1. The Environmental Protection Agency (EPA) warns communities when their tap water is contaminated with too much lead. Drinking water is considered unsafe if the mean concentration of lead is 15.1 parts per billion or greater. The EPA would like to conduct a hypothesis test at the 10% level of significance to determine whether there is significant evidence that the tap water in one particular community is safe. They randomly select 26 water samples from the community and calculate a mean lead concentration of 14.67 parts per billion. Lead concentrations in the community are known to follow a normal distribution wtih standard deviation 2.41 parts per billion.
(a) What are the hypotheses for the appropriate test of significance?
(b) What is the value of the test statistic (to two decimal places)?
(c) What is the P-value of the test (to four decimal places)?
(d) What is the appropriate conclusion for this test?
2. A city's fire department would like to conduct a hypothesis test at the 1% level of significance to determine if their mean response time is greater than the target time of 7 minutes. A random sample of 49 responses is timed, resulting in a mean of 8.4 minutes. Response times are known to follow some right-skewed distribution with standard deviation 3.57 minutes.
(a) Despite the fact that response times do not follow a normal distribution, it is still appropriate to use inference methods which rely on the assumption of normality. This is because, of the Central Limit Theorem What does the Central Limit Theorem states?
(b) What are the hypotheses for the appropriate test of significance?
(c) What is the test statistic (to two decimal places)?
(d) What is the P-value of the test (to four decimal places)?
(e) What is the correct conclusion of the test?
3. A machine is designed to fill automobile tires to a mean air pressure of 30 pounds per square inch (psi). The manufacturer tests the machine on a random sample of 12 tires. The air pressures for these tires are shown below:
30.7 31.0 29.5 30.4 31.6 28.6 32.2 29.6 29.4 31.9 30.3 30.8
Fill pressures for the machine are known to follow a normal distribution with standard deviation 1.2 psi.
(a) Construct a 95% confidence interval for the true mean fill pressure for this machine. Round your answers to two decimal places.
(b) Provide an interpretation of the confidence interval in (a).
(c) Use JMP to help you conduct a hypothesis test at the 5% level of significance to determine whether the true mean fill pressure for the machine differs from 30 psi. To get the appropriate output in JMP, enter the data in a column titled Fill Pressure. Go to Analyze > Distribution and select Fill Pressure as Y. Click OK. Under the red arrow, select Test Mean. Enter 30 for the hypothesized mean and 1.2 for the true standard deviation. Click OK. Use the results from the output to conduct an appropriate hypothesis test. Show all of your steps, but you do not need to show the calculation of the test statistic and the P-value given in JMP. Just report their values in the test.
(d) Provide an interpretation of the P-value of the test in (c).
(e) Could you have used the confidence interval in (a) to conduct the test in (c)? If no, explain why not. If yes, explain why, and explain what your conclusion would have been and why.