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Chapter 9: Q 42. (page 582)

Significance tests A test of H0:p=0.65 against Ha:p<0.65

based on a sample of size 400 yields the standardized test statistic z=−1.78 .

a. Find and interpret the P-value.

b. What conclusion would you make at the α=0.10 significance level? Would

your conclusion change if you used α=0.05 instead? Explain your reasoning.

c. Determine the value of p⏜= the sample proportion of successes.

Short Answer

Expert verified

a. Null hypothesis is rejected.

b. α=0.1:fail to reject the null hypothesis.and α=0.05:reject the null hypothesis

c.p^=0.6510

Step by step solution

01

Part (a) Step 1: Given Information

It is given that z=1.78

n=400,α=0.65

H0:p=0.65

Ha:p<0.65

02

Part (a) Step 2: Explanation

Value of zscore, z=1.78is:

P(x<z)=0.96246

P(x>z)=1-0.96246=0.037538

If Pvalue>α, null hypothesis is rejected.

P=0.037538<0.65, null hypothesis is rejected here.

H0:p=0.65

03

Part (b) Step 1: Given Information

It is given thatα=0.1,α=0.05

04

Part (b) Step 2: Explanation

As studied above, for α=0.65, hypothesis is rejected.

For α=0.1,P=0.037538<α, null hypothesis is rejected. H0:p=0.65

For α=0.05,P=0.037538<α, null hypothesis is rejected. H0:p=0.5

Pvalue changes due to significance level changes.

05

Part (c) Step 1: Given Information

It is given that n=400

p=0.65

z=1.78

06

Part (c) Step 2: Explanation

Test statistic is calculated as:

Z=p^-pp(1-p)n

1.78=p^-0.650.65(1-0.65)400

1.78=p^-0.650.65(0.35)400

1.78=p^-0.655.6875×10-4

Hence,p^=0.6510

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