Question

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. A random sample of 501 satellite radio subscribers are surveyed and 396 of them had a satellite receiver in their car. Consider a test of the NAB spokesperson’s claim.

a. Set up the null and alternative hypotheses if you believe that the spokesperson’s claim is too high. Be sure to define the parameter used in your hypotheses.

b. What is the test statistic? Compute the test statistic.

c. Determine the rejection region using α = 0.10?

d. What is your conclusion on whether the spokesperson’s claim is too high?

e. Compute the p-value of the test. Discuss whether you will draw a different or the same conclusion if making decision based on the p-value

Answer 1

a) Null hypothesis

H0:p=0.80

Alternative hypothesis

H1:p<0.80

b) We have for given example,

Population proportion value is =0.8

x=396

n=501

Estimate for sample proportion = 0.79042

Z test statistic formula for proportion

$Z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}$

$=\frac{0.79042-0.80}{\sqrt{\frac{0.80(1-0.80)}{501}}}$

= -0.536

c) Decision rule : Reject H0 if Z calculated value < -1.28 ....................z critical value by using z table.

d) Conclusion: We do not have sufficient evidence to say that , the spokesperson’s claim is too high.

e)  

P value for Z test statistic is =0.296....................by using NORMSDIST(-0.536)

P value > 0.10

Therefore, we fail to reject H0.