Question

The physical plant at the main campus of a large state university receives daily requests to replace florecent lightbulbs. The distribution of the number daily requests is bell shaped and has a mean of 51 and a standard deviation of 6. Using the empirical rule what is the approximate percentage of lightbulbs replacement requests numbering between 33 and 51?

Answer 1

The distribution of the number of daily requests is bell-shaped and has a mean of 51 and a standard deviation of 6.

`\begin{gathered} \mu=51 \\ \sigma=6 \end{gathered}`

Empirical Rule:

The empirical rule states that

68% of all the observed data will fall within 1 standard deviation from the mean.

95% of all the observed data will fall within 2 standard deviations from the mean.

99.7% of all the observed data will fall within 3 standard deviations from the mean.

Using the empirical rule what is the approximate percentage of lightbulbs replacement requests numbering between 33 and 51.

`lower\: limit=\mu-z\cdot\sigma`

Using the above equation, let us find z (that is how many standard deviations away is out data from the mean)

`\begin{gathered} 33=51-z\cdot6 \\ z\cdot6=51-33 \\ z=(51-33)/(6) \\ z=3 \end{gathered}`

That means that 99.7% of the lightbulbs replacement requests are found between 33 and 51 daily requests.

Since we have a bell-shaped distribution that is symmetrical around the mean,

99.7%/2 = 49.85%

49.85% of lightbulbs replacement requests are found between 33 and 51.