# mathematics

The senior management at Canine Kernels Company (CKC) is concerned with the existing capacity limitation, so they want to accept the mix of orders that maximizes the company’s profits. Traditionally, CKC has utilized a method whereby decisions are made to produce as much of the product with the highest contribution margin as possible (up to the limit of its demand), followed by the next highest contribution margin product, and so on until no more capacity is available. Because capacity is limited, choosing the proper product mix is crucial. Troy Hendrix, the newly hired production supervisor, is an avid follower of the theory of constraints philosophy and the bottleneck method for scheduling. He believes that profitability can indeed be approved if bottleneck resources are exploited to determine the product mix. a. What is the profit if the traditional contribution margin method is used for determining CKC’s product mix? b. What is the profit if the bottleneck method advocated by Troy is used for selecting the product mix? c. Calculate the profit gain, both in absolute dollars as well as in terms of percentage gains, by using TOC principles for determining product mix.

For all hypothesis test and confidence interval questions state any assumptions needed for your tests and C.I.’s to be valid.
1. The amount of shaft wear(.0001 in.) after a fixed mileage was determined for each of n = 8 internal combustion engines having copper lead as a bearing material, resulting in an average of 3.72.
Assuming that the distribution is normal with a known standard deviation of 1.25
a) Test at significance level α = 0.05, whether there is sufficient evidence to conclude that mean shaft wear is more than 3.50.
b) What is the p-value of the test?
c) What is the probability of a Type II error if the true mean shaft wear is actually 4.00?
d) What is the power of the test if the true mean shaft wear is 4.00?

2. Many college and university professors have been accused of grade inflation over the past several years. If grade inflation has occurred, the grade-point average of today’s students should exceed the mean of 10 years ago. Based on the following data
Present 10 Years Ago
y 1 = 3.04 y 2 = 2.82
s
2
1 = 0.38 s
2
2 = 0.43
n1 = 75 n2 = 60
a) Test the grade inflation theory using a level of significance of 0.05.
b) What is the p-value of the test?
c) Find a 99% confidence interval estimate for the average GPA of today’s students.
d) What is the interpretation of the confidence interval in part (c)?

3. A marketing analyst for a company which distributes decaffeinated coffee claims that the proportion of coffee drinkers who drink just decaffeinated coffee is greater than 0.2. A random sample of 1000 coffee drinkers contained 190 that drink just decaffeinated coffee.
a) Does this sample provide sufficient evidence to refute the company’s claim? Use α = 0.10.
b) What is the probability of not rejecting the null hypothesis if the true proportion of those who just drink decaffeinated coffee is 0.195?

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