640:481 - Mathematical Theory of Statistics
Review Problems - Final
Chapters: 7-13
1. (a) Suppose that X1 , . . . , Xn are independent random variables having chi-square distribution with 1 degree of freedom. Show that X1 + · · · + Xn has chi-square distribution with n degrees of freedom.
(b) Suppose that Y is a random variable having gamma distribution with parameters α and β, and that c is a positive number. Identify the distribution of cY as some known distribution.
2. A certain machine can ill soft drink cans with a (programmable) mean ill of µ luid ounces; the amount put in each can is an independent random variable with standard deviation 0 .1 luid ounce.
(a) The company needs to ill a large order, which requires that with probability 95% the cans contain at least 12 luid ounces of soft drink. Product engineer A, who likes Chebyshev’s inequality, uses it to calculate an appropriate value µA to which the machine should be set to accomplish this; product engineer B assumes that the distribution of the amount of soft drink in a can is a normal random variable, and on that basis calculates a value µB . Which number is larger? No calculation should be needed!
(b) Check your answer to (a) by calculating µA and µB .
(c) On a diferent occasion the company needs to ill an order for 10 , 000 cans of soft drink, and sets µ = 12.1 luid ounces. Engineers A and B each calculate how much bulk soft drink must be available so that the probability that the order can be illed is at least 99%; B proceeds as above, while A uses the Central Limit Theorem. What are their answers?
3. Nine independent observations of a normal population are to be used to estimate the mean and variance of this population, using X and S2 .
(a) Suppose that the variance of the population is known to be equal to 15. What is the probability that the estimate of the mean will be correct to within 3 units?
(b) Suppose again that the variance is not known, but that S2 = 15. Find a number b such that there is at most a 5% chance that the estimate of the mean will be wrong by more than b units.
4. Use the Central Limit Theorem to indan approximate value of χ2 ,ν for ν = 50 and for α = 0.01, α = 0.99.
5. Six observations of a normal population of mean µ and variance σ2 yield the values 20, 24, 25, 21, 23, and 25. Find 99% conidence interval for the mean of this population.
6. Random samples of size 10 from two independent normal populations, each having variance 5, yield 1 = 4 and 2 = 4.9. Find a 95% conidence interval for the diference µ1 — µ2 of the populations means.
7. The chi-square distribution with k degrees of freedom has probability density function
and has moment generating function
Show that the mean of a chi-square distribution is k.
8. Suppose we take a random sample of size 400 from a population distribution with mean 0 and variance 25. Approximate the probability that the sample mean is between — 1 and 1. Justify.
9. Let X1 , X2 , . . . , X10 be a random sample from a normal population with mean 5 and variance 20. Let Y1 , Y2 , . . . , Y6 be a random sample from a normal population with mean 4 and variance 18. What is the distribution of the diference of the sample means, X — Y?
10. Suppose our population distribution is chi-square with 1 degree of freedom. From this distribution, we take a random sample X1 , X2 , . . . , X100 .
(a) What is the distribution (including the parameter) of the sum X1 + · · · + X100 ?
(b) Since n = 100 is largish, we can use the Central Limit Theorem. Using the table of probabilities of a standard normal distribution, ind an approximation to the probability that the sample mean exceeds 1.1.
11. A random sample of size n is drawn from a population uniformly distributed on the interval [0, β]. Let Y = Yn- 1 denote the second largest sample value.
(a) Suppose that n = 4. Find the expected value of Y.
(b) Suppose that n = 4. Find a number c such that cY is an unbiased estimator of β .
(c) Suppose that n = 4. With c as in (b), ind the efficiency of cY relative to the standard unbiased estimator 5Y4 /4.
(d) Show that Y is an asymptotically unbiased estimator of β .
(e) Show that Y is a consistent estimator of β .
12. If X1 and X2 constitute a random sample of size n = 2 from a Poisson population, show that the mean of the sample is a su伍cient estimator of the parameter λ .
13. If X1 , X2 , and X3 constitute a random sample of size n = 3 from a Bernoulli population, show that Y = x1 + 2X2 + X3 is not a sufficient estimator of θ. (Hint: Consider special values of X1 , X2 , and X3.)
14. A random sample of size n is drawn from a beta distribution for which it is known that α = 1. Find the estimators of β obtained by the method of moments and the method of maximum likelihood.
15. Samples are taken from two normal populations with variances σ 1(2) and σ2(2); the irst population yields values 10, 12, 15, 13, and 10, while the second yields values 20, 18, 17, 17. Find a 90% conidence interval for σ 1(2)/σ2(2).
16. A food inspector, examining 12 jars of a certain brand of peanut butter, obtained the following percentages of impurities: 2.3, 1.9, 2.1, 2.8, 2.3, 3.6, 1.4, 1.8, 2.1, 3.2, 2.0, and 1.9. Construct a 90% conidence interval for the standard deviation of the population sampled, that is, for the percentage of impurities in the given brand of peanut butter.
17. Consider two simple hypotheses about the parameter θ of some population: the null hypothesis H0 is θ = θ0 , the alternative hypothesis H1 is θ = θ 1 .
(a) Suppose that C is the critical region for some test of these hypotheses, based on a sample of size n. Deine carefully, in terms of C , θ0 , and θ1 , (i) the size of the critical region, (ii) the probability α of
a type I error, (iii) the probability β of a type II error, and (iv) the power of the test.
(b) What would it mean to say that C is a most powerful critical region of size 0.05?
18. An airline wants to test the null hypothesis that 60 percent of its passengers object to smoking inside the plane. Explain under what conditions they would be committing a type I error and under what conditions they would be committing a type II error.
19. The number of successes in n trials is to be used to test the null hypothesis that the parameter θ of a binomial population equals 1/2 against the alternative that it does not equal 1/2. Find an expression for the likelihood ratio statistic.
20. Five measurements of the tar content of a certain kind of cigarette yielded 14 .5, 14.2, 14.4, 14.3, and 14.6 mg/ciagrette. Assume that the data are a random sample from a normal population, show that at the 0.05 level of signiicance the null hypothesis μ = 14 must be rejected in favor of the alternative μ 14.
21. To ind out whether the inhabitants of two South Paciic islands maybe regarded as having the same racial ancestry, an anthropologist determines the cephalic indices of six adults males from each island, getting 1 = 77.4 and 2 = 72.2 and the corresponding standard deviations s1 = 3.3 and s2 = 2.1. Decide, using the 0.01 level of signiicance, whether the diference between the two sample means can be reasonably attributed to chance. Assume that the populations samples are normal and have equal variances.
22. With reference to the previous problem, test at the 0 .1 level of signiicance whether it is reasonable to assume that the two populations sampled have equal variances.
23. A food processor wants to know whether the probability is really 0 .60 that a customer will prefer a new kind of packaging to the old kind. If, in a random sample, 7 of 18 customers prefer the new kind of packagung to the old kind, test the null hypothesis θ = 0.60 against the alternative hypothesis θ ≠ 0.60 at the 0.05 level of signiicance.
24. In random samples, 74 of 250 persons who watched a certain television program on a small TV set and 92 of 250 persons who watched the same program on a large set remembered 2 hours later what products were advertised. Use the χ2 statistic to test the null hypothesis θ1 = θ2 against the alternative hypothesis θ 1 ≠ θ2 at the 0.01 level of signiicance.