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 69 men’s step lengths is taken.

Step 2 of 2: Find the probability that the mean of the sample taken is less than 2.3 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.

Explanation:

For this, we will be using:

Normal Probability Distribution:

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

In a set with mean and standard deviation , the z-score of a measure X is given by:

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

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

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.

Answer 2

Explanation:

We can use the central limit theorem to approximate the sampling distribution of the sample means as normal with a mean of the population mean and a standard deviation of the population standard deviation divided by the square root of the sample size:

mean = 2.7 feet

standard deviation = 0.4 feet / sqrt(69) ≈ 0.048 feet

Now, we want to find the probability that the sample mean is less than 2.3 feet:

z = (2.3 - 2.7) / 0.048 ≈ -8.33

Using a standard normal distribution table or calculator, we can find the probability that a standard normal random variable is less than -8.33, which is essentially 0:

P(z < -8.33) ≈ 0

Therefore, the probability that the mean of the sample taken is less than 2.3 feet is approximately 0.