Question

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car. g

Answer 1

38.40% probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.

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 = 3, \sigma = 1`

Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.

This is the pvalue of Z when X = 3.5 subtracted by the pvalue of Z when X = 2.5

X = 3.5

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

`Z = (3.5 - 3)/(1)`

`Z = 0.5`

`Z = 0.5` has a pvalue of 0.6915

X = 2.5

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

`Z = (2.5 - 3)/(1)`

`Z = -0.5`

`Z = -0.5` has a pvalue of 0.3075

0.6915 - 0.3075 = 0.3840

38.40% probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.