A company surveyed adult Americans about their consumer debt. They reported that 48% of Millennials (those born
between 1981 and 1996) and 59% of Gen Xers (those born between 1965 and 1980) 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.
Find the test statistic. (Round your answer to two decimal places.)
Use technology to find the p-value. (Round your answer to four decimal places.)
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Null hypothesis: The proportion of Gen Xers who do not pay off their credit cards each month is equal to the proportion of Millennials who do not pay off their credit cards each month
H0: p1 = p2
Alternative hypothesis: The proportion of Gen Xers who do not pay off their credit cards each month is greater than the proportion of Millennials who do not pay off their credit cards each month
Ha: p1 > p2
The test statistic can be calculated using the formula:
test statistic = (p1 - p2) / sqrt(p * (1-p) * (1/n1 + 1/n2))
where p = (n1 * p1 + n2 * p2) / (n1 + n2)
For Millennials:
n1 = 450
p1 = 0.48
For Gen Xers:
n2 = 300
p2 = 0.59
Calculating p:
p = (450 * 0.48 + 300 * 0.59) / (450 + 300)
p = 387 / 750
p ≈ 0.516
Calculating the test statistic:
test statistic = (0.59 - 0.48) / sqrt(0.516 * (1-0.516) * (1/450 + 1/300))
Using technology (calculator or software) to find the p-value, the test statistic can be compared to the critical value at a significance level of 0.05 to determine if there is convincing evidence to reject the null hypothesis.
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