STAT 235 Homework 9 Problems

1. The mean drying time of a brand of spray paint is known to be 90 seconds. The research division of the company that produces this paint contemplates that adding a new chemical ingredient to the paint will accelerate the drying process. To investigate this conjecture, the paint with the chemical additive is sprayed on 15 surfaces and the drying times are recorded. The mean and standard deviation computed from these measurements are 86 and 4.5 seconds, respectively. Assume the measurements constitute a random sample from a normal population. Answer the following questions.
(a) Define μ in the context of this problem.
(b) What is the point estimate for the mean drying time of the spray paint with the new chemical ingredient?
(c) Construct a 98% confidence interval for the mean drying time of the paint with the chemical additive. Is μ included in this interval?
(d) Do these data provide strong evidence that the mean drying time is reduced by the addition of the new chemical? Perform the hypothesis testing (at the significance level α = .02) by answering the questions arranged in the following.
i. Formulate the null and alternative hypotheses.
ii. State the Type I error and Type II error in the context of the problem and tell why Type I error is
more serious.
iii. State the test statistic and its distribution.
iv. Determine the critical value for α = .025 and state the decision rule.
v. Calculate the observed value of the test statistic from the data.
vi. State whether H0 is rejected and tell why.

2. Rural and urban students are to be compared on the basis of their scores on a nationwide musical aptitude test. Two random sample of sizes 90 and 100 are selected from rural and urban seventh grade students. The summary statistics from the test scores are
Sample size
Mean
Standard deviation
Rural Urban
90 100 76.4 81.2 8.2 7.6
Answer the following questions.
(a) Define μ1 and μ2 in the context of the problem.
(b) What is the point estimate for μ1 − μ2?
(c) Construct a 97% confidence interval for μ1 − μ2. Is μ included in this interval?
(d) Do these data provide strong evidence that there is a difference in population mean scores between urban and rural students? Perform the hypothesis testing (at the significance level α = .01) by answering the questions arranged in the following.
i. Formulate the null and alternative hypotheses. ii. State the test statistic and its distribution.
iii. Determine the critical value for α = .01 and state the decision rule. iv. Calculate the observed value of the test statistic from the data.
v. State whether H0 is rejected and tell why.