# mathematics

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?