91Ó°ÊÓ

The paper "Debt Literacy, Financial Experiences and Over-Indebtedness" (Social Science Research Network, Working paper WI4808, 2008 ) included analysis of data from a national sample of 1000 Americans. One question on the survey was: "You owe \(\$ 3000\) 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 354 of the 1000 respondents chose the correct answer of never. For purposes of this exercise, you can 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 \(.40(40 \%) ?\) Use \(\alpha=.05\) to test the appropriate hypotheses. b. The paper also reported that \(37.8 \%\) of those in the sample chose one of the wrong answers \((a, b\), and \(c)\) 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 \(\alpha=.05\).

Step by step solution

01

Set up the hypotheses for part a

The null hypothesis, denoted \(H_0\), assumes that the population proportion (\(p\)) is 0.40. So, \(H_0 : p = 0.40\). The alternative hypothesis, denoted \(H_a\), is what the test is attempting to prove, that less than 40% can answer the question correctly. So, \(H_a : p < 0.40\).

02

Calculate the test statistic for part a

The test statistic for a hypothesis test about a proportion is given by the formula: \(Z = ( \hat{p} - p0 )/ \sqrt{p0 * ( 1 - p0 ) / n }\), where \(\hat{p}\) is the sample proportion, \(p0\) is the proportion under the null hypothesis, and \(n\) is the sample size. Plugging in the values from the exercise: \(Z = ( 0.354 - 0.40 ) / \sqrt{0.40 * ( 1 - 0.40 ) / 1000 }\).

03

Determine the p-value for part a

The p-value is the probability of getting a test statistic as extreme or more extreme than the observed test statistic, given that the null hypothesis is true. We would use a standard normal distribution (Z-distribution) table or use software to find the p-value corresponding to the calculated z-score.

04

Make a decision for part a

If the p-value is less than the significant level (\(0.05\)), we reject the null hypothesis, providing evidence to support the alternative hypothesis. Otherwise, we fail to reject the null hypothesis.

05

Set up the hypotheses for part b

The null hypothesis, \(H_0\), assumes that the population proportion (\(p\)) is 0.33 (one third). So, \(H_0 : p = 0.33\). The alternative hypothesis, \(H_a\), is aiming to prove that more than a third of Americans would select a wrong answer. Therefore, \(H_a : p > 0.33\).

06

Calculate the test statistic for part b

The test statistic can be calculated similar to step 2, but this time, we have \(\hat{p} = 0.378\) and \(p0 = 0.33\). So, \(Z = (0.378 - 0.33) / \sqrt{0.33 * (1-0.33) / 1000 }\).

07

Determine the p-value for part b

Like in step 3, find the p-value corresponding to the calculated z-score. But note that this time, because it’s an upper-tail test, the p-value is the proportion in the tail above the calculated z-score.

08

Make a decision for part b

Using the same criteria as Step 4, if the \(p-value < 0.05\), reject the null hypothesis, providing evidence that more than a third of American adults would select a wrong answer. Else, fail to reject the null hypothesis.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Hypothesis Test

A proportion hypothesis test is a statistical method used to test if the proportion of a certain category in a population differs from a specified value. In the context of the given exercise, we're looking at whether the proportion of Americans who can correctly answer a financial literacy question is less than 40%.
The process starts with setting up two hypotheses:

• The null hypothesis (\(H_0\)) supposes there is no effect or difference, and for our exercise, it's that the true proportion \(p = 0.40\).
• The alternative hypothesis (\(H_a\)) is what we are testing for. Here, it states that the proportion is less than 0.40 (i.e., \(p < 0.40\)).

These hypotheses form the basis of our test, driving us to use sample data to infer about the population. This approach helps to draw meaningful conclusions about the underlying population behavior.

Z-Score Calculation

The z-score is a crucial element in hypothesis testing, used to measure the number of standard deviations an element is from the mean. For proportion tests, it's a tool to convert our proportion difference into a standardized form, allowing us to assess how unusual our sample data is under the null hypothesis.
To calculate the z-score for our problem, use the formula:\[Z = \frac{\hat{p} - p_0}{\sqrt{p_0 (1 - p_0) / n}} \]Here:

• \(\hat{p}\) is the sample proportion. In the exercise, it is 0.354 for part a.
• \(p_0\) is the hypothesized population proportion (0.40 in the example).
• \(n\) is the sample size, which is 1000.

After plugging in, you calculate a z-score that shows how far the observed sample proportion deviates from the hypothesized population proportion, relative to the standard deviation. This helps in understanding how significant the deviation is.

P-Value Interpretation

The p-value plays a pivotal role in hypothesis testing, providing a metric for deciding whether to reject the null hypothesis. It represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.
Once the z-score is found, we utilize statistical tables or software to find the corresponding p-value. A smaller p-value indicates stronger evidence against the null hypothesis.

• If the p-value < 0.05 (our significance level), we reject the null hypothesis, confirming the alternative hypothesis. For example, if the p-value for our exercise is less than 0.05, it suggests that less than 40% of Americans can correctly answer the question.
• If the p-value is higher, we do not reject the null hypothesis, implying insufficient evidence to say the proportion is different from 40%.

This method ensures a systematic approach to decision-making based on statistical evidence.