A company surveyed adult Americans about their consumer debt. They reported that 48% 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

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jtellezd jtellezd · Answer 1 · 2021-04-20T18:20:33+02:00

Answer:

We reject H₀ at 95 % of CI the group of Gen Xers who do not pay their credit card each month, is greater than this proportion for Millennians

Step-by-step explanation:

Sample proportion 1: Born between 1980 and 1996

sample size n₁ = 450

p₁ = 0,48 p₁ = 48 %

x₁ = 0,48 * 450 x₁ = 216

Sample proportion 2: Born between 1965 and 1971

sample size n₂ = 300

p₂ = 0,60 p₂ = 60%

x₂ = 0,6* 300 x₂ = 180

Test Hypothesis:

Null Hypothesis H₀ p₂ = p₁

Alternative Hypothesis H p₂ > p₁

Samples both big enough to use the normal distribution as an approximation to the binomial distribution.

We assume CI = 95 % then significance level α = 5 % the alternative hypothesis tells us that we need to develop a one-tail test to the right.

z-score is z(c) for α = 0,05 from z-table z(c) = 1,64

To calculate z(s)

z(s) = ( p₂ - p₁ ) / √ p*q ( 1 /n₁ + 1 / n₂)

p₂ - p₁ = 0,60 - 0,48

p₂ - p₁ = 0,12

p = ( x₁ + x₂ ) / n₁ + n₂

p = ( 216 + 300) / 450 + 300

p = 516 / 750 p = 0,688 then q = 1 - p q = 0,312

z(s) = 0,12 / √ 0,688*0,312* ( 1/450) + 1/ 300)

z(s) = 0,12 / √ 0,2146* ( 0,0022 + 0,0033)

z(s) = 0,12 / √ 0,001192

z(s) = 0,12/ 0,03452

z(s) = 3,47

Comparing |z(c)| and |z(s)|

z(s) > z(c)

Then z(s) is in the rejection region. We reject H₀.

We can support the claim that the proportion of the group Gen Xers who do not pay their credit card is greater than Millennians