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?
4. The marketing manager of National Hardware is conducting an economic feasibility study for construction of a new store in a rapidly expanding suburban neighbourhood. A sample of 150 families, randomly selected from the area, yielded a mean family income of $55,000 with a standard deviation of $1000.
a) Find a 90% confidence interval estimate for the true average family income of all families in the potential market area.
b) If it is desired to be 95% confident that the sample mean is within $100 of the true population mean income of all families in the area, how many additional families should be surveyed?
5. A 99% confidence interval for mean battery life (in hours) of a certain type of battery, based on a random sample of size 60 is (150, 160).
a) Which of the following is a correct interpretation of this interval?
I) A battery will last between 150 and 160 hours 99% of the time.
II) The sample tested suggests average battery life is between 150 and 160 hours.
In repeated sampling, 99% of all the intervals constructed would bracket the true average battery life.
III) 99% of the batteries in the population have lives between 150 and 160 hours.
b) If the sample size had been 80 instead of 60, would the 99% confidence for mean battery life have been wider or narrower than (150, 160)?
6. Let X1 ,....,Xn be a random sample from the exponential distribution with
1 f(x : ) = - x/θ
θ
θ .
a) Find the maximum likelihood estimator for θ .
b) What is the maximum likelihood estimator of VAR(X) = θ2 ?
c) Is this MLE for θ2 an unbiased estimator of θ2 ? If not find a function of it that is unbiased.
d) Find the MVUE of VAR(X).
e) Find the MME for θ .
7. The desired percentage of SiO2 in a certain type of aluminous cement is 5.5. To test whether the true mean percentage is 5.5 for a particular production facility, 16 independently obtained samples were analyzed and gave a mean of 5.25 and a standard deviation of 0.3 Does this sample provide sufficient evidence at the 0.01 level of significance, to conclude that the true average percentage is not 5.5?