## Assume that the heights of adult Caucasian women have a mean of 63.6 inches and a standard deviation of 2.5 inches. If 100 women are randoml

Question

Assume that the heights of adult Caucasian women have a mean of 63.6 inches and a standard deviation of 2.5 inches. If 100 women are randomly​ selected, find the probability that they have a mean height greater than 63.0 inches. Round to four decimal places.

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2 hours 2021-10-14T13:31:45+00:00 1 Answer 0

0.9918 = 99.18% probability that they have a mean height greater than 63.0 inches.

Step-by-step explanation:

To solve this question, we have 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 random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation, which is also called standard error In this problem, we have that: Find the probability that they have a mean height greater than 63.0 inches.

This is 1 subtracted by the pvalue of Z when X = 63. So By the Central Limit Theorem    has a pvalue of 0.0082

1 – 0.0082 = 0.9918

0.9918 = 99.18% probability that they have a mean height greater than 63.0 inches.