Question

The physical plant at the main campus of a large state university recieves daily requests to replace fluorescent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 49 and a standard deviation of 9. Using the empirical rule, what is the approximate percentage of lightbulb replacement requests numbering between 49 and 76?

Do not enter the percent symbol.
ans = %

2) The number of potholes in any given 1 mile stretch of freeway pavement in Pennsylvania has a bell-shaped distribution. This distribution has a mean of 43 and a standard deviation of 5. Using the empirical rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 38 and 53?

Do not enter the percent symbol.
ans = %

Answer 1

1)

Z value = (X - $\mu$ ) / $\sigma$

Z value of 49 is (49 - 49) / 9 = 0

Z value of 76 is (76 - 49) / 9 = 3

By Empirical rule, 99.7% of the data lies between 3 standard deviations from mean (z = -3 to z = +3)

Due to symmetry of bell-shaped distribution, percentage of lightbulb replacement requests numbering between 49 and 76 (z = 0 to z = +3)

= 99.7 / 2 = 49.85%

2)

Z value = (X - $\mu$ ) / $\sigma$

Z value of 38 is (38 - 43) / 5 = -1

Z value of 53 is (53 - 43) / 5 = 2

By Empirical rule, 68% of the data lies between 1 standard deviations from mean (z = -1 to z = +1)

Due to symmetry of bell-shaped distribution, 68% / 2 = 34% of the data lies mean and 1 standard deviation from mean on one side (z = -1 to z = 0)

By Empirical rule, 95% of the data lies between 2 standard deviations from mean (z = -2 to z = +2)

Due to symmetry of bell-shaped distribution, 95% / 2 = 47.5% of the data lies mean and 2 standard deviation from mean on one side (z = 0 to z = +2)

The approximate percentage of 1-mile long roadways with potholes numbering between 38 and 53 (z = -1 to z = +2)

= 34 + 47.5

= 81.5 %