The physical plant at the main campus of a large state university receives daily requests to replace florecent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 46 and a standard deviation of 9. Using the empirical rule (as presented in the book), what is the approximate percentage of lightbulb replacement requests numbering between 46 and 73?
Z = (score-mean)/SD = (73-46)/9 = ?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between mean and that Z score.
To find the approximate percentage of lightbulb replacement requests numbering between 46 and 73 using the empirical rule, you can follow these steps:
1. Understand the empirical rule: The empirical rule states that for a bell-shaped, normally distributed data set, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations of the mean, and approximately 99.7% falls within three standard deviations of the mean.
2. Calculate the standard deviation: Since the standard deviation is given as 9, we know that one standard deviation in this case is equal to 9.
3. Determine the range: To find the percentage of lightbulb replacement requests between 46 and 73, we need to calculate how many standard deviations away these values are from the mean.
a. To find how many standard deviations away 46 is from the mean of 46, we use the formula (46 - 46) / 9 = 0. This means 46 is exactly at the mean, so it is 0 standard deviations away from the mean.
b. To find how many standard deviations away 73 is from the mean of 46, we use the formula (73 - 46) / 9 = 3. This means 73 is three standard deviations away from the mean.
4. Apply the empirical rule: Since 68% of the data falls within one standard deviation of the mean (which is 46), we know that 34% lie on each side of the mean. As 46 is exactly at the mean, we can say that approximately 34% of the requests fall below 46.
5. Since 95% of the data falls within two standard deviations of the mean, and 46 is at the mean (0 standard deviations away), we know that approximately 95% - 34% = 61% of the requests fall between 46 and 73.
Therefore, the approximate percentage of lightbulb replacement requests numbering between 46 and 73 is approximately 61%.