6.8: Practice

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A bottle of water contains 12.05 fluid ounces with a standard deviation of 0.01 ounces. Define the random variable X in words. X = ____________.

A normal distribution has a mean of 61 and a standard deviation of 15. What is the median?

X ~ N(1, 2)

σ = _______

A company manufactures rubber balls. The mean diameter of a ball is 12 cm with a standard deviation of 0.2 cm. Define the random variable X in words. X = ______________.

X ~ N(–4, 1)

What is the median?

X ~ N(3, 5)

σ = _______

X ~ N(–2, 1)

μ = _______

What does a z-score measure?

What does standardizing a normal distribution do to the mean?

10.

Is X ~ N(0, 1) a standardized normal distribution? Why or why not?

11.

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

12.

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

13.

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

14.

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

15.

Suppose X ~ N(2, 6). What value of x has a z-score of three?

16.

Suppose X ~ N(8, 1). What value of x has a z-score of –2.25?

17.

Suppose X ~ N(9, 5). What value of x has a z-score of –0.5?

18.

Suppose X ~ N(2, 3). What value of x has a z-score of –0.67?

19.

Suppose X ~ N(4, 2). What value of x is 1.5 standard deviations to the left of the mean?

20.

Suppose X ~ N(4, 2). What value of x is two standard deviations to the right of the mean?

21.

Suppose X ~ N(8, 9). What value of x is 0.67 standard deviations to the left of the mean?

22.

Suppose X ~ N(–1, 2). What is the z-score of x = 2?

23.

Suppose X ~ N(12, 6). What is the z-score of x = 2?

24.

Suppose X ~ N(9, 3). What is the z-score of x = 9?

25.

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?

26.

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.

27.

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.

28.

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.

29.

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.

30.

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.

31.

About what percent of x values from a normal distribution lie within one standard deviation (left and right) of the mean of that distribution?

32.

About what percent of the x values from a normal distribution lie within two standard deviations (left and right) of the mean of that distribution?

33.

About what percent of x values lie between the second and third standard deviations (both sides)?

34.

Suppose X ~ N(15, 3). Between what x values does 68.27% of the data lie? The range of x values is centered at the mean of the distribution (i.e., 15).

35.

Suppose X ~ N(–3, 1). Between what x values does 95.45% of the data lie? The range of x values is centered at the mean of the distribution(i.e., –3).

36.

Suppose X ~ N(–3, 1). Between what x values does 34.14% of the data lie?

37.

About what percent of x values lie between the mean and three standard deviations?

38.

About what percent of x values lie between the mean and one standard deviation?

39.

About what percent of x values lie between the first and second standard deviations from the mean (both sides)?

40.

About what percent of x values lie between the first and third standard deviations(both sides)?

Use the following information to answer the next two exercises: The life of wearable fitness devices is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A wearable fitness device is guaranteed for three years. We are interested in the length of time a wearable fitness device lasts.

41.

Define the random variable X in words. X = _______________.

42.

X ~ _____(_____,_____)

43.

How would you represent the area to the left of one in a probability statement?

Figure 6.12

44.

What is the area to the right of one?

Figure 6.13

45.

Is P(x < 1) equal to P(x ≤ 1)? Why?

46.

How would you represent the area to the left of three in a probability statement?

Figure 6.14

47.

What is the area to the right of three?

Figure 6.15

48.

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

49.

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

Use the following information to answer the next four exercises:

X ~ N(54, 8)

50.

Find the probability that x > 56.

51.

Find the probability that x < 30.

52.

Find the 80^th percentile.

53.

Find the 60^th percentile.

54.

X ~ N(6, 2)

Find the probability that x is between three and nine.

55.

X ~ N(–3, 4)

Find the probability that x is between one and four.

56.

X ~ N(4, 5)

Find the maximum of x in the bottom quartile.

57.

Use the following information to answer the next three exercise: The life of wearable fitness devices is normally distributed with a mean of 4.1 years and a standard deviation of 1.3 years. A wearable fitness device is guaranteed for three years. We are interested in the length of time a wearable fitness device lasts. Find the probability that a wearable fitness device will break down during the guarantee period.

Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.

Figure 6.16
P(0 < x < ____________) = ___________ (Use zero for the minimum value of x.)

58.

Find the probability that a wearable fitness device will last between 2.8 and six years.

Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
Figure 6.17
P(__________ < x < __________) = __________

59.

Find the 70^th percentile of the distribution for the time a wearable fitness device lasts.

Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70%.
Figure 6.18
P(x < k) = __________ Therefore, k = _________

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