# Central Limit Theorem and single sample hypothesis test

A spokesman for a popular television game show claims that contestants on the show win an average of $1200. In a random sample, 35 contestants were questioned on the amount of money they had won in order to test the hypothesis.

The null hypothesis or Ho: the population mean equals $1200,

The alternative Hypothesis or Ha: the population mean is not equal to $1200.

A. Suppose the mean win for the sample of 35 contestants turns out to be $1045. Assuming the null hypothesis is true, what is the probability of obtaining a sample mean value of $1045 or more? (Use 325 as the population standard deviation.)

B. Does a sample mean of $1045 provide evidence that the null hypothesis is false? Explain your answer.

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#### Solution Preview

This problem deals with the distribution of the sample mean. I have attached a Word document explaining how the sample mean is distributed according to the Central Limit Theorem. Once the distribution of the sample mean is determined, the probability of interest can be calculated using normal probability tables. ...

#### Solution Summary

In this problem, we use the central limit theorem to find the distribution of a sample mean. Then we use that distribution to find the probability that the sample mean from a random sample exceeds a particular value. The probability we find is converted into the p-value then used to test the hypothesis of interest.