Question

Test the claim that the proportion of people who own cats is larger than 20% at the 0.005 significance level.

The null and alternative hypothesis would be:

H0:μ≤0.2H0:μ≤0.2
Ha:μ>0.2Ha:μ>0.2

H0:μ≥0.2H0:μ≥0.2
Ha:μ<0.2Ha:μ<0.2

H0:p≤0.2H0:p≤0.2
Ha:p>0.2Ha:p>0.2

H0:p≥0.2H0:p≥0.2
Ha:p<0.2Ha:p<0.2

H0:p=0.2H0:p=0.2
Ha:p≠0.2Ha:p≠0.2

H0:μ=0.2H0:μ=0.2
Ha:μ≠0.2Ha:μ≠0.2

The test is:

left-tailed

two-tailed

right-tailed

Based on a sample of 100 people, 26% owned cats

The p-value is:  (to 2 decimals)

Based on this we:

• Fail to reject the null hypothesis
• Reject the null hypothesis

Test the claim that the proportion of people who own cats is larger than 20% at the 0.005 significance level. The null and alternative hypothesis would be The test is left-tailed two-tailed right-tailed Based on a sample of 100 people, 26% owned cats The p-value i Based on this we (to 2 decimals) Fail to reject the null hypothesis Reject the null hypothesis

Answer 1

If p be the proportion of people who own cats.

At 5% level of significance, we are to conduct a test of hypothesis to test whether the proportion is greater than 20% or not

An appropriate hypothesis would be :

                 Ho: p<=0.2 versus Ha: p>0.2               (option c)

Since our concern is mainly to test whether the proportion is greater than 20%, according to the alternate hypothesis Ha, we shall use a right tailed test.      (option c)

Based on a sample of 100 people,26 people owned cats i.e. 26 people.

To obtain the p value for a right tailed test, we calculate the probability P(X>26) where X is the number of people who own cats in a sample of size 100.

X follows a Binomial Distribution with parameters 0.2 and 100.

                             $P(X=x)=\binom{100}{x}0.20^{x}(1-0.20)^{100-x}$

               Therefore,

                              $P(X>x)=\sum_{x&plus;1}^{100}\binom{100}{x}0.20^{x}(1-0.20)^{100-x}$

Hence, p-value is :

                             $P(X>26)=\sum_{27}^{100}\binom{100}{x}0.20^{x}(1-0.20)^{100-x}$

                                                      $=0.0558$

Note that we accept the Null hypothesis Ho if p-value is greater than 0.05 and reject otherwise.

Clearly, 0.0558 > 0.05 so we accept the null hypothesis.

                              ie. we failed to reject the Null hypothesis         (option a)