Question

Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.6 feet and a standard deviation of 0.2 feet. A sample of 53 men's step lengths is taken.

Find the probability that the mean of the sample taken is less than 2.4 feet. Round your answer to 4 decimal places, if necessary.

Answer 1

The probability that the mean of the sample taken is less than 2.4 feet is 0.0013.

Step-by-step explanation:

The question involves using the Central Limit Theorem to find the probability of a sample mean being less than a certain value in a normal distribution. The mean walking step length (μ) for adult males is 2.6 feet, and the standard deviation (σ) is 0.2 feet. The sample size (n) is 53.

The formula for the standard error of the mean (SE) is given by:

SE = σ/√n

Substitute the given values:

SE = 0.2/√53

Calculate the standard error.

Next, calculate the z-score, which tells us how many standard errors the sample mean is from the population mean. The z-score formula is:

z =
`\bar{x}` - μ/SE

Substitute the values, with
`\(\bar{x}\)` being 2.4 feet.

Now, consult a standard normal distribution table or use a calculator to find the probability associated with the calculated z-score. In this case, the probability is 0.0013.

In conclusion, with a mean sample length of 2.4 feet, there is a very low probability (0.0013) that the sample mean is less than this value, indicating that such a result is highly unlikely under the given normal distribution.