Question

at a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 24 minutes and a standard deviation of 5 minutes. using the empirical rule, what percentage of customers have to wait between 19 minutes and 29 minutes?

Explanation:

Step1: Recall empirical rule

The empirical rule for a normal - distribution states that about 68% of the data lies within 1 standard deviation of the mean, about 95% lies within 2 standard deviations of the mean, and about 99.7% lies within 3 standard deviations of the mean. The formula for the range within \(k\) standard deviations of the mean is \(\mu\pm k\sigma\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation.

Step2: Calculate \(k\) values

Given \(\mu = 24\) minutes and \(\sigma=5\) minutes. For \(x_1 = 19\) minutes, we calculate \(k_1=\frac{\mu - x_1}{\sigma}=\frac{24 - 19}{5}=1\). For \(x_2 = 29\) minutes, we calculate \(k_2=\frac{x_2-\mu}{\sigma}=\frac{29 - 24}{5}=1\).

Step3: Apply empirical rule

Since the values 19 and 29 are 1 standard - deviation below and above the mean respectively (\(\mu - \sigma=24 - 5 = 19\) and \(\mu+\sigma=24 + 5 = 29\)), by the empirical rule, the percentage of data within 1 standard deviation of the mean is 68%.

Answer:

68%