Question

the physical plant at the main campus of a large state university recieves 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 4. using the empirical rule (as presented in the book), what is the approximate percentage of lightbulb replacement requests numbering between 46 and 58? do not enter the percent symbol. ans = i % question help: message instructor post to forum

Explanation:

Step1: Calculate the number of standard - deviations from the mean

First, find the difference between the mean $\mu = 58$ and the value $x = 46$. The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $\sigma = 4$. So, $z=\frac{46 - 58}{4}=\frac{- 12}{4}=-3$.

Step2: Recall the empirical rule

The empirical rule for a normal distribution states that approximately 68% of the data lies within 1 standard - deviation of the mean ($\mu\pm\sigma$), approximately 95% lies within 2 standard - deviations ($\mu\pm2\sigma$), and approximately 99.7% lies within 3 standard - deviations ($\mu\pm3\sigma$). The distribution is symmetric about the mean. The percentage of data between $\mu - 3\sigma$ and $\mu$ is half of the percentage of data between $\mu - 3\sigma$ and $\mu+3\sigma$. Since the percentage of data between $\mu - 3\sigma$ and $\mu + 3\sigma$ is 99.7%, the percentage of data between $\mu - 3\sigma$ and $\mu$ is $\frac{99.7}{2}=49.85$.

Answer:

49.85