Question

A well-known brokerage firm executive claimed that 70% of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that in a sample of 800 people, 67% of them said they are confident of meeting their goals.

Test the claim that the proportion of people who are confident is smaller than 70% at the 0.005 significance level.

The null and alternative hypothesis would be:

H0:p≥0.7H0:p≥0.7
H1:p<0.7H1:p<0.7

H0:μ=0.7H0:μ=0.7
H1:μ≠0.7H1:μ≠0.7

H0:μ≥0.7H0:μ≥0.7
H1:μ<0.7H1:μ<0.7

H0:p=0.7H0:p=0.7
H1:p≠0.7H1:p≠0.7

H0:p≤0.7H0:p≤0.7
H1:p>0.7H1:p>0.7

H0:μ≤0.7H0:μ≤0.7
H1:μ>0.7H1:μ>0.7

The test is:

left-tailed

two-tailed

right-tailed

The test statistic is:  (to 3 decimals)

The p-value is:  (to 4 decimals)

Based on this we:

Fail to reject the null hypothesis

Reject the null hypothesis

2) Test the claim that the proportion of men who own cats is significantly different than 60% at the 0.05 significance level.

The null and alternative hypothesis would be:

H0:p=0.6H0:p=0.6
H1:p<0.6H1:p<0.6

H0:μ=0.6H0:μ=0.6
H1:μ<0.6H1:μ<0.6

H0:μ=0.6H0:μ=0.6
H1:μ≠0.6H1:μ≠0.6

H0:p=0.6H0:p=0.6
H1:p≠0.6H1:p≠0.6

H0:μ=0.6H0:μ=0.6
H1:μ>0.6H1:μ>0.6

H0:p=0.6H0:p=0.6
H1:p>0.6H1:p>0.6

The test is:

two-tailed

left-tailed

right-tailed

Based on a sample of 20 people, 65% owned cats

The test statistic is:  (to 2 decimals)

The positive critical value is:  (to 2 decimals)

Based on this we:

Fail to reject the null hypothesis

Reject the null hypothesis

Answer 1

(1)

Claim: The proportion of people who are confident is smaller than 70% that is p < 0.7

The claim comes under the alternative hypothesis and the null and alternative hypothesis are:

That is the first option is correct.

The alternative hypothesis contains less than symbol so the test is left tailed test.

The formula of test statistics is,

The test statistics is -1.852

The formula of P-value for a left tailed test is,

P-value = P(Z < test statistics) = P(Z < -1.852)

By using z table the probability for z is, 0.0320

Therefore, the P-value is 0.0320

Decision rule: If P-value > alpha (level of significance) then we fail to reject the null hypothesis otherwise we reject the null hypothesis.

Here P-value (0.0320) is > alpha (0.005) so we fail to reject the null hypothesis.

(2)

Claim: The proportion of men who own cats is significantly different than 60% that is

The claim comes under the alternative hypothesis and the null is exactly the opposite of alternative.

The null and alternative hypothesis are:

The alternative hypothesis contains not equal to symbol so the test is a two-tailed test.

The test is: two-tailed test

The formula of test statistics is,

The test statistics is: 0.46

Critical value:

The test is a two-tailed and alpha (level of significance) is 0.05

To find positive critical value need area 1 - (alpha/2)

1 - (alpha/2) = 1 - (0.05/2) = 1 - 0.025 = 0.975

And the critical value using z table for area 0.975 is 1.96

The positive critical value is: 1.96

Decision rule: If critical value > test statistics then we fail to reject the null hypothesis otherwise reject the null hypothesis.

Here critical value 1.96 is > 0.46 so we fail to reject the null hypothesis.

Based on this we: Fail to reject the null hypothesis.