課程名稱︰高等統計二
課程性質︰
課程教師︰黃芳玫
開課學院：生農學院
開課系所︰農經系
考試日期（年月日）︰ 2010/6/14
考試時限（分鐘）：3hr
是否需發放獎勵金： 是
（如未明確表示，則不予發放）
試題 :
1. Let independent random samples of sizes n and m be taken respectively from
two normal distributions with unknown means μx and μy and unknown variances
σ^2x and σ^2y
Show that the likelihood ratio for testing H0:μx=μy against H1:μx≠μy
when σ^2x = σ^2y (var(x)=var(y)) is a function of the usual two-sample t
statistics. (|t|>=t(α/2) (n+m-2))
2. Let X1, X2,.... Xn be a random sample from a normal distribution N(0, μ^2)
(a) Show that C={(X1, X2... Xn): Σ(i=1~n) xi^2 ≧c} is a best critical region
for testing H0: σ^2=4 against H1:σ^2=16.
(b) If n=15, find the value of c so that α=.05
Hint: recall that the Σ(i=1~n) xi^2/ σ^2 is χ^2(n)
(c) If n=5 and c is the value found in part (b), find the approximate value of
β=P(Σ(i=1~n) xi^2 <c; σ^2=16)
3. A certain size bag is designed to hold 25 pounds of potatoes. A farmer
fills such bags in the field. Assume that the weight X of potatoes in a bag
is N(μ,9). We shall test the null hypothesis. H0: μ=25 against the
alternative hypothesis H1:μ<25. Let X1, X2, X3, X4 be a random sample of size
4 from this distribution , and let the critical region C for this test be
defined by x bar ≦22.5, where x-bar is the observed value of X-bar.
(a) What is the power function of K(μ) of this test?
In particular, what is the significance level α=K(25) for your test?
(b) if the randome sample of four bags of potatoes yielded the values x1=20.14,
x2=23.11, x3=22.62, x4=26.82, would you accept or reject H0 using your test?
(c) What is the p-value assoiated with the x-bar in part (b)
4. While testing a computer tape for bad records, the computer operator counted
the number of flaws per 100 feet. Let X equal this number and test the
hypothesis that the distribution of X is Possion with a mean of λ=2.4 using
the following 40 observations of X. (Using kolmogorov-smirnov goodness of fit
test) Let α=.1, approximately.
X frequency
0 5
1 7
2 12
3 9
4 5
5 1
6 1
5. For the following set of data show that the computed SSE/n-m=1 and SST/m-1
=75. This suggests that the unbiased estimate of σ^2 based on SST is usually
greater than σ^2 when the true means are unequal.
X1: 4, 5, 6
X2: 9, 10, 11
X3: 14, 15, 16
6. It has seen claimed that, when spinning a penny minted in 1999 or earlier ,
the possibility of observing heads in p=.3. Three students got together and
they would each spin a penny and record the number of heads out of the
three spins , say X. They repeated this n=200 times, observing 0, 1, 2 and 3
heads 57, 95, 38 and 10 times, respectively. Use these data to test the
hypotheses that X is b(3,0.3).
Give limits for the p-value of this test. In addition, out of the 600 spins,
calculate the number of heads occurring and then a 95% confidence interval
for p.
7. Let X and y equal the cpu time in days per month for two different computers
on a college campus. Let mx and my equal the medians of X and Y. Using
n1=n2=14 observations of each of X and Y, test the null hypothesis H0: mx=my,
against the alternative hypothesis H1: mx≠my. Use a normal approximation and
the two-sample wilcoxon test. Let α=.05, approximately.
X: 7.1 8.5 9.9 13.8 2.2 2.4 2.3 1.0 3.9 4.6 5.3 4.0 3.9 6.6
Y: 7.9 15.4 16.2 9.6 4.9 8.9 9.4 15.2 12.1 4.3 5.0 2.4 2.3 5.4
8. Let X and Y equal the number of milligrams of tar in filtered and
nonfiltered cigarettes, respectively. Assume that the distribution of X and Y
are N(μx, σ^2x) and N(μy, σ^2y), respectively. We shall test the null
hypothesis H0: μx-μy=0 against the H1:μx-μy≠0 using random samples of
size of n=9 and m=11 observations of X and Y respectively.
(a) define the test statistics and a critical region that has an α=.01
significance level. Sketch a figure illustrating this critical region.
(b) Given the n=9 observations of X,
0.9 1.1 0.1 0.7 0.4 1.0 0.8 1.0 0.4
and the m=11 observations of Y,
1.5 0.9 1.6 0.5 1.4 1.9 1.0 1.2 1.3 1.6 2.1
calculate the value of the test statistic and clearly state your conclusion.
Locate the value of test statistic on your figure.
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