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 42 and a standard deviation of 9. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 33 and 42?
well, how many std's between 33 and 42?
To find the approximate percentage of lightbulb replacement requests numbering between 33 and 42, we can use the 68-95-99.7 rule, also known as the empirical rule or the three-sigma rule.
The 68-95-99.7 rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case, we have a bell-shaped distribution with a mean of 42 and a standard deviation of 9.
To find the approximate percentage of lightbulb replacement requests numbering between 33 and 42, we need to calculate the z-scores for these values.
The z-score formula is:
z = (x - μ) / σ
Where:
- x is the value we want to find the z-score for (in this case, 33 and 42).
- μ is the mean of the distribution (42).
- σ is the standard deviation of the distribution (9).
For x = 33:
z = (33 - 42) / 9 = -1
For x = 42:
z = (42 - 42) / 9 = 0
Now that we have the z-scores, we can refer to the 68-95-99.7 rule.
Approximately 68% of the data falls within one standard deviation of the mean. Since our range of interest falls within one standard deviation (from -1 to 0), we know that the approximate percentage is around 68%.
Therefore, the approximate percentage of lightbulb replacement requests numbering between 33 and 42 is approximately 68%.