Question 1

a) An analyst remarked, "When the sample size is large, the sample mean is as likely to lie above the population mean as below it." Do you agree? Explain.

b) The trees in a timber tract form the population of interest to a forester. Suppose that unknown to the forester, the mean growth of these trees during the previous decade, as measured by their last 10 annual growth rings, is µ=8.38 centimetres. A random sample of four trees from the tract yields growth measurements of 8.76, 7.93, 7.88 and 8.55 centimetres.

i. The sample mean is to be used to estimate the population mean.

- What is the value of the parameter being estimated?
- What is the estimator here? To what does the sampling distribution of the estimator refer here?
- What is the value of the estimate here? What is the magnitude of the sampling error in this estimate?

ii. The sample median and the sample midrange are alternative estimators of the population mean.

- What are their values for this sample?
- Which estimate - mean, median, or midrange - has the smallest sampling error for this sample?
- Would this estimator necessarily have the smallest sampling error for another sample?

Question 2

a) From the information given, indicate if a correct decision, a Type I error or a Type II error was made.

i. H₀ : µ = 1.5 litres. The decision was to not reject H0 and µ is actually 1.5 litres.

ii. H₀ : µ = 1.5 litres. The decision was to reject H0 and µ is actually 1.5 litres.

iii. H₀ : µ = 1.5 litres. The decision was to reject H0 and µ is actually 1.6 litres.

b) A Gallop survey found the mean charitable contribution on federal tax returns was $1075. Assume a sample of 2012 tax returns is used to conduct a hypothesis test to determine whether any change occurred in the mean charitable contributions.

i. Formulate the null and alternative hypotheses.

ii. Using α = 0.05, what is the critical value for the test statistic? State the rejection rule.

iii. Assume that a sample of 200 tax returns shows a sample mean of $1160 and a sample standard deviation of $840. What is the value of the test statistic?

iv. Using the p-value what is your conclusion?

Question 3

a) Distinguish between regression analysis and correlation analysis

b) The management of a building society in Darwin would like to know how house sales vary with interest rates. The following table shows the average home mortgage interest rates and corresponding number of house sales within the community during ten randomly selected months.

Number of houses sold         Interest rate
          ('00) (Y)                 (average %) (X)

             11                             14
             23                             12
             38                             11.5
             45                             10.5
              9                              15
             30                             11
             16                             13
             18                             13.5
             39                             11
             41                             11.9

          ΣY = 270      ΣX = 123.4 ΣXY = 3172.4
          ΣY² = 8882   ΣX²= 1542.36 

You have been employed by the Society's management to examine the relationship between house sales and interest rates using regression analysis. Carry out an appropriate analysis and write a report to management stating and interpreting your findings, and indicating any limitations of the analysis.

c) How many houses would be sold if the interest rate was 5 percent per annum? Explain any reservations you might have about this prediction (use 95 percent confidence level).

Question 4

a) Distinguish between an estimator and an estimate. State and briefly explain three properties of a reliable estimator.

b) An airline researcher studied reservation records for a random sample of 100 days to estimate µ, the mean number of no-shows (persons who fail to keep their reservations) on the daily 4pm commuter flight to Canberra. The records revealed the following:

Using the above data, construct a 99 percent confidence interval for µ. Interpret the confidence interval. Would a different random sample of 100 days have provided the same interval estimate as the one you just constructed? Explain.

c) Find the sample size needed to estimate the population mean to within 1/5 of a standard deviation with 99 percent confidence.

Question 5

a) Describe a normal distribution. What two parameters determine its location and shape?

b) Consider the family of normal probability distributions.

i. Identify the probability distributions denoted by N(20,9), N(0,9), N(5,1).

ii. For each of the normal random variables in part (i.) obtain expression for the standardised normal variable Z.

c) NT Trucking Company determined that on an annual basis the distance travelled per truck is normally distributed with the mean of 100,000 kilometres and a standard deviation of 20,000 kilometres.

i. What proportion of trucks can be expected to travel between 80,000 and 120,000 kilometres in the year?

ii. What percentage of trucks can be expected to travel either below 60,000 or above 140,000 kilometres in the year?

iii. How many kilometres will be travelled by at least 80 percent of the trucks?