Problem Description:
In this SPSS homework, we will perform a one-way analysis of variance (ANOVA) to compare the means of religiosity scores among a sample of participants identifying as conservative, liberal, or independent. The goal is to determine if there are significant differences in religiosity levels among these three groups.
Solution
Below are data measuring religiosity (on a 1-5 scale) among a sample of participants identifying as conservative (µ1), liberal (µ2), or independent (µ3).
| Conservative | Liberal | Independent | |||
|---|---|---|---|---|---|
| X | X2 | X | X2 | X | X2 |
| 4 | 16 | 3 | 9 | 2 | 4 |
| 3 | 9 | 2 | 4 | 1 | 1 |
| 3 | 9 | 1 | 1 | 3 | 9 |
| 4 | 16 | 2 | 4 | 2 | 4 |
| 3 | 9 | 1 | 1 | 2 | 4 |
| 2 | 4 | 2 | 4 | 1 | 1 |
| 3 | 9 | 1 | 1 | ||
| 2 | 4 | 2 | 4 | ||
| Total | |
|---|---|
| ΣX | ΣX2 |
| 49 | 127 |
| N = 22 | |
1. State the null and alternative hypotheses.
H0: All the treatment means are the same (μ_1=μ_2=μ_3)
H1: At least one of the treatment means is different from others μ_i≠μ_j for at least one i≠j, i=1,2,3)
2. Calculate and report the degrees of freedom for the F test.
df_Between=3-1=2
df_Total=22-1=21
df_Within=21-2=19
The degrees of freedom for the F test is (df_Between,df_Within )=(2,19)
3. Determine and report the critical F value using .05 significance level.
Critical F value= 3.5219
4. Complete the table on the previous page.
5. Using the values from the table, calculate the total sum of squares (SS) for the scores. Show your work!
SST=∑▒X^2 -〖(∑▒〖X)〗〗^2/N
=127-49^2/22
=17.8636
6. Next, calculate the between-groups sum of squares (SS). Show your work!
SSB=∑((T_j^2)/n)-〖(∑▒〖X)〗〗^2/N,
where T_j=∑▒X
SSB=24^2/8+14^2/8+11^2/6-49^2/22
=7.5303
7. Subtract the between-groups sum of squares from the total sum of squares to obtain the within-groups sum of squares.
SSW=SST-SSB
SSW=17.8636-7.5303
=10.3333
8. Calculate and report the mean of squares (MS) for the between-groups variability and the MS for the within-groups variability.
MSB=SSB/(k-1)
=7.5303/(3-1)
=3.76515
MSB=SSW/(N-k)
=10.3333/(22-3)
=0.5438579
9. Calculate the F statistic
F=MSB/MSW=3.76515/0.5438579= 6.92304
10. Should we reject the null hypothesis?
Since the F statistic (6.9230) is greater than the critical value (3.5219), we reject the null hypothesis and conclude that at least one of the treatment means is significantly different from others.
SPSS
Well done! Now let’s explore how effective pain relievers like acetaminophen (Tylenol) and ibuprofen (Advil) are. Dr. Douglas, a sensory psychologist, wanted to see if medications such as these affect a person’s pain threshold (how long it takes to perceive a stimulus as painful). Twelve male undergraduates volunteered for the study, and she randomly assigned them to three groups. Each participant was given a pill – either a placebo, ibuprofen, or acetaminophen. Participants waited one hour for the medication to take effect, then placed his hand in a bucket of ice water. He was instructed to remove his hand from the ice water as soon as it became painful.
- Open the “PainResponse” file in the ‘Homework Data Files’ folder on Canvas.
- Copy and paste the data into SPSS
11. Conduct an ANOVA with these data to determine if the amount of time it took participants to remove their hand significantly differed across the three groups. Copy and paste the output of the ANOVA into the assignment sheet below. (2 points). (“Analyze” > “General Linear Model” > “Univariate”)
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 |