Question

Borachio eats at the same fast food restaurant every day. Suppose the time X between the moment Borachio enters the restaurant and the moment he is served his food is normally distributed with mean 4.2 minutes and standard deviation 1.3 minutes. Find the probability that when he enters the restaurant today it will be at least 5 minutes until he is served.

Answer 1

26.94% probability that when he enters the restaurant today it will be at least 5 minutes until he is served.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

`\mu = 4.2, \sigma = 1.3`

Find the probability that when he enters the restaurant today it will be at least 5 minutes until he is served.

This is 1 subtracted by the pvalue of Z when X = 5. So

`Z = (X - \mu)/(\sigma)`

`Z = (5 - 4.2)/(1.3)`

`Z = 0.615`

`Z = 0.615` has a pvalue of 0.7308.

1 - 0.7308 = 0.2694

26.94% probability that when he enters the restaurant today it will be at least 5 minutes until he is served.