Problem 10 - An Introduction to Psychological Statistics

Question

You want to know if college students have more stress in their daily lives than the general population \((\mu=12)\), so you gather data from 25 people to test your hypothesis. Your sample has an average stress score of \(\bar{X}=13.11\) and a standard deviation of \(s=3.89 .\) Use a 1-sample \(t\) -test to see if there is a difference.

Step-by-Step Solution

Answer

The stress level of college students is not significantly different from the general population.

Step 1: Define the Hypotheses

Define the null hypothesis \((H_0)\) and the alternative hypothesis \((H_a)\). The null hypothesis states that the mean stress score of college students is the same as the general population, \( \mu = 12 \). The alternative hypothesis states that the mean stress score of college students is different from the general population, \( \mu eq 12 \).

Step 2: Calculate the Test Statistic

Use the formula for the t-statistic: \[ t = \frac{\bar{X} - \mu}{\frac{s}{\sqrt{n}}} \]Substitute the given values: \( \bar{X} = 13.11 \), \( \mu = 12 \), \( s = 3.89 \), and \( n = 25 \).Calculate: \[ t = \frac{13.11 - 12}{\frac{3.89}{\sqrt{25}}} = \frac{1.11}{0.778} \approx 1.426 \]

Step 3: Determine the Degrees of Freedom

The degrees of freedom \((df)\) for a 1-sample t-test is \(n - 1\). Here, \(n = 25\), so \(df = 24\).

Step 4: Identify the Critical Value

Using a t-table and a significance level \(\alpha = 0.05\) for a two-tailed test with \(df = 24\), the critical t-value is approximately \(2.064\).

Step 5: Compare Test Statistic to Critical Value

Compare the calculated t-statistic \(1.426\) to the critical t-value \(2.064\).

Step 6: Make a Decision

Since the absolute value of the t-statistic (1.426) is less than the critical t-value (2.064), we fail to reject the null hypothesis.

Step 7: Interpret the Results

The data does not provide sufficient evidence to conclude that the mean stress score of college students is different from the general population mean of 12 with a significance level of 0.05.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter. In this context, we are deciding whether the average stress levels of college students differ from those of the general population. The process begins by stating two hypotheses: the null hypothesis and the alternative hypothesis.

• The null hypothesis, denoted as \(H_0\), posits that there is no difference, meaning the stress level of college students is equal to that of the general population (\( \mu = 12 \)).
• The alternative hypothesis, denoted as \(H_a\), argues that there is a difference, implying that college students' stress levels are not equal to those of the general population (\( \mu eq 12 \)).

The primary goal here is to gather evidence to either support or reject the null hypothesis by using statistical tests. This is an essential process in research, as it helps determine the validity of a claim based on sample data.

Degrees of Freedom

Degrees of freedom refer to the number of values in a statistical calculation that are free to vary. They play a crucial role in accurately reflecting the variability of the dataset under consideration. In the context of the 1-sample t-test, degrees of freedom are calculated as the sample size \( n \) minus one (\( df = n - 1 \)).
In our exercise, we analyzed data from 25 participants. Therefore, the degrees of freedom is setup as \( df = 25 - 1 = 24 \).

• Degrees of freedom influence the shape of the t-distribution, which affects the critical values used to decide whether to reject or accept the null hypothesis.
• A higher number of degrees of freedom generally result in a distribution that more closely resembles the normal distribution.

Understanding degrees of freedom helps in selecting the appropriate statistical table needed for identifying critical values during hypothesis testing.

Significance Level

The significance level, often denoted by \( \alpha \), is a threshold set by the researcher that defines the probability of rejecting the null hypothesis when it is actually true. It represents the risk of a Type I error, which is the incorrect rejection of a true null hypothesis.
In our scenario, a significance level of 0.05 was used, depicting a 5% risk of concluding that there is a difference in stress levels when there is none. This is a conventional choice in many scientific studies and indicates a moderate level of confidence in the test results.

• The 0.05 significance level implies that results obtaining a p-value lower than 0.05 would lead researchers to reject the null hypothesis.
• The smaller the significance level, the stronger the evidence must be to reject the null hypothesis, meaning stricter criteria for claiming a significant difference.

Carefully choosing an appropriate significance level is a vital part of planning a statistical test, influencing the reliability and interpretation of results.

Null Hypothesis

The null hypothesis is a foundational concept in statistical testing and provides a baseline or "default position." It asserts that there is no effect or no difference between groups or conditions being compared. In the stress level example, the null hypothesis \( H_0 \) is that the mean stress score for college students equals that of the general population, \( \mu = 12 \).

• The null hypothesis is tested against the alternative hypothesis \( H_a \), which suggests a deviation from the specified population mean.
• In hypothesis testing, we often assume the null hypothesis is true and use sample data to determine whether there is sufficient evidence to reject it.

Understanding the null hypothesis is crucial because it underpins the entire hypothesis testing procedure, aiding in drawing valid conclusions from the test results. By failing to reject the null hypothesis in our example, we conclude that there isn’t enough evidence to say college students have different stress levels than the general population.