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Chapter 2 Discrete Random Variables and Probability Distributions
approximately, Þ = 0:05 and þ = 0:10: If we assume that the best critical region is of the form {x | x ≥ k}; then
Þ =
n
X
x=k
n x
. 0:20
x
. 0:80
n−x
= 0:05 and
þ =
k−1
X
x=0
n x
. 0:30
x
. 0:70
n−x
= 0:10: These equations are difficult to solve without the aid of extensive
binomial tables or a computer algebra system. We find that Þ
=
156
P
x=40
156 x
. 0:20
x
. 0:80
156−x
= 0:05145 and
þ =
39
P
x=0
156 x
. 0:30
x
. 0:70
156−x
= 0:09962; so n ≈ 156 and k ≈ 40. These values are probably close enough for all
practical purposes. Other solutions are possible, of course, depending upon the closeness with which we want to solve the equations for Þ and þ. It
may well be that we cannot carry out an experiment with this large sample size; such a restriction would obviously have implications for the sizes of
Þ
and þ that can be entertained.
b
Exercises 2.7
1. It is thought that 80 of VCR owners do not know how to program their VCR for
taping a TV program. To test this hypothesis, a sample of 20 VCR owners is chosen and the proportion, p, who can program a VCR is recorded. The hypotheses are
H
o
: p = 0:80 H
a
: p 0:80:
a. Find Þ if the critical region is X 14 where X is the number in the sample
who cannot program a VCR.
b. Find þ for the alternative H
a
: p = 0:70:
c. Graph þ as a function of p; 0 ≤ p ≤ 0:80:
2.7 Hypothesis Testing: Binomial Random Variables
119 2. A researcher speculates that 20 of the people in a very large group under study
are left-handed, a proportion much larger than the 10 of people in the population who are left-handed. A sample is chosen to test
H
o
: p = 0:10 H
a
: p = 0:20: The critical region is X ≥ k; where X is the number of left-handed people in the
sample. It is desired to have Þ = 0:07 and þ = 0:13; approximately. How large a sample should be chosen?
3. In exercise 2, show that þ is larger for the critical region X ≤ c where c is chosen
so that the test has size Þ:
4. A drug is thought to cure
2 3
of the patients with a disease; without the drug,
1 3
of the patients recover. The hypothesis
H
o
: p =
1 3
is tested against H
a
: p =
2 3
on the basis of a sample of 12 patients. H
o
is rejected if X, the number of patients in the sample who recover, is greater than 5. Find Þ and þ for this test.
5. In exercise 4, find the sample size for which Þ = 0:05 and þ = 0:13; approxi-
mately.
6. A recent survey showed that 46 of Americans feel that they are “being left
behind by technology.” To test this hypothesis, a sample of 36 Americans showed that 18 of them agreed that they were being left behind by technology. Do the data
support the hypothesis H
o
: p = 0:46 against the alternative H
a
: p 0:46 ? Use Þ
= 0:05:
7. A publisher thinks that 57 of the magazines on newsstands are unsold. To test
this hypothesis, a sample of 1000 magazines put on the newsstand resulted in 495 unsold magazines. Do these data support H
o
: p = 0:57 or the alternative H
a
: p 0:57 if Þ = 0:05?
8. A survey indicates that 41 of the people interviewed think that holders of Ph.D.
degrees have attended medical school. In a sample of 88 people, 50 agreed that Ph.D.’s attended medical school. Is this evidence, using Þ = 0:05; that the per-
centage of people thinking that Ph.D.’s are M.D.’s is greater than 41?
9. In a survey of questions concerning health issues, 59 of the respondents thought
that at some time in their lives they would develop cancer. If a sample of 200 people showed that 89 agreed that they would develop cancer at some time, is this
evidence to support the hypothesis that the percentage thinking they will develop cancer is less than 59? Use Þ = 0:05:
10. Among Americans earning more than 50,000 per year,
2 3
agree that Americans
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Chapter 2 Discrete Random Variables and Probability Distributions
