Question

A researcher claimed that less than 20% of adults smoke cigarettes. A Gallup survey of 1016
randomly selected adults showed that 17% of the respondents smoke. Is this evidence to support the
researcher’s claim?
(a) What is the sample proportion, ˆp, for this problem?
(b) State the null and alternative hypothesis.
H0 : p =
H1 : p <
Note that you should have written the same number in the null and alternative hypotheses.
This is the value that the researcher is claiming. The sample proportion should never go in the
null or alternative hypotheses. Therefore,the number you have in (a) should not be in the null
or alternative hypothesis.
(c) Calculate the test statistic using the following formula:

z =
pˆ− p
√pq
n
=

where p is the value you stated in the null hypothesis and q is 1 − p.

(d) Now, using either your calculator or the normal distribution table, look up the area that cor-
responds to the z-score you calculated in (c). You don’t need to subtract from 1 because the

alternative hypothesis has < in it. This value is called the p-value.
p-value=
The p-value represents the probability of getting a ˆp of .17 or smaller if the null hypothesis

is true (p=.20). Therefore, if this probability is small then we would doubt that the null hy-
pothesis is true (i.e. if it is not very likely to get ˆp if the true proportion is 20% then this

gives us reason to doubt that p = .2). Thus, small p-values support the alternative hypothesis.
P-values less than .10 or .05 are generally considered small.
(e) Circle the correct answer:
i. The p-value was small which gives us evidence that the actual percent of adults who smoke
is less than 20%.
ii. The p-value was large so we do not have evidence that the actual percent of adults who
smoke is less than 20%.

The answer your circled in (e) is the conclusion to the hypothesis test. We always state conclusions in
context of the problem and we state whether we did or did not have evidence for the alternative hypothesis.
In this case, we did have evidence for the alternative hypothesis that the percentage of adult smokers is
less than 20%.

Answer 1

Solution:

Given:

Claim: less than 20% of adults smoke cigarettes.

p = 0.20

q= 1 - p = 1 - 0.20 = 0.80

Sample size = n = 1016

Sample proportion=

a) What is the sample proportion, ˆp, for this problem?

Sample proportion=

b) State the null and alternative hypothesis.
H0 : p = 0.20
H1 : p < 0.20

c) Calculate the test statistic using the following formula:

d) Find p-value

p-value = P( Z< z )

p-value = P( Z< -2.39)

Look in z table for z = -2.3 and 0.09 and find corresponding area.

P( Z< -2.39) = 0.0084

Thus

p-value = P( Z< -2.39)

p-value = 0.0084

e) Circle the correct answer:

P-values less than .10 or .05 are generally considered small.

i. The p-value was small which gives us evidence that the actual percent of adults who smoke
is less than 20%.