At a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 20 minutes and a standard deviation of 2 minutes. Using the empirical rule, what percentage of customers have to wait between 14 minutes and 26 minutes?
By using the empirical rule (also known as the 68-95-99.7 rule), we know that:
- Approximately 68% of the data fall within 1 standard deviation of the mean
- Approximately 95% of the data fall within 2 standard deviations of the mean
- Approximately 99.7% of the data fall within 3 standard deviations of the mean
In this case, we want to find the percentage of customers that have to wait between 14 minutes and 26 minutes, which is within 3 standard deviations of the mean.
First, we need to calculate the z-scores for 14 minutes and 26 minutes using the formula:
z = (X - mean) / standard deviation
For 14 minutes:
z = (14 - 20) / 2 = -3
For 26 minutes:
z = (26 - 20) / 2 = 3
Next, we can look up the percentages associated with z-scores of -3 and 3 in a standard normal distribution table or use a calculator.
For z = -3, the percentage is close to 0.15%.
For z = 3, the percentage is close to 99.85%.
Therefore, the percentage of customers who have to wait between 14 minutes and 26 minutes is approximately 99.85% - 0.15% = 99.7%. This falls within the 99.7% range predicted by the empirical rule.