2.  The Computer Anxiety Rating Scale (CARS) measures an individual’s level of computer anxiety on a scale from 20 (no anxiety) to 100 (highest level of anxiety).  Researchers at one university administered the CARS to 172 business students.  One of the objectives of the study was to determine whether there is a difference in the level of computer anxiety experienced by female and male students.  They found the following:


		Males:		n = 100	Mean = 41.26		Std Dev = 13.35
		Females:	n =   72	Mean = 36.85		Std Dev =   9.42

(a)  At a 0.05 level of significance and assuming the population variances are equal, is there evidence of a difference in mean computer anxiety experienced by female and male business students?

(b)  Interpret the p-value and the 95% confidence interval estimate of the difference between the population means.


Let's break this down and solve each part of the question step by step.

### Part (a) Hypothesis Testing

We are asked to test whether there is a **difference in the mean computer anxiety experienced by female and male students**. The appropriate test here is a **two-sample t-test**, assuming equal population variances (as stated in the problem).

#### Step 1: State the Hypotheses

We want to test if there's a difference between the means of two populations. The hypotheses are:

- **Null Hypothesis (H₀):** There is no difference in the mean computer anxiety between males and females.
  \[
  H₀: \mu_{\text{males}} = \mu_{\text{females}} \quad \text{or equivalently} \quad \mu_{\text{males}} - \mu_{\text{females}} = 0
  \]
  
- **Alternative Hypothesis (H₁):** There is a difference in the mean computer anxiety between males and females.
  \[
  H₁: \mu_{\text{males}} \neq \mu_{\text{females}} \quad \text{or equivalently} \quad \mu_{\text{males}} - \mu_{\text{females}} \neq 0
  \]

#### Step 2: Calculate the Test Statistic

To perform the two-sample t-test, we use the formula for the t-statistic:
\[
t = \frac{\bar{X}_1 - \bar{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}
\]
where:
- \(\bar{X}_1 = 41.26\) (mean for males),
- \(\bar{X}_2 = 36.85\) (mean for females),
- \(n_1 = 100\) (sample size for males),
- \(n_2 = 72\) (sample size for females),
- \(S_p\) is the pooled standard deviation.

The pooled standard deviation \(S_p\) is calculated as:
\[
S_p = \sqrt{\frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}}
\]
where \(S_1 = 13.35\) (standard deviation for males) and \(S_2 = 9.42\) (standard deviation for females).

Let me calculate the values.The pooled standard deviation \( S_p \) is approximately 11.87, and the calculated t-statistic is approximately \( t = 2.40 \) with 170 degrees of freedom.

#### Step 3: Determine the Critical Value or p-value

We will now compare the calculated t-statistic with the critical value from the t-distribution for a two-tailed test at a significance level of \( \alpha = 0.05 \).

Let me calculate the p-value next.It seems like I can't do more advanced data analysis right now. Please try again later.

However, you can find the p-value using a t-distribution table or software by looking up a two-tailed test with a t-statistic of 2.40 and 170 degrees of freedom. If the p-value is less than 0.05, you would reject the null hypothesis, indicating a significant difference between the computer anxiety levels of males and females.

### Part (b) Interpretation

1. **P-value Interpretation**:
   - The p-value helps us decide whether to reject the null hypothesis. If the p-value is smaller than 0.05, we reject the null hypothesis, meaning there is enough evidence to suggest a difference between male and female students' computer anxiety levels.

2. **95% Confidence Interval**:
   The confidence interval estimates the range within which the true difference in population means lies. If the interval does not include 0, it suggests a significant difference between male and female students' anxiety levels.

---

Let me know if you need more details or have any questions. Here are some related questions:

1. How do you interpret confidence intervals in hypothesis testing?
2. How can we calculate p-values manually?
3. Why is the assumption of equal variances important in t-tests?
4. How does changing the level of significance (α) affect hypothesis testing?
5. What is the impact of sample size on the reliability of t-tests?

**Tip**: Always visualize your data to get an initial understanding of patterns before conducting hypothesis tests.

2. The Computer Anxiety Rating Scale (CARS) measures an individual’s level of computer anxiety on a scale from 20 (no anxiety) to 100 (highest level of anxiety). Researchers at one university administered the CARS to 172 business students. One of the objectives of the study was to determine whether there is a difference in the level of computer anxiety experienced by female and male students. They found the following: Males: n = 100 Mean = 41.26 Std Dev = 13.35 Females: n = 72 Mean = 36.85 Std Dev = 9.42 (a) At a 0.05 level of significance and assuming the population variances are equal, is there evidence of a difference in mean computer anxiety experienced by female and male business students? (b) Interpret the p-value and the 95% confidence interval estimate of the difference between the population means.

Explore the statistical analysis of the difference in computer anxiety between male and female business students using a two-sample t-test. Learn how to calculate the test statistic, interpret the p-value, and construct a 95% confidence interval for the difference in population means. Suitable for undergraduate students in statistics or business analytics courses.

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