Description
1. Let X1, X2, · · · , Xn be a random sample from N(µ, σ2
), then the pivotal quantity
(n−1)S
2
σ2 ∼ χ
2
(n − 1), and we can make use of its quantiles a, b to construct a 100(1 − α)% confidence
interval for σ. The quantiles a, b need to satisfy the constraint
G(b) − G(a) = P r{
a ≤
(n − 1)S
2
σ
2
≤ b
}
= 1 − α
where G is the CDF of χ
2
(n − 1). Obviously there are many possible choices for a and b.
(a) Construct the 100(1 − α)% confidence interval for σ in terms of the quantiles a, b defined
above. Let k be the length of the confidence interval. Express k in terms of n, s2
, a and b.
(b) Show that the k is minimized when a, b also satisfy
a
n
2 e
− a
2 − b
n
2 e
− b
2 = 0.
Combining with the constraint above, we can numerically solve for the optimal pair of
quantiles a, b to minimize the length of the confidence interval.
2. It is reported that in a telephone poll of 2000 adult, 1325 of them are nonsmokers. Also, y1 = 650
of nonsmokers and y2 = 425 of smokers said yes to a particular question. Let p1, p2 equal the
proportions of nonsmokers and smokers that would say yes to this question respectively
(a) Find a two-sided 95% confidence interval for p1 − p2.
(b) Find a two-sided 95% confidence interval for p, the proportion of adult who would say yes to
this question.
3. Let Y be Binomial(50, p). To test H0 : p = 0.08 against H1 : p < 0.08, we reject H0 if and only if
Y ≤ 7.
(a) Determine the significance level α of the test
(b) Calculate the value of the power function if in fact p = 0.05.
4. The mean birth weight in the United States is µ = 3320 grams, with a standard deviation of
σ = 580. Let X equal the birth weight in Rwanda. Assume that the distribution of X is N(µ, σ2
).
We shall test the hypothesis H0 : σ = 580 against the alternative hypothesis H1 : σ < 580 at an
α = 0.05 significance level.
(a) What is your decision if a random sample of size n = 81 yields X¯ = 2989 and s = 516?
(b) What is the approximate p-value of this test?
5. Assume that IQ scores for a certain population are approximately N(µ, 100). To test
H0 : µ = 110 against H1 : µ > 110
we take random sample of size n = 16 from this population and observe X¯ = 114
(a) Do we accept or reject H0 at the 1% significance level?
(b) Do we accept or reject H0 at the 5% significance level?
(c) What is the p-value of this test?
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6. The following text was shown to a large class of students for 30 seconds, and they were told to
report the number of F’s that they found:
IN FINANCIAL TRANSACTIONS, SIMPLE INTEREST IS OFTEN USED FOR FRACTIONS
OF AN INTEREST PERIOD FOR CONVENIENCE.
Let p equal the proportion of students who find 6F’s. We shall test the null hypothesis
H0 : p = 0.5 against H1 : p < 0.5
(a) Given a sample size of n = 230, define a critical region with an approximate significance level
of α = 0.05.
(b) If y = 110 students report that they found 6F’s, what is your conclusion?
(c) what is the p-value of this test?
7. In 1000 tosses of a coin, 560 heads and 440 tails appear. Using direct calculation or normal
approximation, test whether the coin is fair, at the 5% significance level.
8. For a random sample X1, · · · , Xn of Bernoulli(p) variables, it is desired to test
H0 : p = 0.49 against H1 : p = 0.51
Use the Central Limit Theorem to determine, approximately, the sample size needed so that the
two probabilities of error are both about 0.01. Use a test function that rejects H0 if ∑n
i=1 Xi
is
large. Find the critical value as well.
9. Let X1, · · · , Xn be a random sample from the uniform distribution on (θ, θ + 1). To test H0 : θ = 0
versus H1 : θ > 0, use the test
reject H0 if Yn ≥ 1 or Y1 ≥ k,
where k is a constant, Y1 = min{X1, · · · , Xn}, Yn = max{X1, · · · , Xn}. Determine k, in terms of n
and α, so that the test would have significance level α.
10. In a given city it is assumed that the number of automobile accidents in a given year follows a
Poisson distribution. In past years the average number of accidents per year was 15, and this year
it was 10. Test whether the accident rate has dropped, at the 5% significance level.
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