# 5 probability hw

6.4-6. Find the maximum likelihood estimates for θ1 = μ and θ2 = σ2 if a random sample of size 15 from N(μ, σ2) yielded the following values:

31.5 36.9 33.8 30.1 33.9 35.2 29.6 34.4 30.5 34.2 31.6 36.7 35.8 34.5 32.7

6.4-13. Let X1,X2,…,Xn be a random sample from a uniform distribution on the interval (θ − 1,θ + 1), where −∞ <θ < ∞.

(a) Find the method-of-moments estimator of θ.

(b) Is your estimator in part (a) an unbiased estimator of θ?

(c) Given the following n = 5 observations of X,givea point estimate of θ: 6.61 7.70 6.98 8.36 7.26

(d) The method-of-moments estimator actually has greater variance than the estimator [min(Xi) + max(Xi)]/2, which is a maximum likelihood estima-tor of θ. Compute the value of the latter estimator for the n = 5 observations in (c).

6.4-16. An urn contains 64 balls, of which N1 are orange and N2 are blue. A random sample of n = 8 balls is selected from the urn without replacement, andX is equal to the number of orange balls in the sample. This experi-ment was repeated 30 times (the eight balls being returned to the urn before each repetition), yielding the following data:

300111131120131 010211232243112

Using these data, guess the value of N1 and give a reason for your guess.

6.5-1. Show that the residuals, Yi − Yi (i = 1, 2,…, n),

6.5-8. The data in the following table, part of a set of data collected by Ledolter and Hogg (see References), provide the number of miles per gallon (mpg) for city and high-way driving of 2007 midsize-model cars as well as the curb weight of the carsType Ford Fusion V6 SE

mpg mpg Curb City Hwy Weight

20 28

Chevrolet Sebring Sedan Base 24 32 Toyota Camry Solara SE

Honda Accord Sedan Audi A6 3.2

BMW 5-series 525i Sedan Chrysler PT Cruiser Base

24 34 20 29 21 29 20 29 22 29

Mercedes E-Class E350 Sedan 19 26 Volkswagen Passat Sedan 2.0T 23 32

Nissan Altima 2.5 KiaOptimaLX

26 35 24 34

3230 3287 3240 3344 3825 3450 3076 3740 3305 3055 3142

(a) Find the least squares regression line for highway mpg (y)and city mpg(x).

(b) Plot the points and the least squares regression line on the same graph.

(c) Repeat parts (a) and (b) fo

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