91Ó°ÊÓ

In a study entitled How Undergraduate Students Use Credit Cards, it was reported that undergraduate students have a mean credit card balance of \(\$ 3173\) (Sallie Mae, April 2009 ). This figure was an all-time high and had increased \(44 \%\) over the previous five years. Assume that a current study is being conducted to determine if it can be concluded that the mean credit card balance for undergraduate students has continued to increase compared to the April 2009 report. Based on previous studies, use a population standard deviation \(\sigma=\) \(\$ 1000\) a. State the null and alternative hypotheses. b. What is the \(p\) -value for a sample of 180 undergraduate students with a sample mean credit card balance of \(\$ 3325 ?\) c. Using a .05 level of significance, what is your conclusion?

Step by step solution

01

State the Null and Alternative Hypotheses

The null hypothesis (H_0) represents a statement of no effect or no difference. In this case, we assume:\[ H_0: \mu = 3173 \]The alternative hypothesis (H_a) is what we want to test for. Since we want to determine if the mean credit card balance has increased, we have:\[ H_a: \mu > 3173 \]

02

Calculate the Test Statistic

The sample mean (\bar{x}) is 3325, the population mean (\mu_0) is 3173, the population standard deviation (\sigma) is 1000, and the sample size (n) is 180. We use the formula for the z-test statistic:\[ z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} \]Plug in the values:\[ z = \frac{3325 - 3173}{\frac{1000}{\sqrt{180}}} \]First calculate \( \sqrt{180} \), then find \( \frac{1000}{\sqrt{180}} \), and then compute \( z \).

03

Compute the z-value

Calculate \( \sqrt{180} \):\[ \sqrt{180} \approx 13.416 \]Calculate \( \frac{1000}{13.416} \):\[ \frac{1000}{13.416} \approx 74.53 \]Now compute the z-value:\[ z = \frac{3325 - 3173}{74.53} \approx \frac{152}{74.53} \approx 2.04 \]

04

Calculate the p-value

Using the z-value of 2.04, find the p-value from a standard normal distribution table or a calculator that provides cumulative probabilities. You will find:\[ p \approx 0.0207 \]

05

Conclusion Based on the p-value and Significance Level

The significance level (\alpha) is 0.05. Compare the p-value to the significance level:\[ p = 0.0207 < 0.05 \]Since the p-value is less than the significance level, we reject the null hypothesis. This suggests there is sufficient evidence to conclude that the mean credit card balance has increased since April 2009.

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Key Concepts

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

Null Hypothesis

In statistical hypothesis testing, the null hypothesis is a fundamental concept. It serves as a default or starting assumption that there is no effect or change. Specifically, it posits that the parameter of interest remains equal to a benchmark or historical value.
In our example, the null hypothesis (\(H_0\)) is set as the mean credit card balance being \(3173\). This suggests that the current mean balance has not risen above the previously reported value of \(3173\).

• The null hypothesis is an assumption that any observed difference in data is due to sampling or random variation.
• It is often considered the position of no change or "status quo" until evidence suggests otherwise.

Rejecting the null hypothesis, when evidence is strong enough, allows researchers to support their alternative hypothesis, which suggests a new outlook.

Alternative Hypothesis

The alternative hypothesis (\(H_a\)) represents what researchers intend to prove or find evidence for. It is the statement that indicates there is a change or effect being tested.
In our scenario, the alternative hypothesis is formulated as \(H_a: \mu > 3173\).
This aims to show that the current mean credit card balance has, indeed, increased beyond \(3173\).

• The alternative hypothesis challenges the null and seeks to provide sufficient evidence to support claims of change.
• It is reliant on statistical testing to validate itself above the assumptions of the null hypothesis.

Understanding and defining these hypotheses clearly is crucial before performing any statistical tests.

P-Value

The p-value is a key concept in hypothesis testing because it helps determine the statistical significance of the observed results. It quantifies how likely it is to observe the data, or something more extreme, assuming the null hypothesis is true.

• A low p-value (\(< \alpha\)), typically below 0.05, suggests the observed data is improbable under the null hypothesis, leading us to reject it.
• A high p-value (\(\ge \alpha\)) suggests the observed data fits well under the null hypothesis, hence failing to reject it.

For this exercise, the p-value was calculated as approximately \(0.0207\), indicating a significant difference from the null hypothesis at the 0.05 significance level. This p-value suggests sufficient evidence against the null hypothesis.

Z-Test

The z-test is a statistical method used to test hypotheses about the mean of a population when the variance is known and the sample size is large. It compares the sample mean with the population mean to determine if there is a significant difference.
In this exercise, the z-test statistic was employed to assess if the change in undergraduate credit card balances was significant:
\[ z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}\]

• \(\bar{x}\) is the sample mean, \(\mu_0\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
• A higher absolute value of the z-score indicates a greater deviation from the null hypothesis.

For the given sample with a mean balance of \(3325\), the calculated z-score (\(2.04\)) provided statistical evidence to conduct further analysis using the p-value.

Level of Significance

The level of significance, denoted by \(\alpha\), is the threshold set by the researcher to determine whether to accept or reject the null hypothesis. It quantifies the probability of rejecting the null hypothesis when it is actually true (Type I error).

• A commonly used level of significance is 0.05, implying a 5% risk of concluding a difference when there is none.
• The level of significance acts as a benchmark for determining whether the observed p-value indicates a significant effect.

In this study, we used an \(\alpha\) of 0.05. Since the p-value (\(0.0207\)) was less than this significance level, it led us to reject the null hypothesis, suggesting strong evidence of an increase in the mean credit card balance for undergraduates.