Problem 1
SmithCo is a supplier to many brand-name makers of mobile phones. SmithCo manufactures the external shells that enclose mobile phones. The strength of a shell is measured by applying increasing pressure to the shell and recording the pressure at which the shell breaks. Shell strengths are approximately normally distributed. A random sample of 10 shells made by SmithCo showed an average strength of 45.0 pounds and a standard deviation of 2.0 pounds. Use StatTools, or conduct calculations by hand, to construct a 90% confidence interval for the mean shell strength. The appropriate multiplier is 1.833.
State the confidence interval in the form a ± b.
State the confidence interval in the form [c, d].
Problem 2
A consumer products firm has recently introduced a new brand. The firm would like to estimate the proportion of people in its target market segment who are aware of the new brand. As part of a larger market research study, it was found that in a sample of 125 randomly selected individuals from the target market segment, 84 individuals were aware of the firm’s new brand.
a.Construct a 95% confidence interval for the proportion of individuals in the target market segment who are aware of the firm’s new brand. If you use Excel and/or StatTools, please specify any functions you use and all the inputs.
b.The manager in charge of the new brand has stated that the brand awareness is greater than .75, meaning that more than 75% of the population is aware of the brand. He would like to use hypothesis testing to prove his claim. At the 5% significance level, conduct a hypothesis test with the goal of proving his claim. In particular
i) specify the null hypothesis and the alternative hypothesis,
ii) state whether you are using a one- or two-tailed test,
iii) specify the p-value of the test, and
iv) provide the results of the test in “plain English”.
Hints: 1) Think carefully about what is the null hypothesis, and what is the alternative. 2) If you want to use StatTools for the analysis, you need to create the survey data in Excel first; then you can run the analysis.
For future polls, the firm is interested in minimizing their marketing-research costs. The margin of error (MOE) in their pools should be no larger than B (i.e., ± 100B percentage points). To simplify the analysis, we will assume their marketing study only has a single question that is used to estimate p, at the 5% significance level.
a.If the firm had no prior knowledge of p, how large would the sample size have to be to ensure MOE ≤ B? (Hint: your answer will be a formula containing B).
b.The firm has prior knowledge that p will likely be somewhere between .10 and .20 (that is between 10% and 20%). How large should the sample be if they would like to ensure that MOE (=B) is no larger than 0.01 or 1%?
Problem 3
Financial analysts specializing in credit markets are often interested in creating models to predict whether a firm will go bankrupt within some fixed period of time. If there is a good chance that a particular firm will go bankrupt, then the firm will have to pay a very high interest rate on any debt (bonds) that it may issue.
In practice, statistical models to predict bankruptcy are fairly difficult to construct. One of the variables that may be useful in distinguishing between firms that go bankrupt and firms that stay solvent is the return on assets (ROA). The accompanying file Bankruptcy.xls (Links to an external site.) contains financial data on 44 firms. Of these 44 firms, 20 firms went bankrupt within 1 year after the data were collected; the other 24 firms remained solvent after 1 year. For this assignment, ignore all financial measures other than ROA.
As a first step, unstack the ROA variable (see Data Utilities menu in StatTools) using “Bankrupt” as the code variable (in the menu, set Bankrupt as the “cat” variable and ROA as the “Val”). Now you should have two columns of ROA data in a new worksheet. For the purpose of this exercise we will assume that firms’ ROA is, in general, normally distributed.
Construct a 90% confidence interval for the average ROA of firms that remained solvent.
Attach the StatTools output as Exhibit A.
State the confidence interval in the form: point estimate ± margin of error.
Test the hypothesis that the average ROA of firms that went bankrupt is less than -5%.
Attach the StatTools output as Exhibit B.
State the p-value for this hypothesis test.
At the 10% level of significance, is the average ROA of firms that went bankrupt less than -5%? Yes or No
