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STAT 4202 Introduction to Mathematical Statistics II

Homework 3

[11.1] If x is a value of a random variable having an exponential distribution (i.e. the pdf of the random variable X is fθ (x) = e−北/θ for x > 0 and fθ (x) = 0 for x ≤ 0), find k so that the interval from 0 to kx is a (1 − α) × 100% confidence interval for the parameter θ .

[11.3] It can be shown that for a random sample of size n = 2 from a continuous uniform density with lower bound 0 and upper bound θ, the distribution of the sample range R is given by

fθ (r) =〈 θ 2 (θ − r),

(0,

0 < r < θ

otherwise

Use the result to find c so that

r < θ < cr

is a (1 − α)100% confidence interval for θ .

[11.22 (modified)] A medical research worker intends to use the mean of a random sample of size n = 120 to estimate the mean blood pressure of women in their fifties. If, based on experience, he knows that σ = 10.5 mm of mercury, what can he assert with 99% confidence about the maximum error?

[11.29] In a study of television viewing habits, it is desired to estimate the average number of hours that teenagers spend watching per week. If it is reasonable to assume that σ = 3.2 hours, how large a sample is needed so that it will be possible to assert with 95% confidence that the sample mean is off by less than 20 minutes? You may assume that the distribution of the sample average is approximately normal.

[11.30] The length of the skulls of 10 fossil skeletons of an extinct species of birds has a mean of 5.68 cm and a standard deviation of 0.29 cm. Assuming that such measurements are normally distributed, find a 95% confidence interval for the mean length of the skulls of this species of bird.

[11.33] Independent random samples of sizes n1 = 16 and n2 = 25 from normal populations with σ 1 = 4.8 and σ2 = 3.5 have the means 1 = 18.2 and 2 = 23.4. Find a 90% confidence interval for µ 1 − µ2 .

[11.42] Among 100 fish caught in a certain lake, 18 were inedible as a result of chemical pollution. Construct a 99% confidence interval for the corresponding true proportion. State any necessary assumptions.

[11.50] Among 500 marriage license applications chosen at random in a given year, there were 48 in which the woman was at least one year older than the man, and among 400 marriage license applications chosen at random six years later, there were 68 in which the woman was at least one year older than the man. Construct a 99% confidence interval for the difference between the corresponding true proportions of marriage license applications in which the woman was at least one year older than the man.

[11.53] With reference to Exercise 11.30, construct a 95% confidence interval for the true variance of the skull length of the given species of bird.

[11.57] A study of two kinds of photocopying equipment shows that 61 failures of the first kind of equipment took on the average 80.7 minutes to repair with a standard deviation of 19.4 minutes, whereas 61 failures of the second kind of equipment took on the average 88.1 minutes to repair with a standard deviation of 18.8 minutes. Construct a 98% confidence interval for the ratio of the variances of the two populations sampled.