when levi commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of…

when levi commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 53 minutes and a standard deviation of 3.5 minutes. using the empirical rule, what percentage of his commutes will be between 46 and 60 minutes?

when levi commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 53 minutes and a standard deviation of 3.5 minutes. using the empirical rule, what percentage of his commutes will be between 46 and 60 minutes?

Answer

Explanation:

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

First, find how many standard deviations 46 and 60 are from the mean. Let $\mu = 53$ be the mean and $\sigma=3.5$ be the standard deviation. For $x = 46$, $z_1=\frac{46 - 53}{3.5}=\frac{-7}{3.5}=- 2$. For $x = 60$, $z_2=\frac{60 - 53}{3.5}=\frac{7}{3.5}=2$.

Step2: Apply the empirical rule

The empirical rule for a normal distribution states that approximately 95% of the data lies within 2 standard deviations of the mean. That is, $P(\mu - 2\sigma<X<\mu + 2\sigma)\approx95%$. Since $46=\mu - 2\sigma$ and $60=\mu + 2\sigma$, the percentage of his commutes between 46 and 60 minutes is approximately 95%.

Answer:

95%