A half-century ago, the mean height of women in a particular country in their 20s was 62.1 inches. Assume that the heights of today's women in their 20s are approximately normally distributed with a standard deviation of 2.85 inches. If the mean height today is the same as that of a half-century ago, what percentage of all samples of 24 of today's women in their 20s have mean heights of at least 63.42 inches? About nothing% of all samples have mean heights of at least 63.42 inches.

About 1.16% of all samples have mean heights of at least 63.42 inches.

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 normally distributed samples are 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 sample means with size n can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

What percentage of all samples of 24 of today's women in their 20s have mean heights of at least 63.42 inches?

This is 1 subtracted by the pvalue of Z when X = 63.42. So

By the Central Limit Theorem

has a pvalue of 0.9884

1 - 0.9884 = 0.0116

About 1.16% of all samples have mean heights of at least 63.42 inches.

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