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 57 and a standard deviation of 6 . Using the empirical rule (as presented in the book), what is the approximate percentage of lightbulb replacement requests numbering between 57 and 75 ?
Do not enter the percent symbol.
ans =
$\%$

Step 1 ：The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution, almost all values lie within 3 standard deviations of the mean. More specifically, 68% of the data falls within the first standard deviation, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

Step 2 ：In this case, the mean is 57 and the standard deviation is 6. We are asked to find the percentage of requests between 57 and 75.

Step 3 ：First, we need to calculate how many standard deviations away 75 is from the mean. The formula to calculate the number of standard deviations is \((value - mean) / std_dev\).

Step 4 ：Substituting the given values into the formula, we get \((75 - 57) / 6 = 3.0\). So, the value 75 is 3 standard deviations away from the mean.

Step 5 ：According to the empirical rule, 99.7% of the data falls within three standard deviations. However, since we are only interested in the data between the mean and 75, we only need to consider half of this percentage.

Step 6 ：So, the percentage of lightbulb replacement requests numbering between 57 and 75 is \(99.7 / 2 = 49.85\)%.

Step 7 ：\(\boxed{49.85}\) is the final answer.