Business statistics

Question

Test the claim that the proportion of people who own cats is larger than 40% at the 0.10 significance level.

The null and alternative hypothesis would be:

H0:p≥0.4H0:p≥0.4

H1:p<0.4H1:p<0.4

H0:p≤0.4H0:p≤0.4

H1:p>0.4H1:p>0.4

H0:μ=0.4H0:μ=0.4

H1:μ≠0.4H1:μ≠0.4

H0:μ≤0.4H0:μ≤0.4

H1:μ>0.4H1:μ>0.4

H0:μ≥0.4H0:μ≥0.4

H1:μ<0.4H1:μ<0.4

H0:p=0.4H0:p=0.4

H1:p≠0.4H1:p≠0.4

The test is:left-tailed

two-tailed

right-tailed

Correct

Based on a sample of 200 people, 42% owned cats

The p-value is :______ t (to 2 decimals)

Answer 1

Hi Salma,

(a) None of the answers are correct! Remember that null hypothesis means exactly that: nothing. In this case, no difference from 0.4, which is a proportion, not a mean. Symbolically:

H₀: p=0.4

Now, for the alternative hypothesis, the question specifically said larger than 0.4, so that would mean:

H_A: p>0.4

Final answer should be:

H₀: p=0.4

H_A: p>0.4

(b) With 200 people, we know that np and n(1-p) will exceed 10, so we can use z. z-test statistic:

z=(p-hat - p)/sqrt(p(1-p)/n)

p-hat=probability from sample

p=hypothesized value

n=sample size

For our problem:

p-hat=0.42

p=0.4

n=200

z=(0.42-0.4)/sqrt(0.4*0.6)/200)

z=0.58

To get p-value, proceed to the z-table and look up 0.5 in the column, 0.08 in the row. Note that this is a "greater than" alternative, which means we will need the complement rule or "one minus" trick.

P(Z<0.58)=0.7190

P(Z>0.58)=1-0.7190

P(Z>0.58)=0.281

This means we fail to reject H₀; conclude that population of people who own cats does not significantly exceed 40.

Finally, because we explored the greater than alternative, this is a right-tailed test.

I hope this helps.