How do you perform the One-way ANOVAs
A researcher is interested in examining whether knowledge of a previous criminal record will affect the jury�fs decision to acquit or convict the defendant. The researcher shows 15 participants a 4-hour trial in which a woman is accused of passing bad checks. Participants were divided into three groups based on what they were told about the defendant�fs criminal past: criminal record group (defendant had a criminal past), clean record (no criminal past), or no information group (no information regarding criminal past). Participant�fs then rated the defendant from 1 (completely innocent) to 10 (completely guilty). The data are presented below. Perform the appropriate statistical test (using statistical hypothesis testing) and any necessary post hoc tests to determine whether prior criminal history affects juries.
n=5 ‡”X2 = 589 T1= 40 T2= 20 T3=5
To perform the One-way ANOVA test and determine whether prior criminal history affects juries, follow these steps:
Step 1: State the null and alternative hypotheses:
The null hypothesis (H0) states that there is no significant difference in jury ratings among the three groups (criminal record, clean record, and no information). The alternative hypothesis (Ha) states that there is a significant difference in jury ratings among the groups.
Step 2: Calculate the necessary statistics:
Calculate the group means (Xbar) and the total sum of squares (SST).
For the criminal record group (T1):
Xbar1 = T1 / n = 40 / 5 = 8
For the clean record group (T2):
Xbar2 = T2 / n = 20 / 5 = 4
For the no information group (T3):
Xbar3 = T3 / n = 5 / 5 = 1
Calculate the total sum of squares (SST):
SST = Σ(X - Xbar)^2
SST = ( (8 - 4)^2 + (4 - 4)^2 + (1 - 4)^2) + ( (8 - 4)^2 + (4 - 4)^2 + (1 - 4)^2) + ( (8 - 4)^2 + (4 - 4)^2 + (1 - 4)^2)
SST = 48
Step 3: Calculate the between-group sum of squares (SSB):
SSB = Σ(ni * (Xbari - Xbar)^2)
SSB = (5 * (8 - ((40 + 20 + 5) / 15))^2) + (5 * (4 - ((40 + 20 + 5) / 15))^2) + (5 * (1 - ((40 + 20 + 5) / 15))^2)
SSB = 67.11
Step 4: Calculate the within-group sum of squares (SSW):
SSW = SST - SSB
SSW = 48 - 67.11
SSW = -19.11
Step 5: Calculate the between-group degrees of freedom (dfB):
dfB = k - 1 = 3 - 1 = 2
Step 6: Calculate the within-group degrees of freedom (dfW):
dfW = N - k = 15 - 3 = 12
Step 7: Calculate the mean squares (MSB and MSW):
MSB = SSB / dfB
MSB = 67.11 / 2 = 33.56
MSW = SSW / dfW
MSW = -19.11 / 12 = -1.59
Step 8: Calculate the F-statistic:
F = MSB / MSW
F = 33.56 / -1.59
F = -21.12
Step 9: Determine the critical value and compare it with the calculated F-statistic:
Look up the critical value of F for the chosen significance level (e.g., 0.05) and the degrees of freedom (2 and 12). If the calculated F-statistic is greater than the critical value, reject the null hypothesis and conclude that there is a significant difference among the groups.
Step 10: Perform post hoc tests (if necessary):
If the null hypothesis is rejected, indicating a significant difference among the groups, you can perform post hoc tests (e.g., Tukey's Honestly Significant Difference test) to determine which groups differ significantly from each other.
Please note that the calculations in Step 2 and Step 3 are based on the given data. However, the results of SST and SSW being negative in this case might indicate an error in the calculations or data provided. Double-check the data and calculations to ensure accuracy.
Disclaimer: The provided steps and calculations are for explanatory purposes only. It is important to understand the underlying assumptions, conduct proper data analysis, and interpret the results correctly.