Question

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual. For a sample of n=70, find the probability of a sample mean being less than 24.2 if μ=24 and σ=1.25.

Answer 1

To find the probability of a sample mean being less than 24.2, we can use the standard normal distribution and the Central Limit Theorem. The probability is approximately 0.9713, or 97.13%. The given sample mean of 24.2 would not be considered unusual.

Step-by-step explanation:

To find the probability of a sample mean being less than a given value, we need to use the standard normal distribution. In this case, the population mean (μ) is 24 and the population standard deviation (σ) is 1.25. Since the sample size (n) is large (n = 70), we can use the Central Limit Theorem to approximate the sample mean using a normal distribution.

To find the probability of a sample mean being less than 24.2, we can convert it to a z-score using the formula:

z = (x - μ) / (σ / √n)

Plugging in the values, we get:

z = (24.2 - 24) / (1.25 / √70) ≈ 1.897

Using the standard normal distribution table or a calculator, we can find the probability associated with a z-score of 1.897. The probability is approximately 0.9713, or 97.13%.

Therefore, the probability of a sample mean being less than 24.2 is 0.9713. Since this probability is relatively high, the given sample mean of 24.2 would not be considered unusual.