Question

A random sample of 200 books purchased at a local bookstore showed that 72 of the books were murder mysteries. Let ?? be the true proportion of books sold by this store that are murder mysteries. Test the claim at the 1% level of significance that half of the books are sold.
a) Details given in the problem:
b) Assumptions (if any):
c) Null, Alternate Hypotheses and the tail test?
H0:
H1:
Tail?
d) Find the Critical Statistic (Table value) and Illustrate:
Critical Statistic:
e) State and Compute Test Statistic:
f) Compute p-value:
g) Conclusion using critical method:
h) Conclusion using p-value method:

Answer 1

a) Details given in the problem:

Here given ,n = total number of books purchased at a local bookstore showed = 200

             x = number of the books were murder mysteries= 72

Let ?? be the true proportion of books sold by this store that are murder mysteries.

So here sample proportion ,p^ = x/n = 72/200 = 0.36

Claim : half of the books are sold.

So here claim : P =0.5

b) Assumptions (if any) :

If np 10 and n(1-p) 10. then we use Normal approximation .

so here ,200 *0.5 = 100 and 200* ( 1-0.5 ) = 100 , Since here we use normal approximation .

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c) Null, Alternate Hypotheses and the tail test :

claim : P =0.5

Null hypothesis contains equality and alternative hypothesis contains inequality .

Null and alternative hypothesis oppositive of each other .

So here , null hypothesis : H0 :P=0.5    vs   alternative hypothesis :H1: P 0.5.

So here alternative hypothesis contains, sign . Since test is " Two tailed " test .

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d) Find the Critical Statistic (Table value) and Illustrate:
Critical Statistic:

Here test is two tailed test , since we get two critical values : and

Normal distribution is symmetric ,since both values are same.But just change in sign .

= significance level = 1% = 0.01 ,

Use negative normal distribution table :

So another critical value +

Since here critical values are ,    and

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e) State and Compute Test Statistic:

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f) Compute p-value:

Here test is two tailed test . So p- value for two tailed test is as ,

p- value = 2 * P( Z < -3.96 ) that is find , P(Z< 3.96) using normal distribution table then multiply by 2.

so , p- value = 2 * P( Z < -3.96 ) = 2 * 0.0000 = 0.0000

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g) Conclusion using critical method:

Decision : If test statistics value <       or   test statistics value >     ; then reject H0 .

                     So here , Z =- 3.96 < ; since reject H0 .

Conclusion : There is not sufficient eveidence to conclude that , the claim at the 1% level of significance that half of the books are sold.

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h) Conclusion using p-value method:

Decision Rule Based on P Values

1. If p is less than or equal to alpha, then reject the null hypothesis

      2. If p is greater than alpha, the fail to reject the null hypothesis.

So here p- value = 0.0000 < 0.01 ; since Reject H0 .

Conclusion : There is not sufficient eveidence to conclude that , the claim at the 1% level of significance that half of the books are sold.