5. The average age of men at the time of their first marriage is 24.8 years. Suppose the standard deviation is 2.8 years. Forty-nine married males are selected at random and asked the
age at which they were first married. Find the probability that the sample mean will be more than 26.

0.0013 = 0.13% probability that the sample mean will be more than 26.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

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

In a set with mean and standard deviation , the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

The average age of men at the time of their first marriage is 24.8 years. Suppose the standard deviation is 2.8 years.

This means that

Forty-nine married males are selected at random and asked the age at which they were first married.

This means that

Find the probability that the sample mean will be more than 26.

This is 1 subtracted by the pvalue of Z when X = 26. So

By the Central Limit Theorem

has a pvalue of 0.9987

1 - 0.9987 = 0.0013

0.0013 = 0.13% probability that the sample mean will be more than 26.