Question

(Refer to HW6 #4) A past survey shows that 49% of eighth-graders in the United States Believe
that smoking one pack of cigarettes a day does not pose a serious risk to their health. A recent
sample of survey of 400 eighth-graders from the United States showed that 176 of them believe that
smoking one pack of cigarettes a day does not pose a serious rick to their health.
Is the currant proportion of all U.S. eighth-graders who hold this opinion is less than 49%? Perform
the hypothesis testing at 1% significance level by answering the following questions.
(a) Define the p in the context of the problem.
(b) Formulate the null and alternative hypothesis.
(c) State the test statistic and its approximate distribution.
(d) Determine the critical value for α = .01 and state the decision rule.
(e) Calculate the observed value of the test statistic from the sample.
(f) State whether H0 is rejected and tell why.
(g) Calculate the P-value and do the hypothesis testing based on the P-value.
(h) Express the conclusion in the context of the problem, using common English.

Answer 1

SolutionA:

p is the population proportion  of eighth-graders in the United States Believe
that smoking one pack of cigarettes a day does not pose a serious risk to their health

p=0.49

SolutionA:

Hp:p=0.49

Ha:p<0.49

(c) State the test statistic and its approximate distribution.

z=p^-p/(sqrt(p*(1-p)/n)

p^=x/n=176/400= 0.44

z=( 0.44-0.49)/sqrt(0.49*(1-0.49/400)

z=-2.000

since n=400,n>30

sample follows normal distribution according to central limit theorem.

Solutiond:

alpha=0.01

z critical in excel

==NORMINV(0.01;0;1)

=-2.33

z critical=Zc=-2.33

(e) Calculate the observed value of the test statistic from the sample.

p^=x/n=176/400=0.44

(f) State whether H0 is rejected and tell why.

it is observed that

z =-2.00>zc=-2.33

null hypothesis is not rejected

(g) Calculate the P-value and do the hypothesis testing based on the P-value.

p value==NORM.S.DIST(-2;TRUE)

=0.022750132

alpha=0.01

p>0.01

Fail to reject null hypothesis

Accept null hypothesis.

(h) Express the conclusion in the context of the problem, using common English.

there is no sufficient evidence at 1% level of significance that he currant proportion of all U.S. eighth-graders who hold this opinion is less than 49%.