Statistics: Testing the Null Hypothesis

Please show all calculations on every problem.

1. A machine produces 3-inch nails. A sample of 12 nails was selected and the lengths determined. The results are as follows:
2.89 2.95 3.00 3.05 2.99 2.96 3.10 3.06 3.00 3.12 3.00 2.95

Use these results to test H0: u = 3 and Ha: u does not = 3 at alpha = 0.05. Give the critical region, the computed test statistic, and the conclusion.

2. A sample of size n = 20 is selected from a normal population to construct a 95% confidence interval estimate for a population mean. The interval was computed to be (8.20 to 9.80). Determine the sample standard deviation.

3. A random sample of 46 observations was selected from a normally distributed population. The sample mean was = 81, and the sample variance was s^2 = 35.0. Does the sample show sufficient reason to conclude that the population standard deviation is not equal to 7 at the 0.05 level of significance? Use the p-value method.

4. An insurance company states that 75% of its claims are settled within 5 weeks. A consumer group selected a random sample of 50 of the company's claims and found 35 of the claims were settled within 5 weeks. Is there enough evidence to support the consumer group's claim that fewer than 75% of the claims were settled within 5 weeks? Test using the traditional approach with alpha = 0.05.

5. A teacher wishes to compare two different groups of students with respect to their mean time to complete a standardized test. The time required is determined for each group. The data summary is given below. Test the claim at alpha = 0.05, that there is no difference in variance. Give the critical region, test statistic value, and conclusion for the F test.
n1=60, s1=24
n2=120, s2=28
alpha = 0.05

6. A machine produces 9 inch latex gloves. A sample of 80 gloves is selected, and it is found that 20 are shorter than they should be. Find the 99% confidence interval on the proportion of all such gloves that are shorter than 9 inches.

7. The pulse rates below were recorded over a 30-second time period, both before and after a physical fitness regimen. The data is shown below for 10 randomly selected participants. Is there sufficient evidence to conclude that a significant amount of improvement took place? Assume pulse rates are normally distributed. Test using alpha = 0.05.

Before: 31 25 30 31 33 25 32 45
After: 32 29 30 37 40 35 34 52

8. Determine the p-value for each of the following hypothesis-testing situations:
a. H0: p = 0.30 and Ha: p does not = 0.30; z test value = 1.65
b. H0: u >/= 30 and Ha: u < 30; t test value = - 1.7 d.f. = 15