Problem Description
In a comprehensive statistical analysis assignment using SAS of midterm grades, we explore the influence of race and gender on students' performance across various subjects. We aim to answer whether race or gender significantly impacts midterm grades. Using SAS software, we analyze race-based grade distributions, test for gender-based proportions, assess normality in reading scores, and conduct t-tests to compare subject scores by gender. Our goal is to understand the factors that affect academic performance and draw meaningful conclusions.
Part 1: Analysis by Race
Statistics for Midterm Grades Based on Race
| Race Code | Frequency | Percent | Test Percent | Cumulative Frequency | Cumulative Percent |
|---|---|---|---|---|---|
| 1 | 24 | 43.64 | 45.00 | 24 | 43.64 |
| 2 | 11 | 20.00 | 20.00 | 35 | 63.64 |
| 3 | 20 | 36.36 | 35.00 | 55 | 100.00 |
Table 1: Race-Based Midterm Grade Distribution
Chi-Square Test for Specified Proportions
- Chi-Square: 0.0519
- Degrees of Freedom (DF): 2
- p-value (Pr>ChiSq): 0.9744
The p-value (0.9744) is greater than the significance level (5%), indicating that there is no significant difference between the race percentages and the specified values (0.45, 0.2, and 0.35, respectively).
Part 2: Analysis by Gender
Statistics for Midterm Grades Based on Gender
| Race Code | Frequency | Percent | Test Percent | Cumulative Frequency | Cumulative Percent |
|---|---|---|---|---|---|
| 1 | 24 | 43.64 | 45.00 | 24 | 43.64 |
| 2 | 11 | 20.00 | 20.00 | 35 | 63.64 |
| 3 | 20 | 36.36 | 35.00 | 55 | 100.00 |
Table 2: Gender-Based Midterm Grade Distribution
Chi-Square Test for Equal Proportions
- Chi-Square: 1.4727
- Degrees of Freedom (DF): 1
- p-value (Pr>ChiSq): 0.2249
The p-value (0.2249) is greater than the significance level (5%), indicating that there is no significant difference between the proportions of males and females.
Part 3: Normal Distribution Test
Fitted Normal Distribution for Reading Scores
| Goodness-of-Fit Tests for Normal Distribution | ||||
|---|---|---|---|---|
| Test | DF | Statistic | p Value | |
| Chi-squared | 4 | 3.489507482 | Pr> | 0.479476 |
Table 3: Fitted Normal Distribution for Reading Scores
Goodness-of-Fit Tests for Normal Distribution
- Chi-squared Test
- Degrees of Freedom (DF): 4
- Statistic: 3.4895
- p-value (Pr> χ^2): 0.4795
The p-value (0.4795) is greater than the significance level (5%), suggesting that student reading scores follow a normal distribution with a mean of 47.8 and a standard deviation of 8.8.
Part 4: Independent Sample t-Test
Comparison of Scores by Subject and Gender
| Subjects | Method | Variances | DF | t Value | Pr > |t| |
|---|---|---|---|---|---|
| Read |
Pooled | Equal | 53 | 0.04 | 0.9650 |
| Satterthwaite | Unequal | 52.414 | 0.05 | 0.9636 | |
| write | Pooled | Equal | 53 | -1.85 | 0.0693 |
| Satterthwaite | Unequal | 45.648 | -1.83 | 0.0731 | |
| math | Pooled | Equal | 53 | 0.09 | 0.9325 |
| Satterthwaite | Unequal | 52.272 | 0.09 | 0.9301 | |
| science | Pooled | Equal | 53 | 0.82 | 0.4172 |
| Satterthwaite | Unequal | 48.004 | 0.82 | 0.4164 | |
| socst | Pooled | Equal | 53 | -1.34 | 0.1868 |
| Satterthwaite | Unequal | 50.542 | -1.36 | 0.1787 |
Table 4: Comparison of Scores by Subject and Gender (Pooled and Satterthwaite Variance)
Null Hypotheses:
- There is no difference between the mean scores of (read, write, math, science, socst) for males and females.
Alternative Hypotheses:
- There is a difference between the mean scores of (read, write, math, science, socst) for males and females.
For each subject (read, write, math, science, socst), two types of t-tests were conducted: Pooled (Equal Variance) and Satterthwaite (Unequal Variance).
The p-value for all t-tests conducted is greater than the significance level (0.05), indicating that there is no significant difference between the mean scores across genders.
Part 5: Test for Equality of Means
Comparison of Sciences and Mathematics Scores of Females
| Method | Variances | DF | t Value | Pr > |t| |
|---|---|---|---|---|
| Pooled | Equal | 44 | 0.97 | 0.3390 |
| Satterthwaite | Unequal | 42.021 | 0.97 | 0.3393 |
Table 5: Comparison of Sciences and Mathematics Scores of Females (Pooled and Satterthwaite Variance)
Sciences and Mathematics Scores of Females
- Pooled (Equal Variance)
- Degrees of Freedom (DF): 44
- t Value: 0.97
- p-value (Pr> |t|): 0.3390
- Satterthwaite (Unequal Variance)
- Degrees of Freedom (DF): 42.021
- t Value: 0.97
- p-value (Pr> |t|): 0.3393
The p-values (0.3390 and 0.3393) are both greater than the significance level (0.05), indicating that there is no significant difference between the population mean of mathematics and science scores among females.
Writing and Social Sciences Scores of Females
- Pooled (Equal Variance)
- Degrees of Freedom (DF): 44
- t Value: 0.11
- p-value (Pr> |t|): 0.9129
- Satterthwaite (Unequal Variance)
- Degrees of Freedom (DF): 43.84
- t Value: 0.11
- p-value (Pr> |t|): 0.9129
The p-values (0.9129 for both tests) are greater than the significance level (0.05), indicating no significant difference between the population mean of writing and social science scores among females.
In conclusion, the analysis shows that there are no significant differences in midterm grades based on race or gender. Additionally, the scores follow a normal distribution, and there are no significant differences in mean scores across subjects and genders.