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Practice Midterm Exam; Stat 431
Instructions : Closed Book. The “Facts and Formulas” sheets will be included with the exam. Calculators are allowed for numerical calculations. On the regular exam you will be asked to write answers on the test pages along with your work. (Space, and additional pages if necessary, will be allowed for this.) When performing hypothesis tests, clearly state the null and alternative hypotheses and show the critical value and/or P-value (from the table) where appropriate. Normal and t tables will be provided. Time = 90 minutes.
(24) 1. In any bottling process a manufacturer will lose money if the bottles contain either more or less than is claimed on the label. Suppose a quality manager for a ketchup company is interested in testing whether the mean number of ounces per family size bottle differs from the labeled amount of 24 ounces. The manager samples 90 bottles, measures the weight of their contents, and finds that x = 23. 96 and s =. 32.
i) Does the sample evidence indicate that the ketchup dispensing machine needs adjustment? State null and alternative hypotheses and test at α = .05. ii) Find a 99% confidence interval for the mean number of ounces of ketchup being dispensed. iii) Bottles having net weight less than 23.76 ounces are more than 1% underweight, and hence can be cited by the FDA as being mislabeled. Using the data provided, give the best available estimate of the proportion of ketchup bottles that will fail to meet the FDA standard of weighing at least 23.76 ounces.
(25) 2. Wansley, Roenfeldt, and Cooley (1983) compared the profiles of a sample of 8 firms that merged during 1975-1976 with those of a separate sample of 12 firms that did not merge. They wished to compare the logarithms of the firms’ price-to-earnings ratios. Selected JMP plots and tables for the Log(Price/Earnings) are given below. (These firms may be considered a random sample of all firms during the 70s. In answering questions i) - iv) below you may assume the relevant populations are satisfactorily normal.)
Oneway Analysis of Log(Price/Earnings) By Type
Log(Price/Earnings)
-1.
-0.
0
1
2
MERGED NONMERGED
Type
Means and Std Deviations
Level Number Mean Std Dev Std Err Mean
MERGED 8 0.441 1.158 0.
NONMERGED 12 0.942 0.655 0.
i) Give a 95% confidence interval for the mean Log(Price/Earnings) of all Merged Firms.
ii) Construct a similar 95% confidence interval for the mean Log(Price/Earnings) of all Nonmerged Firms.
iii) Consider testing
H 0 : μMerged = μNonmerged versus Ha: μMerged ≠ μNonmerged
Does a test at α = .05 reject H 0 or fail to reject H 0? Show your work or explain your reasoning. (Here μMerged denotes the population mean of the value of Log(Price/Earnings) for the population of all Merged Firms, and similarly for μNonmerged.)
iv) Give the approximate P-value for the test in part iii). (Make the best statement you can based on the t-tables in the text.)
(12) 3. A randomized experiment was performed to test the food value of a new variety of high-lysine corn. 20 sibling-pairs of one-day-old chicks were chosen for the experiment. (The two members of each pair come from the same “mother” and “father” chickens.) One member of each pair was fed a diet based on the new corn; the other member of the pair was fed a similar diet based on normal corn. The data are the weight gains in grams after 21 days. Here are the summary data for the chicks fed the Control (=normal) and Experimental diets, and for the difference in weights between the control and experimental chick in each sibling pair:
N Mean StDev Control 20 362.8 50. Experimental 20 392.9 42. Difference 20 omitted 46.
i) Use this data to construct the most appropriate test at level alpha = .01 to discover
Ketchup costs the producer 1¢ per ounce and empty bottles cost 5¢ each. Filled bottles
that pass the weight test can be sold for 50¢ each. Using the available data, what is the best
estimate for the profit the manufacturer will make on each 1000 bottles produced?
{Note: This problem is different from any we’ve done in class. It involves some principles you should be carrying forward from Stat 430, or equivalent. This illustrates that you may expect to see something on the actual exam that goes beyond what we’ve explicitly covered in class. Of course, the actual exam won’t contain this problem – or one just like it.}
