Question

Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.5 feet and a standard deviation of 0.4 feet. A sample of 45 men’s step lengths is taken. Step 1 of 2 : Find the probability that an individual man’s step length is less than 1.9 feet. Round your answer to 4 decimal places, if necessary.

Answer 1

0.0668 = 6.68% probability that an individual man’s step length is less than 1.9 feet.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Normally distributed with a mean of 2.5 feet and a standard deviation of 0.4 feet.

This means that
`\mu = 2.5, \sigma = 0.4`

Find the probability that an individual man’s step length is less than 1.9 feet.

This is the p-value of Z when X = 1.9. So

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

`Z = (1.9 - 2.5)/(0.4)`

`Z = -1.5`

`Z = -1.5` has a p-value of 0.0668

0.0668 = 6.68% probability that an individual man’s step length is less than 1.9 feet.