. If a male friend of yours said he thought his systolic blood pressure was 2.5 standard deviations below the mean, but that he believed his blood pressure was between 100 and 150 millimeters, what would you say to him?

19. A variable is normally distributed with mean 2 and standard deviation 6. What value of x has a z-score of three?

20. Suppose a variable has mean 8, standard deviation 1, and is normally distributed. What value of x has a z-score of –2.25?

21. Suppose a normally distributed variable has mean 4 and standard deviation 2. What value of x is 1.5 standard deviations to the left of the mean?

22. A variable is normally distributed with mean 4 and standard deviation 2. What value of x is two standard deviations to the right of the mean?

23. The life of Sunshine DVD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. How long does a DVD player last if its z-score is -2.0?

24. The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005–2006 season. The heights of basketball players have an approximately normal distribution with mean 79 inches and a standard deviation 3.89 inches.

a. If an NBA player reported his height had a z-score of 3.5, would you believe him? Explain your answer.

b. If an NBA player reported his height had a z-score of -3.5, would you believe him? Explain your answer.

25. The systolic blood pressure (given in millimeters) of males has an approximately normal distribution with mean 125 and standard deviation 14.

a. Find the systolic blood pressure of a man with a z-score of 0.75. Is this an unusual systolic blood pressure? Why or why not?

b. Find the systolic blood pressure of a man with a z-score of 3.2. Is this an unusual systolic blood pressure? Why or why not?

26. If the area to the left of X in a normal distribution is 0.123, what is the area to the right of x?

27. If the area to the right of x in a normal distribution is 0.543, what is the area to the left of x?

28. A variable is normally distributed with mean 16 and standard deviation 3. Find each of the following. Draw the associated normal curve, shading the solution region.

a. Find the area to the left of 12.
b. Find the area to the left of 20.
c. Find the area to the right of 15.
d. Find the area to the right of 24.
e. Find the area between 12 and 20.

29. A variable is normally distributed with mean 84.6 and standard deviation 19.1. Find each of the following. Draw the associated normal curve, shading the solution region.

a. Find the area to the left of 60.5.
b. Find the area to the right of 60.5.
c. Find the area to the left of 128.0.
d. Find the area to the right of 119.3.
e. Find the area between 80 and 90.

30. A variable is normally distributed with mean 45 and standard deviation 4.6. Find each of the following. Draw the associated normal curve, shading the solution region.

a. Find the area to the left of 27.
b. Find the area to the left of 45.
c. Find the area to the right of 52.
d. Find the area to the right of 38.
e. Find the area between 33 and 48.

31. The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.

a. What is the probability of spending less than two days in recovery?
b. What is the probability of spending more than two days in recovery?
c. What is the probability of spending more than 10 days in recovery?

32. According to a study done by De Anza students, the height for Asian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Asian adult male is randomly chosen. Let X = height of the individual.

a. Find the probability that the person is between 65 and 69 inches. Include a sketch of the graph.
b. Would you expect to meet many Asian adult males over 72 inches? Explain why or why not, and justify your answer numerically.

33. IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual.

a. Find the probability that the person has an IQ greater than 120. Include a sketch of the graph.
b. Find the probability that the person has an IQ less than 60. Include a sketch of the graph.

34. The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of 10. Suppose that one individual is randomly chosen. Let X = percent of fat calories.

a. Find the probability that the percent of fat calories a person consumes is more than 40. Graph the situation. Shade in the area to be determined.

b. Find the probability that the percent of fat calories a person consumes is between 10 and 20. Graph the situation. Shade in the area to be determined.

35. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet.

a. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? Sketch the graph. Shade the area to be determined.

b. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled more than 350 feet? Sketch the graph, and shade the area to be determined.