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?

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