7.6 Large-Sample Test of Hypothesis about a Population p

7.6 Large-Sample Test of Hypothesis about a Population
Proportion p
Mathematics: Recall that for large n (i.e. when np ≥ 15 and nq ≥ 15, where q = 1-p) the sampling
distribution of the sample proportion ̂ is approximately normal with the mean ̂
and the standard
deviation
̂
√
. Hence
̂
√
is approximately standard normal N(0,1).
P-value – computation depends on the form of an alternative hypothesis. If we denote the
observed value of the test statistic by zobs, then
Ha
Ha: p > p0
p-value (= area of shaded region)
p-value = P(z > zobs )
zobs
Ha: p < p0
p-value = P(z < zobs)
zobs
Ha: p ≠ p0
p-value = P(|z| >|zobs|)
|zobs|
Example (Ex. 7.10, p. 389)
Conclusion:
Exercise 1 [Example 7.64, page 354] Suppose a random sample of 100 observations from a binomial population (=
population with unknown proportion p of items of some type) gives a value of ̂ = .63 and you wish to test the null
hypothesis that the population parameter p is equal to .70 against the alternative hypothesis that p is less than .70. Use
large sample z-test
a. Write the null and alternative hypotheses,
H0: ……….
Ha: ……………..
b. Give and graph the rejection region at α= .05.
c. Find the observed value of the test statistic.
d. State the conclusion for the hypothesis test.
e. Find and interpret the observed significance level (p-value) of the test you conducted.
Exercise 2 [7.70, p.392] A spokesperson for the National Association of Broadcasters (NAB) claims that 80% of all
satellite radio subscribers have a satellite radio receiver in their car. In a recent survey of 501 satellite radio
subscribers, 396 have a satellite receiver in their car. Consider a test of the NAB spokesperson's claim.
a. Define the parameter of interest to the NAB spokesperson.
b. Set up the null hypothesis for testing the claim. H0: ……….
Ha: ……………..
c. Specify the alternative hypothesis if you believe that the spokesperson's claim is too high.
d. Compute the value of the test statistic.
e. Determine the rejection region for the test using α=.10.
f. Compute the p-value of the test.
g. Make the appropriate conclusion. Show that the decision based on the rejection region agrees with the decision
based on the p-value.