Question

Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.7 feet and a standard deviation of 0.4 feet. A sample of 74 men’s step lengths is taken. Step 1 of 2 : Find the probability that an individual man’s step length is less than 2.5 feet. Round your answer to 4 decimal places, if necessary.

Answer 1

Answer:

0.3085 = 30.85% probability that an individual man’s step length is less than 2.5 feet.

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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.

Mean of 2.7 feet and a standard deviation of 0.4 feet.

This means that $\mu = 2.7, \sigma = 0.4$

Find the probability that an individual man’s step length is less than 2.5 feet.

This is the p-value of Z when X = 2.5. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{2.5 - 2.7}{0.4}$

$Z = -0.5$

$Z = -0.5$ has a p-value of 0.3085

0.3085 = 30.85% probability that an individual man’s step length is less than 2.5 feet.