Problem 9 - An Introduction to Psychological Statistics

Question

Administrators at a university want to know if students in different majors are more or less extroverted than others. They provide you with descriptive statistics they have for English majors (coded as 1 ) and History majors (coded as 2) and ask you to create a confidence interval of the difference between them. Does this confidence interval suggest that the students from the majors differ? $$\overline{X_{1}}=3.78, \overline{X_{2}}=2.23, s_{1}=2.60, s_{2}=1.15, n_{1}=45, n_{2}=40$$

Step-by-Step Solution

Answer

Yes, the confidence interval suggests a significant difference in extroversion between the majors.

Step 1: Identify key information

The problem involves two groups: English majors and History majors. We're given the sample means \(\overline{X_{1}} = 3.78\) for English majors, \(\overline{X_{2}} = 2.23\) for History majors, the sample standard deviations \(s_{1} = 2.60\) and \(s_{2} = 1.15\), and the sample sizes \(n_{1} = 45\) and \(n_{2} = 40\). Our task is to create a confidence interval for the difference in extroversion.

Step 2: Calculate the standard error for the difference in means

The standard error (SE) for the difference in means is calculated using the formula:\[SE = \sqrt{\frac{s_{1}^2}{n_{1}} + \frac{s_{2}^2}{n_{2}}}\]Substitute the values into the formula:\[SE = \sqrt{\frac{2.60^2}{45} + \frac{1.15^2}{40}} = \sqrt{\frac{6.76}{45} + \frac{1.3225}{40}} = \sqrt{0.1502 + 0.0331} = \sqrt{0.1833} = 0.4283\]

Step 3: Determine the critical value for a 95% confidence interval

For a 95% confidence interval, we'll assume a normal distribution since the sample sizes are relatively large. The critical value \(z\) for a two-tailed test at the 95% level is approximately 1.96.

Step 4: Calculate the confidence interval for the difference

The formula for the confidence interval is:\[(\overline{X_{1}} - \overline{X_{2}}) \pm z \cdot SE\]Compute the difference in means:\[(3.78 - 2.23) = 1.55\]Calculate the confidence interval:\[1.55 \pm 1.96 \cdot 0.4283 = 1.55 \pm 0.8394 = (0.7106, 2.2894)\]

Step 5: Interpret the results

The confidence interval for the difference between the means is (0.7106, 2.2894). Since the interval does not contain zero, it suggests that there is a statistically significant difference in extroversion between English majors and History majors.

Key Concepts

Descriptive Statistics

Descriptive statistics are crucial for understanding and summarizing data. They provide a snapshot of data through measures such as mean, median, standard deviation, and range. In our exercise, descriptive statistics help us understand the extroversion levels of students in different majors.

The mean score, represented as \(\overline{X}\), is a measure of central tendency that indicates the average extroversion score for each group. For English majors, this is \(3.78\), while for History majors, it’s \(2.23\).

Standard deviation, notated as \(s\), tells us about the spread of scores around the mean. A larger standard deviation indicates more variability among the scores. In our exercise, English majors have a higher standard deviation (\(2.60\)) than History majors (\(1.15\)), implying greater variability in extroversion scores among English majors.

Sample size, denoted by \(n\), represents the number of observations in each group. Larger sample sizes usually lead to more reliable statistics. Here, we have 45 English majors and 40 History majors. Together, these descriptive statistics set the stage for more detailed analysis, like confidence intervals.

Extroversion Measurement

Measuring extroversion can offer insights into how outgoing or socially inclined a group of individuals may be. In psychological research, extroversion is often assessed using standardized surveys or questionnaires. The numerical results from these assessments can then be summarized through descriptive statistics.

In the given exercise, the scores reported for English and History majors serve as indicators of their respective extroversion levels. The average score (mean) gives a general idea of the typical extroversion level for each group, while the standard deviation provides insight into how consistent these levels are within each group.

• A higher mean score suggests a group is generally more extroverted.
• The higher standard deviation implies more varied personality types within that group.

Overall, analyzing extroversion scores with these statistical tools helps us compare personality trends across different demographics, like university majors in this scenario.

Statistical Significance

Understanding statistical significance is key in distinguishing meaningful differences or changes within data. A confidence interval is one method used to infer statistical significance. It tells us the range within which we can expect the true population parameter to fall, given our sample data.

In the exercise, we calculated a confidence interval for the difference in extroversion between English and History majors. The interval \((0.7106, 2.2894)\) does not include zero, indicating that the observed difference is statistically significant. This means that there is likely a real difference in extroversion levels between the two groups, not just a difference emerging by chance.

Key points to remember about statistical significance include:

• An interval that excludes zero suggests a true difference between groups.
• The choice of confidence level (here 95%) affects the width of the interval and the likelihood of accurately capturing the true parameter.
• Significance informs us whether differences are enough to be considered impactful, beyond random variation.

By applying these principles, we can make well-founded conclusions about extroversion differences between these student groups.