Question 1

Find the z-score for an IQ test score of 118 when the mean is 100 and the standard deviation is 15.

Accepted Answer

Z-score = (x − mean)/std dev = (118 − 100)/15 = 18/15 = 1.2

Question 2

Using Chebyshev's theorem, approximate the minimum proportion of the data that will be within μ ± kσ for k = 1.6.

Accepted Answer

100(1 − 1 /k²)% = 100(1 − 1 /1.6²)% = 61%

Question 3

In a hearing test, randomly selected subjects estimate the loudness (in decibels) of a sound, and the results are (mean = 70): 68, 67, 70, 71, 68, 75, 68, 62, 80, 73, 68. What is the range?

Accepted Answer

62, 67, 68, 68, 68, 68, 70, 71, 73, 75, 80
80 − 62 = 18

Question 4

Quality control is an important issue at ACME Company, which manufactures light bulbs. To test the life-hours of their light bulbs, they randomly sampled nine light bulbs and measured how many hours they lasted: 378, 361, 350, 375, 200, 391, 375, 368, 321. What is the median?

Accepted Answer

To calculate median, put data measurements in ascending order. The median for an odd number of measurements is the middle measurement; median = 368.

Question 5

In a hearing test, randomly selected subjects estimate the loudness (in decibels) of a sound, and the results are (mean = 70): 68, 67, 70, 71, 68, 75, 68, 62, 80, 73, 68. What is the standard deviation?

Accepted Answer

You take the square root of 22.40 = 4.73

Question 6

Find the z-score for an IQ test score of 92 when the mean is 100 and the standard deviation is 15.

Accepted Answer

Z-score = (x − mean)/std dev = (92 − 100)/15 = −8/15 = −.53

Question 7

According to Chebyshev's theorem, a range of how many standard deviations would include at least 80 percent of the values?

Accepted Answer

100(1 − 1 /k²)% = 100(1 − 1 /2.2²)% = 80%

Question 8

In a statistics class, 10 scores were randomly selected with the following results (mean = 71.5): 74, 73, 77, 77, 71, 68, 65, 77, 67, 66. What is the standard deviation?

Accepted Answer

You take the square root of 22.71 = 4.77

Question 9

Using Chebyshev's theorem, approximate the minimum proportion of the data that will be within μ ± kσ for k = 3.2.

Accepted Answer

100(1 − 1 /k²)% = 100(1 − 1 /3.2²)% = 90%

Question 10

Find the z-score for an IQ test score of 142 when the mean is 100 and the standard deviation is 15.

Accepted Answer

Z-score = (x − mean)/std dev = (142 − 100)/15 = 42/15 = 2.8

Question 11

Quality control is an important issue at ACME Company, which manufactures light bulbs. To test the life-hours of their light bulbs, they randomly sampled nine light bulbs and measured how many hours they lasted: 378, 361, 350, 375, 200, 391, 375, 368, 321. What is the mode?

Accepted Answer

Mode is the value(s) that appears most frequently; mode = 375 (occurs two times).

Question 12

Find the coefficient of variation for IQ tests with a mean of 100 and a standard deviation of 15.

Accepted Answer

Coefficient of variation = (Std dev/mean) × 100 = (15/100) × 100 = 15

Question 13

In a statistics class, 10 scores were randomly selected with the following results (mean = 71.5): 74, 73, 77, 77, 71, 68, 65, 77, 67, 66. What is the variance?

Accepted Answer

(74 − 71.5) = 2.5 squared = 6.25
(73 − 71.5) = 1.5 squared = 2.25
(77 − 71.5) = 5.5 squared = 30.25
(77 − 71.5) = 5.5 squared = 30.25
(71 − 71.5) = 0.5 squared = 0.25
(68 − 71.5) = −3.5 squared = 12.25
(65 − 71.5) = −6.5 squared = 42.25
(77 − 71.5) = 5.5 squared = 30.25
(67 − 71.5) = −4.5 squared = 20.25
(66 − 71.5) = −5.5 squared = 30.25
(6.25 + 2.25 + 30.25 + 30.25 + 0.25 + 12.25 +42.25 + 30.25 + 20.25 + 30.25) = 204.5 / 9 = 22.72

Question 14

According to a survey of the top 10 employers in a major city in the Midwest, a worker spends an average of 413 minutes a day on the job. Suppose the standard deviation is 26.8 minutes and the time spent is approximately a normal distribution. What are the times within which approximately 99.73 percent of all workers will fall?

Accepted Answer

3(26.8) − 413 = 332.6
3(26.8) + 413 = 493.4

Question 15

According to a survey of the top 10 employers in a major city in the Midwest, a worker spends an average of 413 minutes a day on the job. Suppose the standard deviation is 26.8 minutes, and the time spent is approximately a normal distribution. What are the times within which approximately 68.26 percent of all workers will fall?

Accepted Answer

413 − 26.8 = 386.2
413 + 26.8 = 439.8

Question 16

In a hearing test, randomly selected subjects estimate the loudness (in decibels) of a sound, and the results are (mean = 70): 68, 67, 70, 71, 68, 75, 68, 62, 80, 73, 68. What is the variance?

Accepted Answer

62, 67, 68, 68, 68, 68, 70, 71, 73, 75, 80
(62 − 70)= -8 squared = 64
(67 − 70)= -3 squared = 9
(68 − 70)= -2 squared = 4
(68 − 70)= -2 squared = 4
(68 − 70)= -2 squared = 4
(68 − 70)= -2 squared = 4
(70 − 70)= 0 squared = 0
(71 − 70)= 1 squared = 1
(73 − 70)= 2 squared = 4
(75 − 70)= 5 squared = 25
(80 − 70)= 10 squared = 100
(64 + 9 + 4 + 4 + 4 + 4 + 0 + 1 + 4 + 25 + 100) = 224 / 10 = 22.40

Question 17

Find the z-score for an IQ test score of 125 when the mean is 100 and the standard deviation is 15.

Accepted Answer

Z-score = (x − mean)/std dev = (125 − 100)/15 = 25/15 = 1.67

Question 18

According to Chebyshev's theorem, at least what proportion of the data will be within μ ± kσ for k = 2?

Accepted Answer

For any values of k greater than 1, at least 100(1 − 1 /k²)% of the population measurements lie in the interval.

Question 19

Using Chebyshev's Theorem, find the interval that contains at least 93.75 percent of all measurements when the mean = 2.549 and s = 1.828.

Accepted Answer

1 − (1/k²) = .9375; 1/k² = 1 − .9375; 1/k =; k = 4

Question 20

In a statistics class, 10 scores were randomly selected with the following results (mean = 71.5): 74, 73, 77, 77, 71, 68, 65, 77, 67, 66. What is the range?

Accepted Answer

65, 66, 67, 68, 71, 73, 74, 77, 77, 77
77 − 65 = 12

Quiz 3: Descriptive Statistics and Analytics: Numerical Methods

Find the z-score for an IQ test score of 118 when the mean is 100 and the standard deviation is 15.

Using Chebyshev's theorem, approximate the minimum proportion of the data that will be within μ ± kσ for k = 1.6.

In a hearing test, randomly selected subjects estimate the loudness (in decibels) of a sound, and the results are (mean = 70) : 68, 67, 70, 71, 68, 75, 68, 62, 80, 73, 68. What is the range?

Quality control is an important issue at ACME Company, which manufactures light bulbs. To test the life-hours of their light bulbs, they randomly sampled nine light bulbs and measured how many hours they lasted: 378, 361, 350, 375, 200, 391, 375, 368, 321. What is the median?

In a hearing test, randomly selected subjects estimate the loudness (in decibels) of a sound, and the results are (mean = 70) : 68, 67, 70, 71, 68, 75, 68, 62, 80, 73, 68. What is the standard deviation?

Find the z-score for an IQ test score of 92 when the mean is 100 and the standard deviation is 15.

According to Chebyshev's theorem, a range of how many standard deviations would include at least 80 percent of the values?

In a statistics class, 10 scores were randomly selected with the following results (mean = 71.5) : 74, 73, 77, 77, 71, 68, 65, 77, 67, 66. What is the standard deviation?