Assume that the heights of men are normally distributed with a mean of "71.3" inches and a standard deviation of 2.1 inches. If 36 men are randomly​ selected, find the probability that they have a mean height greater than 72.3 inches. Round to four decimal places.

0.0021 = 0.21% probability that they have a mean height greater than 72.3 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 normal distributions can be solved using the z-score formula.

In a set with mean and standard deviation , the z-score 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 p-value, 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.

Assume that the heights of men are normally distributed with a mean of "71.3" inches and a standard deviation of 2.1 inches.

This means that

Sample of 36:

This means that

Find the probability that they have a mean height greater than 72.3 inches.

This is 1 subtracted by the pvalue of Z when X = 72.3. So

By the Central Limit Theorem

has a pvalue of 0.9979

1 - 0.9979 = 0.0021

0.0021 = 0.21% probability that they have a mean height greater than 72.3 inches.

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step-by-step explanation:

alright so i think for this one it would to draw it out. this may be hard for me to try and draw, but i will do my best on here:

        r

  /       |     \

notice how it forms 2 triangles inside one bigger triangle.

so if we know the angle spr is the same as psr that is saying the measure of angle p is the same as angle s. this means the big triangle psr is either isosceles or equilateral. in both cases, the triangle is perfectly symmetrical when split down the middle, which is in this case rq. because we know it is perfectly symmetrical on both sides, we know that the two triangles inside the bigger one are equal.