Hypothesis Test:
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 15.5 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 18 water samples from the community and calculate a mean lead concentration of 14.85 parts per billion. Lead concentrations in the community are known to follow a normal distribution wtih standard deviation 2.6 parts per billion.
(a) What are they hypotheses for the appropriate test of significance? Choose one:
- H0: X BAR = 15.5 vs. Ha: X BAR > 15.5
- H0: u = 14.85 vs. Ha: u > 14.85
- H0: u = 14.85 vs. Ha: u < 14.85
- H0: X BAR = 14.85 vs. Ha: X BAR > 14.85
- H0: X BAR = 15.5 vs. Ha: X BAR < 15.5
- H0: u = 15.5 vs. Ha: u <15.5
- H0: X BAR = 14.85 vs. Ha: X BAR < 14.85
H0: u = 15.5 vs. Ha: u > 15.5
(b) The value of the test statistic (to two decimal places) is z = _____
(c) The P-value of the test (to four decimal places) is _____
(d) What is the appropriate conclusion for this test? Choose one:
- Fail to reject the null hypothesis. There is sufficient evidence that the water is safe.
- Reject the null hypothesis. There is insufficient evidence that the water is unsafe.
- Fail to reject the null hypothesis. There is sufficient evidence that the water is unsafe.
- Fail to reject the null hypothesis. There is insufficient evidence that the water is safe.
- Reject the null hypothesis. There is insufficient evidence that the water is safe.
- Fail to reject the null hypothesis. There is insufficient evidence that the water is unsafe.
- Reject the null hypothesis. There is sufficient evidence that the water is safe.
- Reject the null hypothesis. There is sufficient evidence that the water is unsafe.
A city's fire department would like to conduct a hypothesis test at the 10% level of significance to determine if their mean response time is greater than the target time of 5 minutes. A random sample of 48 responses is timed, resulting in a mean of 6.53 minutes. Response times are known to follow some right-skewed distribution with standard deviation 3.76 minutes.
(e) 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, according to the Central Limit Theorem. Please choose one:
- the normal distribution can always be used when conducting inference for u.
- since n is high, the population distribution of X is exactly normal.
- since n is high, the sampling distribution of X BAR is approximately normal.
- since n is high, the population distribution of X is approximately normal.
- since n is high, the sampling distribution of X BAR is exactly normal.
(f) The hypotheses for the appropriate test of significance are, choose one:
- H0: u = 6.53 vs. Ha: u cannot equal 6.53
- H0: u = 5 vs. Ha: u cannot equal 5
- H0: u = 6.53 vs. Ha: u > 6.53
- H0: u = 5 vs. Ha: u > 5
- H0: u < 6.53 vs. Ha: u > 6.53
- H0: u = 6.53 vs. Ha: u < 6.53
- H0: u < 5 vs. Ha: u > 5
- H0: u = 5 vs. Ha: u < 5
(g) The test statistic (to two decimal places) is _____
(h) The P-value of the test (to four decimal places) is _____
(i) What is the correct conclusion of the test? Choose one:
- Reject H0. We have sufficient evidence that the true mean response time is equal to 5 minutes.
- Reject H0. We have insufficient evidence that the true mean response time is greater than 5 minutes.
- Reject H0. We have sufficient evidence that the true mean response time is greater than 5 minutes.
- Fail to reject H0. We have insufficient evidence that the true mean response time is equal to 5 minutes.
- Fail to reject H0. We have sufficient evidence that the true mean response time is greater than 5 minutes.
- Fail to reject H0. We have sufficient evidence that the true mean response time is equal to 5 minutes.
- Fail to reject H0. We have insufficient evidence that the true mean response time is greater than 5 minutes.
- Reject H0. We have insufficient evidence that the true mean response time is equal to 5 minutes.
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.
(i) Construct a 95% confidence interval for the true mean fill pressure for this machine. Round your answers to two decimal places.
(j) Provide an interpretation of the confidence interval in (a).