Question

1) Suppose a normally distributed set of data has a mean of 179 and a standard deviation of 17. Use the 68-95-99.7 Rule to determine the percent of scores in the data set expected to be below a score of 213. Give your answer as a percent and includeas many decimal places as the 68-95-99.7 rule dictates. (For example, enter 99.7 instead of 0.997.)

2) Suppose a normally distributed set of data with 2800 observations has a mean of 100 and a standard deviation of 15. Use the 68-95-99.7 Rule to determine the number of observations in the data set expected to be below a value of 145. Round your result to the nearest single observation.

3) Suppose a normally distributed set of data with 6100 observations has a mean of 163 and a standard deviation of 16. Use the 68-95-99.7 Rule to determine the number of observations in the data set expected to be above a value of 147. Round your answer to the nearest whole value.

4) In a mid-size company, the distribution of the number of phone calls answered each day by each of the 12 receptionists is bell-shaped and has a mean of 53 and a standard deviation of 8. Using the empirical (68-95-99.7) rule, what is the approximate percentage of daily phone calls numbering between 37 and 69?

5) The physical plant at the main campus of a large state university receives daily requests to replace florescent light bulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 61 and a standard deviation of 8. Using the empirical (68-95-99.7) rule, what is the approximate percentage of light bulb replacement requests numbering between 61 and 77? Do not enter the percent symbol.

Answer 1

(1) A score of 273 is (273 - 179) / 17 = 5.53 standard deviations above the mean.

Therefore we expect 100% of the data to be below the score of 213

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(2) A value of 145 is (145 - 100) / 15 = 3 standard deviations above the mean.

So only (100 - 99.7) / 2 = 0.15% of the data is above this value.

Therefore 100 - 0.15 = 99.85% is below this value of 145

Therefore the number of expected observations = (99.85 / 100) * 2800 = 2795.8    2796

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(3) A value of 147 is (147 - 163) / 16 = -1 i.e 1 standard deviation below the mean.

One standard deviation below the mean has 34% of the observation, and above the mean we have 50% of the values.

Therefore 34 + 50 = 84% of the observations lies above 147

Therefore the number of expected observations = (84 / 100) * 6100 = 5124

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(4) A value of 37 is (37 - 53) / 8 = -2 i.e 2 standard deviations below the mean and

a value of 69 is (69 - 53) / 8 = 2 standard deviations above the mean.

By the empirical rule 95% of the values lie between these 2 values.

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(5) A value of 61 is (61 - 61) / 8 = -0 i.e it is the mean itself

a value of 77 is (77 - 61) / 8 = 2 standard deviations above the mean.

By the empirical rule 95/2 = 47.5% of the values lie between 0 and 2 standard deviations.

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Attached below is the 68 - 95 - 99.7 diagram for the reference