Question

Consider the following two questions designed to assess quantitative literacy.

1. What is 15% of 1000?
2. A store is offering a 15% off sale on all TVs. The most popular television is normally priced at $1000. How much money would a customer save on the television during this sale?

Suppose the first question is asked of 200 randomly selected college students, with 164 answering correctly; the second one is asked of a different random sample of 200 college students, resulting in 147 correct responses. Carry out a test of hypotheses at significance level 0.05 to decide if the true proportion of correct responses to the question without context exceeds that for the one with context. (Use p₁ for the true proportion students who answered the question without context correctly and p₂ for the true proportion of students who answered the question with context correctly.)

State the relevant hypotheses.

- H₀: p₁ − p₂ > 0 H_a: p₁ − p₂ = 0

- H₀: p₁ − p₂ = 0 H_a: p₁ − p₂ < 0    

- H₀: p₁ − p₂ < 0 H_a: p₁ − p₂ = 0

- H₀: p₁ − p₂ = 0 H_a: p₁ − p₂ > 0

- H₀: p₁ − p₂ = 0 H_a: p₁ − p₂ ≠ 0

Calculate 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. The data does not suggest that the true proportion of correct answers to the context-free question is higher than the proportion of right answers to the contextual one.

- Reject H₀. The data suggests that the true proportion of correct answers to the context-free question is higher than the proportion of right answers to the contextual one.

- Fail to reject H₀. The data suggests the true proportion of correct answers to the context-free question is higher than the proportion of right answers to the contextual one.

- Reject H₀. The data does not suggest the true proportion of correct answers to the context-free question is higher than the proportion of right answers to the contextual one.

Answer 1

The statistical software output for this problem is:

From above output:

Hypotheses: H0: p1 − p2 = 0 Ha: p1 − p2 > 0

z = 2.04

P - value = 0.0205

Conclusion: Reject H0. The data suggests that the true proportion of correct answers to the context-free question is higher than the proportion of right answers to the contextual one.