Question

Bicycles arrive at a bike shop as parts in a box. Before they can be sold, they must be unpacked and assembled. Based on past experience, the bike shop owner knows that assembly times follow (roughly) a Normal distribution with a mean of 25 minutes and a standard deviation of 3 minutes. A customer walks into the bike shop and wishes to buy a bike like the one in the window, but in a different color. The shop has one, but it is still in the box, so it will need to be assembled. What is the probability that the bike will be ready within a half hour

Answer 1

95.25% probability that the bike will be ready within a half hour

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 problem, we have that:

`\mu = 25, \sigma = 3`

What is the probability that the bike will be ready within a half hour

This is the pvalue of Z when X = 30. So

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

`Z = (30 - 25)/(3)`

`Z = 1.67`

`Z = 1.67` has a pvalue of 0.9525

95.25% probability that the bike will be ready within a half hour