Question

at a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 46 minutes and a standard deviation of 3 minutes. using the empirical rule, what percentage of customers have to wait between 40 minutes and 52 minutes

Explanation:

Step1: Identify the mean and standard deviation

The mean ($\mu$) is 46 minutes and the standard deviation ($\sigma$) is 3 minutes.

Step2: Calculate the number of standard deviations from the mean for the given values

For 40 minutes:
We calculate how many standard deviations 40 is from the mean using the formula $z=\frac{x - \mu}{\sigma}$.
Substituting the values: $z=\frac{40 - 46}{3}=\frac{- 6}{3}=- 2$. So 40 is 2 standard deviations below the mean.

For 52 minutes:
Using the same formula $z=\frac{x-\mu}{\sigma}$, substituting $x = 52$, $\mu=46$ and $\sigma = 3$: $z=\frac{52 - 46}{3}=\frac{6}{3}=2$. So 52 is 2 standard deviations above the mean.

Step3: Apply the empirical rule

The empirical rule (also known as the 68 - 95 - 99.7 rule) states that for a normal distribution:

• Approximately 68% of the data lies within 1 standard deviation of the mean ($\mu\pm\sigma$).
• Approximately 95% of the data lies within 2 standard deviations of the mean ($\mu\pm2\sigma$).
• Approximately 99.7% of the data lies within 3 standard deviations of the mean ($\mu\pm3\sigma$).

Since 40 is $\mu - 2\sigma$ and 52 is $\mu+2\sigma$, the percentage of data between $\mu - 2\sigma$ and $\mu + 2\sigma$ is approximately 95%.

Answer:

95%