Question

The physical plant at the main campus of a large state university recieves daily requests to replace florecent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 58 and a standard deviation of 10. Using the empirical rule, what is the approximate percentage of lightbulb replacement requests numbering between 38 and 58?

Answer 1

47.5%

Explanation:

Given that :

Mean request (m) = 58

Standard deviation (σ) = 10

Find the approximate percentage of lightbulb replacement requests numbering between 38 and 58

Using the empirical formula :

According to the empirical rule;

68% of data lies within 1 standard deviation (σ) of the mean (m) [(m-σ) and (m+σ)]

95% of data lies within 2 standard deviation of the mean [(m-2σ) and (m+2σ)]

97.5% lies within 3 standard deviation of the mean [(m-3σ) and (m+3σ)]

To obtain the percentage numbering between 38 and 58

Mean of 58 and score of 38 shows the data lies within 2 standard deviations from the mean

Score of 38 lies 2 standard deviations below the mean (m - 2σ)

58 - 2(10) = 58 - 20 = 38

Since 95% = [(m-2σ) and (m+2σ)]

And our data is only 2 standard deviations below the mean and the other half is equal to the mean :

For a normal distribution, distribution is symmetric:

Thus the proportion 95% / 2 should give the percentage lightbulb replacement numbering between 38 and 58

95% / 2= 47.5%