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Statistics Homework 3: Hypothesis Testing and Power Function, Assignments of Mathematical Statistics

University of California - DavisMathematical Statistics

Solutions to selected problems from statistics 131c, homework 3. The problems involve hypothesis testing, determination of power functions, and finding constants for rejection regions. Topics include normal distributions, uniform distributions, and t-tests.

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Spring 2009
Statistics 131C
Homework 3
Due : April 24
1. Problem 8.4.4 : Suppose that X1, . . . , Xnform a random sample from a normal distribu-
tion for which the mean µis unknown and the variance is 1, and it is desired to test the
following hypotheses:
H0: 0.1≤µ≤0.2 against H1:µ < 0.1 or µ > 0.2.
Consider a test procedure δsuch that the hypothesis H0is rejected if either Xn≤c1or
Xn≥c2, and let π(µ|δ) denote the power function of δ. Suppose that the sample size is
n= 25. Determine the values of the constants c1and c2such that π(0.1|δ) = π(0.2|δ) =
0.07.
2. Problem 8.4.5 : Consider again the conditions of Problem 8.4.4, and suppose also that
n= 25. Determine the values of the constants c1and c2such that π(0.1|δ)=0.02 and
π(0.2|δ) = 0.05.
3. Problem 8.4.10 : Suppose that X1, . . . , Xnform a random sample from a uniform distribu-
tion on the interval [0,θ ]. Suppose now that it is desired to test the following hypotheses:
H0:θ= 3 against H1:θ6= 3.
Consider a test procedure δsuch that the hypothesis H0is rejected if either max{X1, . . . , Xn} ≤
c1or max{X1, . . . , Xn} ≥ c2, and let π(θ|δ) denote the power function of δ. Determine
the values of the constants c1and c2such that π(3|δ) = 0.05 and δis unbiased.
4. Problem 8.5.3 : The manufacturer of a certain type of automobile claims that under
typical urban driving conditions the automobile will travel on average at least 20 miles
per gallon of gasoline. The owner of this type of automobile notes the mileages that she
has obtained in her own urban driving when she fills her automobile’s tank with gasoline
on nine different occasions. She finds that the results, in miles per gallon, are as follows:
15.6,18.6,18.3,20.1,21.5,18.4,19.1,20.4,19.0.
Test the manufacturer’s claim by carrying out a test at the level of significance α0= 0.05.
List carefully the assumptions you must make.
5. Problem 8.5.6 : Suppose that the variables X1,. . . , Xnform a random sample from a
normal distribution for which both the mean µand variance σ2are unknown, and a ttest
at a given level of significance α0is to be carried out to test the following hypotheses:
H0:µ≤µ0against H1:µ > µ0.
1
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Spring 2009

Statistics 131C

Homework 3

Due : April 24

  1. Problem 8.4.4 : Suppose that X 1 ,... , Xn form a random sample from a normal distribu- tion for which the mean μ is unknown and the variance is 1, and it is desired to test the following hypotheses: H 0 : 0. 1 ≤ μ ≤ 0. 2 against H 1 : μ < 0 .1 or μ > 0. 2. Consider a test procedure δ such that the hypothesis H 0 is rejected if either Xn ≤ c 1 or Xn ≥ c 2 , and let π(μ|δ) denote the power function of δ. Suppose that the sample size is n = 25. Determine the values of the constants c 1 and c 2 such that π(0. 1 |δ) = π(0. 2 |δ) = 0 .07.
  2. Problem 8.4.5 : Consider again the conditions of Problem 8.4.4, and suppose also that n = 25. Determine the values of the constants c 1 and c 2 such that π(0. 1 |δ) = 0.02 and π(0. 2 |δ) = 0.05.
  3. Problem 8.4.10 : Suppose that X 1 ,... , Xn form a random sample from a uniform distribu- tion on the interval [0, θ]. Suppose now that it is desired to test the following hypotheses: H 0 : θ = 3 against H 1 : θ 6 = 3. Consider a test procedure δ such that the hypothesis H 0 is rejected if either max{X 1 ,... , Xn} ≤ c 1 or max{X 1 ,... , Xn} ≥ c 2 , and let π(θ|δ) denote the power function of δ. Determine the values of the constants c 1 and c 2 such that π(3|δ) = 0.05 and δ is unbiased.
  4. Problem 8.5.3 : The manufacturer of a certain type of automobile claims that under typical urban driving conditions the automobile will travel on average at least 20 miles per gallon of gasoline. The owner of this type of automobile notes the mileages that she has obtained in her own urban driving when she fills her automobile’s tank with gasoline on nine different occasions. She finds that the results, in miles per gallon, are as follows: 15. 6 , 18. 6 , 18. 3 , 20. 1 , 21. 5 , 18. 4 , 19. 1 , 20. 4 , 19. 0. Test the manufacturer’s claim by carrying out a test at the level of significance α 0 = 0.05. List carefully the assumptions you must make.
  5. Problem 8.5.6 : Suppose that the variables X 1 ,... , Xn form a random sample from a normal distribution for which both the mean μ and variance σ^2 are unknown, and a t test at a given level of significance α 0 is to be carried out to test the following hypotheses: H 0 : μ ≤ μ 0 against H 1 : μ > μ 0.

Let π(μ, σ^2 |δ) denote the power function of this t test, and assume that (μ 1 , σ^21 ) and (μ 2 , σ^22 ) are values of the parameters such that

μ 1 − μ 0 σ 1

μ 2 − μ 0 σ 2

Show that π(μ 1 , σ^21 |δ) = π(μ 2 , σ 22 |δ).

  1. Problem 8.5.7 : Consider a normal distribution for which both the mean μ and variance σ^2 are unknown, and suppose that it is desired to test the following hypotheses:

H 0 : μ ≤ μ 0 against H 1 : μ > μ 0.

Suppose that it is possible to observe only a single value of X from this distribution, but that and independent random sample of n observations Y 1 ,... , Yn is available from another normal distribution for which the variance is also σ^2 , and it is known that the mean is 0. Show how to carry out a test of the hypotheses H 0 and H 1 based on the t distribution with n degrees of freedom.

  1. Problem 8.5.17 : Consider a normal distribution for which both the mean μ and variance σ^2 are unknown, and suppose that it is desired to test the following hypotheses:

H 0 : μ = μ 0 against H 1 : μ 6 = μ 0.

Prove that the likelihood ratio test for these hypotheses is the two-sided t test that rejects H 0 if |U | ≥ c, where

U =

n^1 /^2 (Xn − μ) σ′^ with σ′^ =

n − 1

∑^ n

i=

(Xi − Xn)^2

The argument is slightly simpler than, but very similar to, the one given in the text for the one-sided case.