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 58 and a standard deviation of 11. Using the Empirical Rule rule, what is the approximate percentage of lightbulb replacement requests numbering between 58 and 69?
well 69 is 1 std above the mean.
To find the approximate percentage of lightbulb replacement requests numbering between 58 and 69, we can use the Empirical Rule (also known as the 68-95-99.7 rule).
The Empirical Rule states that for a bell-shaped or 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 want to find the percentage of requests between 58 and 69, which falls within one standard deviation of the mean.
First, we calculate the z-scores for the values 58 and 69 using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
For 58: z = (58 - 58) / 11 = 0
For 69: z = (69 - 58) / 11 ≈ 1
A z-score of 0 represents the mean, and a z-score of 1 represents one standard deviation above the mean.
Since the Empirical Rule states that approximately 68% of the data falls within one standard deviation of the mean, we can infer that approximately 68% of the lightbulb replacement requests fall between 58 and 69.
Therefore, the approximate percentage of lightbulb replacement requests numbering between 58 and 69 is approximately 68%.