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Calculating the probability that a mean cost will be less than a specified number given standard deviation.

The average cost of XYZ brand running shoes is $83 per pair, with the standard deviation of $8.00. If 9 pairs of running shoes are selected, find the probability that the mean cost of a pair of shoes will be less than $80. Assume the variable is normally distributed.

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We know that the cost of a pair a running shoes is a random variable that is normally distributed with a mean of $83 and a standard deviation (SD) of $8. Therefore, we know that the variance is SD^2=64. So we will call X to the cost of a pair of shoes, and say that
X ~ N(83,64) (normally distributed with mean 83 and variance 64)

The mean of the cost of 9 pairs of shoes will also be a random variable. We have to find how it is distributed. The mean is defined in this case as:

(X1 + X2 + X3 + ... + X9)/9

where X1,...,X9 is the cost of the first pair through the ninth pair of shoes. We know that X1,...,X9 are all N(83,64), and are independent of each ...

Solution Summary

The solution discusses the probability that the mean cost of a pair of shoes is less than $80.00. It uses explanation, calculations and sources to determine the solution.