Assignment 4
Course: STAT2602 Due: 10 Dec 2024 (11:59pm)
Q1. Let X1 , X2 , · · · , Xn and Y1 , Y2 , · · · , Ym be random samples from distributions N (θ1 , θ3 )
and N (θ2 , θ4 ), respectively. Assume that X1 , X2 , · · · , Xn , Y1 , Y2 , · · · , Ym are independent.
Find the generalized likelihood ratio test statistic for hypotheses:
H0 : θ1 = θ2 and θ3 = θ4 versus H1 : θ1 6= θ2 or θ3 6= θ4 .
Q2. Let X be N (µ, 100).
(i) To test H0 : µ = 230 versus H1 : µ > 230, what is the rejection region specified by the
generalized likelihood ratio test?
(ii) If a random sample of n = 16 yielded x = 232.6, is H0 accepted at a significance level of
α = 0.10? What is the p-value?
Q3. Let X1 , X2 , · · · , Xn be an independent random sample from N (µ, σ 2 ) with unknown
mean µ and σ 2 . Show the details to find the generalized likelihood ratio test for hypotheses
H0 : µ ≥ µ0 versus H1 : µ < µ0 .
Q4. Let X1 , X2 be a random sample of size n = 2 from the distribution having p.d.f.
f (x; θ) = (1/θ)e−x/θ , 0 < x < ∞, zero elsewhere. Consider hypotheses:
H0 : θ = 2 versus H1 : θ = 1.
Find the most powerful test having a size 0.05 for the above hypotheses.
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Q5. Let X1 , X2 , · · · , X12 be an independent random sample for the Poisson distribution with
mean λ. Consider the hypotheses:
1 1
H0 : λ = versus H1 : λ < .
2 2
Suppose that the test for the above hypotheses has the rejection region {X1 + X2 + · · · + X12 ≤
2}, and let K(λ) be the corresponding power function.
(a) Find the powers K(1/2), K(1/3), K(1/4), K(1/6), and K(1/12).
(b) Sketch the graph of K(λ) and state the size of the test.
Q6. Answer the following two questions under the assumption of normal populations with
equal variances.
(i) The following are the numbers of sales which a random sample of nine salesmen of in-
dustrial chemicals in California and a random sample of six salesmen of industrial chemicals in
Oregon made over a fixed period of time:
(California) xi : 40, 46, 61, 38, 55, 63, 36, 60, 51
(Oregon) yi : 33, 62, 44, 54, 23, 42
Using a 5% level of significance, test whether or not Californian salesmen are more efficient in
general. Write down the null and alternative hypothesis first.
(ii) Suppose that we wish to investigate whether males and females earn comparable wages
in a certain industry. Sample data show that 14 randomly surveyed males earn on the average
$213.5 per week with a standard deviation of $16.5, while 18 randomly surveyed females earn
on the average $194.1 per week with a standard deviation of $18.0. Let µ1 denote the average
wage of males and µ2 the average of females. Using a 5% level of significance, test the null
hypothesis that µ1 = µ2 against the alternative that µ1 > µ2 .
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Q7. Management training programs are often instituted to teach supervisory skills and thereby
increase productivity. Suppose a company psychologist administers a set of examinations to each
of 10 supervisors before such a training program begins and then administers similar examina-
tions at the end of the program. The examinations are designed to measure supervisory skills,
with higher scores indicating increased skill. The results of the tests are shown below:
Supervisor Pre-Test Post-Test
1 63 78
2 93 92
3 84 91
4 72 80
5 65 69
6 72 85
7 91 99
8 84 82
9 71 81
10 80 87
Do the data provide evidence that the training program is effective in increasing supervisory
skills, as measured by the examination scores? Set up the appropriate hypotheses and test them
at 5% significance level. State the assumptions made.
Q8. Let X and Y be the times in days required for maturation of Guardiola seeds from narrow-
leaved and broad-leaved parents, respectively. Assume that both X and Y are distributed as
normal and they are independent to each other. A sample of size 13 yield s2X = 9.88 and the
other sample of size 9 yield s2Y = 4.08.
Test the hypothesis that the population variances of X and Y are equal. Use α = 0.05.
