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 0.4 feet. A sample of 61 men’s step lengths is taken.

Step 2 of 2: Find the probability that the mean of the sample taken is less than 2.2 feet. Round your answer to 4 decimal places, if necessary.

Answer 1

Explanation:

We can use the Central Limit Theorem to approximate the distribution of the sample mean. Since the sample size is large (n=61), the sample mean will be approximately normally distributed with mean μ = 2.6 feet and standard deviation σ/√n = 0.4/√61 = 0.0517 feet.

To find the probability that the sample mean is less than 2.2 feet, we standardize the sample mean as follows:

z = (x - μ) / (σ/√n) = (2.2 - 2.6) / (0.4/√61) = -3.692

We look up the probability corresponding to z = -3.692 in the standard normal distribution table or use a calculator to find the area under the standard normal curve to the left of z:

P(z < -3.692) ≈ 0.0001 (rounded to 4 decimal places)

Therefore, the probability that the mean of the sample taken is less than 2.2 feet is approximately 0.0001.