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Statistics Exam: Spring 2009, Statistics 131C, Sample Final, Exams of Mathematical Statistics

University of California - DavisMathematical Statistics

A sample final exam for statistics 131c, spring 2009. The exam covers various topics in statistics, including hypothesis testing, maximum likelihood estimation, and analysis of variance. Students are required to solve problems related to poisson distributions, normal distributions, and linear regression.

Typology: Exams

Pre 2010

Uploaded on 07/30/2009

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Spring 2009
Statistics 131C
Sample Final
(Show all the relevant works)
1. Suppose that X1, . . . , Xnform a random sample from a Poisson distribution with mean
λ. Let λ1> λ0>0. Suppose that you want to test the following hypotheses:
H0:λ=λ0against H1:λ=λ1.
(a) Show that the value of α(δ) + β(δ) is minimized by a test procedure which rejects
H0when Xn> c.
(b) Find the value of c.
(c) Based on your answer in part (a), find (with justification) a nonrandomized UMP
level α0test for
H0:λ≤λ0against H1:λ > λ0.
Can this test be found for all α0∈(0,1) ?
2. Consider two different normal distributions for which both the means µ1and µ2and
the variances σ2
1and σ2
2are unknown. Suppose that a random sample consisting of 16
observations from the first normal population yields P16
i=1 Xi= 84 and P16
i=1 X2
i= 563.
An independent random sample consisting of 10 observations from the second random
sample yields P10
i=1 Yi= 18 and P10
i=1 Y2
i= 72.
(a) What are the MLE’s of σ2
1and σ2
2?
(b) Test the following hypotheses at α0= 0.05:
H0:σ2
1≤σ2
2against H1:σ2
1> σ2
2.
3. Suppose that 300 persons are selected at random from a large population, and each person
in the sample is classified according to blood type, O, A, B or AB; and also according to
Rh factor, positive or negative. The data are given below:
O A B AB
Rh positive 82 89 54 19
Rh negative 13 27 7 9
At 0.05 level of significance, test the hypothesis that the two classifications of blood types
are independent. State clearly the model, hypothesis, test statistic and final conclusion.
4. Data on different varieties of seafood in a certain market for the years 1970 and 1980 are
reported below.
1
pf2

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Download Statistics Exam: Spring 2009, Statistics 131C, Sample Final and more Exams Mathematical Statistics in PDF only on Docsity!

Spring 2009

Statistics 131C

Sample Final

(Show all the relevant works)

  1. Suppose that X 1 ,... , Xn form a random sample from a Poisson distribution with mean λ. Let λ 1 > λ 0 > 0. Suppose that you want to test the following hypotheses:

H 0 : λ = λ 0 against H 1 : λ = λ 1.

(a) Show that the value of α(δ) + β(δ) is minimized by a test procedure which rejects H 0 when Xn > c. (b) Find the value of c. (c) Based on your answer in part (a), find (with justification) a nonrandomized UMP level α 0 test for H 0 : λ ≤ λ 0 against H 1 : λ > λ 0.

Can this test be found for all α 0 ∈ (0, 1)?

  1. Consider two different normal distributions for which both the means μ 1 and μ 2 and the variances σ^21 and σ^22 are unknown. Suppose that a random sample consisting of 16 observations from the first normal population yields

i=1 Xi^ = 84 and^

i=1 X

2 i = 563. An independent random sample consisting of 10 observations from the second random sample yields

i=1 Yi^ = 18 and^

i=1 Y^

2 i = 72.

(a) What are the MLE’s of σ^21 and σ^22? (b) Test the following hypotheses at α 0 = 0.05:

H 0 : σ^21 ≤ σ^22 against H 1 : σ^21 > σ^22.

  1. Suppose that 300 persons are selected at random from a large population, and each person in the sample is classified according to blood type, O, A, B or AB; and also according to Rh factor, positive or negative. The data are given below:

O A B AB

Rh positive 82 89 54 19 Rh negative 13 27 7 9

At 0.05 level of significance, test the hypothesis that the two classifications of blood types are independent. State clearly the model, hypothesis, test statistic and final conclusion.

  1. Data on different varieties of seafood in a certain market for the years 1970 and 1980 are reported below.

Suppose that it is decided to fit a linear regression model to the data to predict the 1980 seafood prices from 1970 seafood prices.

(a) Find the least squares regression coefficients for predicting 1980 prices from 1970 prices. (b) If an additional species sold for 21.4 in 1970, what would you predict for the 1980 selling price? (c) What is the estimated MSE for predicting the 1980 price of a species that sold for 21.4 in 1970?

  1. In 1973, the President of Texaco, Inc. made a statement to a US Senate subcommittee concerned with air and water pollution. The committee was concerned with the noise levels associated with automobile filters. The data cited by him is about a study involving the noise levels for vehicles of three different sizes as reported below.

Vehicle size Noise level measurements Small 810, 820, 820, 835, 835, 835 Medium 840, 840, 840, 845, 855, 850 Large 785, 790, 785, 760, 760, 760

(a) Construct the ANOVA table for the data. (b) Based on the ANOVA table test at 5% level of significance the hypothesis that there is no difference among the average noise levels of the three different types of vehicles.