## Problem Description

In the ANOVA assignment, we delve into the world of statistics and ANOVA testing to examine the impact of political leanings on religiosity. We are presented with data on the religiosity levels of a sample of individuals identifying as conservative, liberal, or independent. Our objective is to determine whether there are significant differences in religiosity among these groups. This analysis involves the application of ANOVA testing, critical value calculations, and various statistical calculations.

**Solution**:

**Step 1: Formulating Hypotheses
**

**Null Hypothesis (H0)**: All treatment means are the same (μ1=μ2=μ3)**Alternative Hypothesis (H1)**: At least one of the treatment means is different from others (μi≠μj for at least one i≠j, i=1,2,3)

**Step 2: Degrees of Freedom for F Test
**

- df_Between = 3 - 1 = 2
- df_Total = 22 - 1 = 21
- df_Within = 21 - 2 = 19
- Degrees of freedom for the F test is (df_Between, df_Within) = (2, 19)

**Step 3: Critical F Value
**

- Critical F value at 0.05 significance level: 3.5219

**Step 4: Calculation of Sums of Squares (SS)
**

- Total Sum of Squares (SST) = ∑X^2 - (∑X)^2/N = 127 - 49^2/22 = 17.8636

**Step 5: Between-Groups Sum of Squares (SSB)
**

- SSB = ∑((Tj^2)/n) - (∑X)^2/N, where Tj = ∑X
- SSB = (24^2/8) + (14^2/8) + (11^2/6) - 49^2/22 = 7.5303

**Step 6: Within-Groups Sum of Squares (SSW)
**

- SSW = SST - SSB
- SSW = 17.8636 - 7.5303 = 10.3333

**Step 7: Mean of Squares (MS)
**

- MSB = SSB / (k-1) = 7.5303 / (3-1) = 3.76515
- MSW = SSW / (N-k) = 10.3333 / (22-3) = 0.5438579

**Step 8: Calculate the F Statistic
**

- F = MSB / MSW = 3.76515 / 0.5438579 = 6.92304

**Step 9: Hypothesis Testing
**

- Since the F statistic (6.9230) is greater than the critical value (3.5219), we reject the null hypothesis. We conclude that at least one of the treatment means is significantly different from others.

Source |
Sum of Squares (SS) |
Degrees of Freedom (df) |
Mean Square (MS) |
F Statistic |
Significance (Sig.) |
---|---|---|---|---|---|

Between-Groups | 7.5303 | 2 | 3.76515 | 6.92304 | < 0.05 (significant) |

Within-Groups | 10.3333 | 19 | 0.5438579 | ||

Total | 17.8636 | 21 |

**Table 1: Summary of ANOVA Test Results**

**Additional Analysis - SPSS:
**

In a separate analysis conducted in SPSS, we explored the effects of pain relievers on an individual's pain threshold. Dr. Douglas, a sensory psychologist, conducted a study with male undergraduates who were given either a placebo, ibuprofen, or acetaminophen. Their pain threshold, measured by the time it took to remove their hand from ice water, was analyzed using ANOVA.

**Results of ANOVA in SPSS:
**

**Mathematica code:
**

Tests of Between-Subjects Effects

Dependent Variable: Time

Source Type III Sum of Squares df Mean Square F Sig.

Corrected Model 62.000a 2 31.000 1.348 .308

Intercept 5043.000 1 5043.000 219.261 .000

Treatment 62.000 2 31.000 1.348 .308

Error 207.000 9 23.000

Total 5312.000 12

Corrected Total 269.000 11

a. R Squared = .230 (Adjusted R Squared = .059)

This analysis suggests that the time taken to perceive pain did not significantly differ across the three groups, with a p-value of 0.308.

In conclusion, these analytical techniques, including ANOVA and statistical tests, are invaluable tools for uncovering meaningful insights and making informed decisions in diverse areas, from religiosity and political leanings to sensory psychology.