Question

Test the claim that the proportion of people who own cats is larger than 20% at the 0.05 significance level.

The null and alternative hypothesis would be:

H0:p≤0.2H0:p≤0.2
H1:p>0.2H1:p>0.2

H0:μ=0.2H0:μ=0.2
H1:μ≠0.2H1:μ≠0.2

H0:μ≤0.2H0:μ≤0.2
H1:μ>0.2H1:μ>0.2

H0:p≥0.2H0:p≥0.2
H1:p<0.2H1:p<0.2

H0:p=0.2H0:p=0.2
H1:p≠0.2H1:p≠0.2

H0:μ≥0.2H0:μ≥0.2
H1:μ<0.2H1:μ<0.2

The test is:

(A) left-tailed

(B) two-tailed

(C) right-tailed

Based on a sample of 100 people, 22% owned cats:

The test statistic is: _____ (to 2 decimals)

The p-value is: _____ (to 2 decimals)

Based on this we:

(A) Fail to reject the null hypothesis

(B) Reject the null hypothesis

Answer 1

given data are:-

sample proportion ($\hat{p}$) = 0.22

sample size (n) = 100

hypothesis:-

$H_{0}:p\leq 0.2$

70 <d:

this test is right-tailed .

the test statistic is:-

$Z=\frac{\hat{p}-p}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}}=\frac{0.22-0.20}{\sqrt{\frac{0.2(1-0.2)}{100}}}\approx \mathbf{0.50}$

the p value is :-

$=P(Z>0.5)$

$=1-P(Z<0.5)$

$=1-\Phi(0.5)$

$=1-0.6915$ [ using standard normal table ]

$=0.3085$

$\approx \mathbf{0.31}$

based in this we :- Fail to reject the null hypothesis (A)

[ p value = 0.31 > 0.05 (alpha) ]

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