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

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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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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.

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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 ...

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