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=69​, find the probability of a sample mean being less than 23.2 if μ=23 and σ=1.21.

Answer 1

To find the probability of a sample mean being less than 23.2, calculate the z-score using the formula z = (x - μ) / (σ / √n). Then, use the standard normal distribution table to find the probability corresponding to the z-score. The required probability is approximately 0.648.

Step-by-step explanation:

To find the probability of a sample mean being less than 23.2, we need to calculate the z-score and use the standard normal distribution table. The formula for calculating the z-score is:

z = (x - μ) / (σ / √n)

Plugging in the values, we get z = (23.2 - 23) / (1.21 / √69) = 0.383.

Looking up the z-score in the standard normal distribution table, we find that the probability of a z-score of 0.383 is approximately 0.648.

Therefore, the required probability of a sample mean being less than 23.2 is 0.648. To determine whether the given sample mean is considered unusual, we need to compare it to the critical values of the normal distribution. If the sample mean falls within the range of the critical values (usually defined as within 2 standard deviations of the mean), then it is considered usual. Otherwise, it is considered unusual.