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# Statistics Problems

1. A machine produces 5-inch nails. A sample of 12 nails was selected and their lengths determined. The results are as follows:
4.85 4.80 4.89 4.82 4.94 4.96 4.96 4.83 4.80 4.84 4.95 4.88
Assuming that alpha = 0.10, test the hypothesis that the population mean is equal to 5.
i) State the null and alternate hypotheses
ii) Calculate the mean and standard deviation
iii) Determine which test statistic applies, and calculate it
iv) Determine the critical value(s).
v) State your decision: Should the null hypothesis be rejected?

2. A sample of size n = 20 is selected from a normal population to construct a 98% confidence interval estimate for a population mean. The interval was computed to be (1.30 to 3.00). Determine the sample standard deviation.

3. A random sample of 51 observations was selected from a normally distributed population. The sample mean was x = 79, and the sample variance was s2 = 29.0. Does the sample show sufficient reason to conclude that the population standard deviation is not equal to 8 at the 0.05 level of significance? Use the p-value method. State the null and alternate hypotheses. Determine which test statistic applies, and calculate it. Determine the corresponding probability, and compare to alpha. State your decision: Should the null hypothesis be rejected?

4. An insurance company states that 85% of its claims are settled within 5 weeks. A consumer group selected a random sample of 90 of the company's claims and found 46 of the claims were settled within 5 weeks. Is there enough evidence to support the consumer group's claim that fewer than 85% of the claims were settled within 5 weeks? Test using the traditional approach with alpha = 0.02.

i) State the null and alternate hypotheses
ii) Calculate the sample proportion
iii) Determine which test statistic applies, and calculate it
iv) Determine the critical value(s).
v) State your decision: Should the null hypothesis be rejected?

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 = 0.10, that there is no difference in variance. Give the critical region, test statistic value, and conclusion for the F test.
n1 = 40 s1 = 34
n2 = 60 s2 = 41
alpha = 0.10

i) State the null and alternate hypothesis
ii) Determine which test statistic applies, and calculate it
iii) Determine the critical region
iv) State your decision: Should the null hypothesis be rejected?

6. A machine produces 9 inch latex gloves. A sample of 17 gloves is selected, and it is found that 5 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 8 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 = 0.10.

i) State the null and alternate hypotheses.
ii) Calculate the mean and standard deviation.
iii) Determine which test statistic applies, and calculate it and determine the critical value(s).
iv) State your decision: Should the null hypothesis be rejected?

Before 31 59 33 33 40 37 48 56
After 31 67 37 34 45 41 56 57

#### Solution Summary

This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Statistics and provides students with a clear perspective of these topics.

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