EC 112 SAMPLING, ESTIMATION AND HYPOTHESIS TESTING. One and One Half Hours (1 1 2 Hours) TWO questions should be answered

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1 ECONOMICS EC 112 SAMPLING, ESTIMATION AND HYPOTHESIS TESTING One and One Half Hours (1 1 2 Hours) TWO questions should be answered Statistical tables are provided. The approved calculator (that is, Casio FX82) may be used. 1. (a) What are the properties of the sampling distribution of the sample mean when the population is normal? Give justification to each property. What is the effect of the Central Limit theorem in relation to the distribution of the sample mean when sampling from non-normal populations? [20%] The mean life of a certain saw blade is 41.5 hours, with a standard deviation of 2.5 hours. What is the probability that a simple random sample of size 50 drawn from this population has a mean larger than 42 hours?

2 2. (a) Define and explain what is meant by maximum likelihood estimation. [20%] Let X have an exponential distribution with the following density function f(x) = θ e - θ x for x > 0 Suppose that a random sample of size 8 selected from this distribution gave values 0.2, 0.5, 0.3, 0.2, 0.4, 0.4, 0.6, 0.7. Find and interpret the estimate of parameter θ applying the maximum likelihood rule. [50%] What are the statistical properties of maximum likelihood estimators and prove one of them for the estimator found in part. [30%] 3. (a) State and explain the desirable properties of a point estimator. [20%] Derive an unbiased estimator of the variance of a normal population. An investigator is interested in the distribution of exam marks of students on a statistics course. She draws 4 samples, each sample containing 10 students. The mean marks were 75, 78, 82, 77 for samples 1 to 4 respectively. Use the estimator derived in to estimate the population variance.

3 4. (a) State the probabalistic and practical interpretation of a confidence interval. [25%] Each of a random sample of 9 cars of a certain make was test driven. The number of miles travelled for each gallon of petrol was recorded for each. The results were as follows: 22, 26, 23, 25, 29, 22, 24, 26, 21 Construct and interpret a 90% confidence interval for the population mean. Explain whether a 95% confidence interval for the population mean would be wider than or narrower than, or the same as that obtained in part. [15%] (d) Does any general relationship exist between the sample size and the length of confidence intervals for the population mean, variance and proportion? [20%]

4 5. (a) Explain the meaning of level of significance and rejection region. What kind of relationship exists between them in hypothesis testing? [20%] In conducting a survey of food prices, two samples of prices of a given food item were collected. Sample I came from a congested city area, and sample II was obtained in the suburbs. The results were: Sample I Sample II n n p i i= 1 1 n 2 p i n i= where p i stands for price recorded in the ith store. (i) At the 10% significance level, test the hypothesis that there is no difference between the mean price of the particular food item in the two areas. (ii) At the 5% significance level, test the hypothesis that the dispersion of prices in the city is smaller than in the suburbs.

5 6. (a) Explain the difference between parametric and non-parametric tests. Give examples of tests belonging to each group, specifying clearly the tested hypothesis. [25%] The following table shows, for independent random samples of men and women, the numbers who watch TV for more or less than 3 hours a day. Less than 3 hours 3 hours or more watched watched Men Women (i) Test at the 5% significance level the hypothesis of no relationship between a person s sex and the amount of TV watched. (ii) At the same significance level test the hypothesis that the proportion of population members watching TV for 3 hours or more a day exceeds 40%. [35%] 7. The following are the scores which 12 students obtained in the mid-term examination (score x) and in the final examination (score y) in a course in statistics: x: y: Compute and interpret the parameters of the regression equation: y = a + bx and predict the final examination score of a student who received a 84 in the mid-term examination.

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