Question

A company surveyed adult Americans about their consumer debt. They reported that 46% of Millennials (those born between 1980 and 1996) and 60% of Gen Xers (those born between 1965 and 1971) did not pay off their credit cards each month and therefore carried a balance from month to month. Suppose that these percentages were based on representative samples of 450 Millennials and 300 Gen Xers. Is there convincing evidence that the proportion of Gen Xers who do not pay off their credit cards each month is greater than this proportion for Millennials? Test the appropriate hypotheses using a significance level of 0.05. (Let p₁ be the proportion of Gen Xers who do not pay off their credit cards each month, and p₂ be the proportion of Millennials who do not pay off their credit cards each month.)

State the appropriate null and alternative 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

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 your conclusion.

We reject H₀. We do not have convincing evidence that the proportion of Gen Xers who do not pay off their credit cards each month is greater than this proportion for Millennials.

We fail to reject H₀. We do not have convincing evidence that the proportion of Gen Xers who do not pay off their credit cards each month is greater than this proportion for Millennials.    

We fail to reject H₀. We have convincing evidence that the proportion of Gen Xers who do not pay off their credit cards each month is greater than this proportion for Millennials.

We reject H₀. We have convincing evidence that the proportion of Gen Xers who do not pay off their credit cards each month is greater than this proportion for Millennials.

Answer 1

Let be the proportion of Gen Xers who do not pay off their credit cards each month

be the proportion of Millennials who do not pay off their credit cards each month.

Here we have to test that

Null hypothesis :

Alternative hypothesis :

Test statistic:

where

z = 3.81 (Round to 2 decimal)

Test statistic = z = 3.81

P value:

Test is right tailed test.

P value = P(z > 3.81)

= 1 - P(z < 3.81)

= 1 - 0.9999 (From statistical table of z values or from excel using function, =NORMSDIST(3.81))

= 0.0001

P value = 0.0001

Significance level = = 0.05

Here p value <

So we reject H0.

Conclusion: There is convincing evidence that the proportion of Gen Xers who do not pay off their credit cards each month is greater than this proportion for Millennials

We reject H0. We have convincing evidence that the proportion of Gen Xers who do not pay off their credit cards each month is greater than this proportion for Millennials.