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# Test hypotheses and sample size requirement

1. A study was conducted to estimate the mean amount spent on birthday gifts for a typical family having two children. A sample of 150 was taken, and the mean amount spent was \$225. Assuming a standard deviation equal to \$50, find the 95% confidence interval form, the mean for all such families.

2. In testing the hypothesis, HO: m greater or equal to 28.7 and Ha: m> 28.7, using the p-value approach, a p-value of 0.0764 was obtained. If alpha = 9.8, find the sample mean which produced this p-value given that the sample of size n= 40 randomly selected.

3. To test the null hypothesis that the average lifetime for a particular brand of bulb is 750 hours versus the alternative that the average lifetime is different from 750 hours, a sample of 75 bulbs is used. If the standard deviation is 50 hours and alpha is equal to 0.01, what values for x will result in rejection of the null hypothesis.

4. By measuring the amount of time it takes a component of a product to move from one workstation to the next, and engineer has estimate that the standard deviation is 4.5 seconds. a. How many measurements should be made in order to be 95% certain that the maximum error estimation will not exceed 1 second? b. What sample size is required for a maximum error of 2 seconds?

5. Determine the critical region and critical values for z that would be used to test the null hypothesis at the given level of significance, as described in each of the following:

a. HO:m=25 and Ha: m not equal 25, alpha = 0.10
b. Ho: m< or equal 32 and Ha:m> 32, alpha =0.01
c. Ho: m > or equal 13 and Ha:m< 13, alpha = 0.05

#### Solution Preview

Please see the attached file for details and missing symbols/formula.

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1. A study was conducted to estimate the mean amount spent on birthday gifts for a typical family having two children. A sample of 150 was taken, and the mena amount spent was \$225. Assuming a standard deviation equal to \$50, find the 95% confidence interval form, the mean for all such families.

Solution. Given . For a 95% confidence level, z=1.96. So, by a formula, we know that the 95% confidence interval form, the mean for all such families is

=

=(217.00, 233.00)

2. In testing the hypothesis, HO: m greater or equal to 28.7 and Ha: m> 28.7, using the p-value approach, a p-value of 0.0764 was obtained. If alpha = 9.8, find the sample mean which produced this p-value given that the sample of size n= 40 randomly selected. ...

#### Solution Summary

The solution provides step by step method for calculating confidence intervals and proving the hypothesis tests. The formula for the calculation and interpretations of the results are also included in an attached Word file.

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