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 10. Using the empirical rule (as presented in the book), what is the approximate percentage of lightbulb replacement requests numbering between 22 and 42?
22 is two SD below the mean, in other words, Z = -2.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score and distance from the mean. Multiply by 100.
To find the approximate percentage of lightbulb replacement requests numbering between 22 and 42 using the empirical rule, we need to determine the z-scores corresponding to these values and then find the area under the normal curve between these two z-scores.
The empirical rule states that for a bell-shaped distribution (also known as a normal distribution):
- Approximately 68% of data falls within one standard deviation of the mean
- Approximately 95% of data falls within two standard deviations of the mean
- Approximately 99.7% of data falls within three standard deviations of the mean
In this case, the mean is 42 and the standard deviation is 10.
Step 1: Calculate the z-scores for 22 and 42.
The z-score formula is z = (x - mean) / standard deviation.
For x = 22:
z₁ = (22 - 42) / 10 = -2
For x = 42:
z₂ = (42 - 42) / 10 = 0
Step 2: Find the area under the normal curve between z₁ and z₂.
Since we're interested in the area between z = -2 and z = 0, we need to find the area to the left of z = 0 minus the area to the left of z = -2.
The area to the left of z = 0 is 0.5 (50%).
The area to the left of z = -2 can be found using a standard normal distribution table or a calculator, which gives approximately 0.0228 (2.28%).
Step 3: Calculate the approximate percentage between 22 and 42.
To find the approximate percentage between 22 and 42, we subtract the area to the left of z = -2 from the area to the left of z = 0:
0.5 - 0.0228 = 0.4772
So, approximately 47.72% of lightbulb replacement requests will number between 22 and 42, based on the empirical rule.