91Ó°ÊÓ

The paper "Debt Literacy, Financial Experiences and Over-Indebtedness" (Social Science Research Network, Working paper W14808, 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

Identify hypotheses for question (a)

Identify the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). Here, \(H_0: p = 0.4\) and \(H_a: p < 0.4\). The parameter of interest, \(p\), is the true proportion of adult Americans who can answer this question correctly.

02

Calculate test statistic for question (a)

Formula for test statistic is (p̂ - p) / sqrt( (p(1 - p)) /n). Given that the sample proportion \(p̂ = 354/1000 = 0.354\), the number of sample \(n = 1000\), and \(p\) is in the null hypothesis \(H_0\), the value of test statistic is calculated as (0.354 - 0.4) / sqrt((0.4 * 0.6) / 1000) = -3.22.

03

Calculate P-value for question (a)

Given one-tailed test in the left direction, the P-value is the probability of getting a test statistic z ≤ -3.22 under the null hypothesis. Here, by using Standard Normal Distribution table or calculator, the P-value is calculated to be near 0.0006.

04

Make decision for question (a)

The rule is: reject \(H_0\) if P-value ≤ 0.05. Here, P-value = 0.0006, which is less than 0.05. Therefore, reject \(H_0\). This indicates that there is strong evidence that the proportion of adult Americans who can answer this question correctly is less than 40%.

05

Identify hypotheses for question (b)

Similar to step 1, here the null hypothesis \(H_0\) is \(H_0: p = 0.33\) and the alternative hypothesis \(H_a\) is \(H_a: p > 0.33\). The parameter of interest, \(p\), is the true proportion of adult Americans who would select a wrong answer to the question.

06

Calculate test statistic for question (b)

Using a similar method, given the sample proportion \(p̂ = 0.378\) and \(n = 1000\), the test statistic is calculated as (0.378 - 0.33) / sqrt((0.33 * 0.67) / 1000) = 2.88.

07

Calculate P-value for question (b)

Given one-tailed test in the right direction, the P-value is the chance of getting a test statistic z ≥ 2.88 under the null hypothesis. Here, by using Standard Normal Distribution table or calculator, the P-value is calculated to be near 0.002.

08

Make decision for question (b)

Similar to step 4, P-value = 0.002, which is less than 0.05. Therefore, reject \(H_0\). This indicates that there is strong evidence more than one-third of adult Americans would select a wrong answer to this question.

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Key Concepts

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

Proportions

In hypothesis testing, proportions are used to represent a fraction or a part of a whole. For instance, in the context of the problem, the proportion could represent the fraction of the total number of respondents who are able to correctly answer a question.

When we say the proportion of a certain outcome is 0.4, it means 40% of the sample has that specific characteristic. In our case, we're assessing if the true proportion of adult Americans who can correctly answer a financial literacy question, based on a sample of 1000, is less than 0.4.

It's important in statistical analyses because:

• Proportions provide a way to express part-to-whole relationships in a dataset.
• They allow for comparison between different groups or populations.

Null Hypothesis

The null hypothesis is a statement used in statistics that suggests no effect or no difference in a study. It's a way to test whether any observed effects, like a difference in proportions, are due to chance.

In hypothesis testing, the null hypothesis (often represented as \(H_0\)) is the hypothesis that there is no effect present, or it's the hypothesis of "no difference." For example:

• In question (a), the null hypothesis could be that the proportion of adults who answer the question correctly \( (p) \) is 0.4.
• In question (b), it could be that the proportion of adults who would select a wrong answer \( (p) \) is 0.33, or exactly one-third.

The purpose of the null hypothesis is to enable a statistical test which can then provide evidence for or against it. If we reject the null hypothesis, it indicates that we see enough evidence to support an alternative hypothesis, suggesting there is an effect or a difference.

P-value

The p-value is a crucial part of hypothesis testing. It helps us determine the strength of the evidence against the null hypothesis. The p-value quantifies the probability of obtaining the observed sample results, or something more extreme, assuming that the null hypothesis is true.

A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so it gets rejected. Here’s how it applies to our case:

• In question (a), a p-value of 0.0006 suggests strong evidence against the null hypothesis that states 40% of the population knows the correct answer.
• In question (b), the p-value of 0.002 provides strong evidence that more than one-third of adults would pick the wrong answer.

Remember, a p-value doesn't provide the probability that the null hypothesis is true or false. It simply indicates the likelihood of the sampled data under this hypothesis.

Test Statistic

The test statistic is a standardized value calculating how far the sample statistic is from the null hypothesis value. This calculation helps in determining whether the null hypothesis should be rejected.

It is determined through a formula that incorporates the sample proportion, the assumed population proportion under the null hypothesis, and the sample size. The formula used is:

• \[ \text{Test Statistic} = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \]

Where \( \hat{p} \) is the sample proportion, \( p \) is the population proportion under the null hypothesis, and \( n \) is the sample size.

For question (a), the test statistic was calculated to be -3.22, while for question (b), it was 2.88. These values, when compared against critical values from the standard normal distribution, help in making a decision about the null hypothesis.

• A test statistic far from zero provides evidence against the null hypothesis.
• The sign of the test statistic helps in determining the direction of the effect.