Question

A paper included analysis of data from a national sample of 1,000 Americans. One question on the survey is given below.

"You owe $3,000 on your credit card. You pay a minimum payment of $30 each month. At an Annual Percentage Rate of 12% (or 1% per month), how many years would it take to eliminate your credit card debt if you made no additional charges?"

Answer options for this question were: (a) less than 5 years; (b) between 5 and 10 years; (c) between 10 and 15 years; (d) never—you will continue to be in debt; (e) don't know; and (f) prefer not to answer.

(a)

Only 358 of the 1,000 respondents chose the correct answer of "never." For purposes of this exercise, assume that the sample is representative of adult Americans. Is there convincing evidence that the proportion of adult Americans who can answer this question correctly is less than 0.40 (40%)? Use

α = 0.05

to test the appropriate hypotheses. (Hint: See Example 10.10.)

State the appropriate null and alternative hypotheses.

H₀: p = 0.40

versus

H_a: p ≠ 0.40

H₀: p = 0.40

versus

H_a: p < 0.40

    

H₀: p < 0.40

versus

H_a: p > 0.40

H₀: p ≠ 0.40

versus

H_a: p = 0.40

H₀: p = 0.40

versus

H_a: p > 0.40

Find the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z=P-value=

State the conclusion in the problem context.

Fail to reject H₀. We do not have convincing evidence that the proportion of all adult Americans who would answer the question correctly is less than 0.40.Fail to reject H₀. We have convincing evidence that the proportion of all adult Americans who would answer the question correctly is less than 0.40.    Reject H₀. We have convincing evidence that the proportion of all adult Americans who would answer the question correctly is less than 0.40.Reject H₀. We do not have convincing evidence that the proportion of all adult Americans who would answer the question correctly is less than 0.40.

(b)

The paper also reported that 37.7% of those in the sample chose one of the wrong answers (a, b, or c) as their response to this question. Is it reasonable to conclude that more than one-third of adult Americans would select a wrong answer to this question? Use

α = 0.05.

(Round your test statistic to two decimal places and your P-value to four decimal places.)

State the appropriate null and alternative hypotheses.

H₀: p ≠

1

3

versus H_a: p =

1

3

H₀: p =

1

3

versus H_a: p >

1

3

    H₀: p =

1

3

versus H_a: p ≠

1

3

H₀: p =

1

3

versus H_a: p <

1

3

H₀: p <

1

3

versus H_a: p >

1

3

Find the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z=

P-value=

State the conclusion in the problem context.

Fail to reject H₀. We have convincing evidence that more than one-third of adult Americans would select a wrong answer.

Fail to reject H₀. We do not have convincing evidence that more than one-third of adult Americans would select a wrong answer.    

Reject H₀. We do not have convincing evidence that more than one-third of adult Americans would select a wrong answer.

Reject H₀. We have convincing evidence that more than one-third of adult Americans would select a wrong answer.

Answer 1

1)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.4
Alternative Hypothesis, Ha: p < 0.4

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.358 - 0.4)/sqrt(0.4*(1-0.4)/1000)
z = -2.71

P-value Approach
P-value = 0.0034
As P-value < 0.05, reject the null hypothesis.

Reject H0. We have convincing evidence that the proportion of all adult Americans who would answer the question correctly is less than 0.40

b)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.3333
Alternative Hypothesis, Ha: p > 0.3333

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.377 - 0.3333)/sqrt(0.3333*(1-0.3333)/1000)
z = 2.93

P-value Approach
P-value = 0.0017
As P-value < 0.05, reject the null hypothesis.

Reject H0. We have convincing evidence that more than one-third of adult Americans would select a wrong answer.