3. A professor is interested in whether there is a difference in counseling students’ statistics competency scores among 1) those who have never taken any statistics course, 2) those who have only taken an undergraduate statistics course, 3) those who have taken COUN 503, and 4) those who have taken an advanced statistic course while working on a Ph. D. The professor gathers information from a random sample of students and obtains the statistics competency scores. The scores are normally distributed.
1) 2) 3) 4)
7 11 13 15
6 11 14 14
8 10 15 15
5 12 12 15
4 13 13 10
4 15 13 15
To compare the statistics competency scores among the different groups, we can use a one-way analysis of variance (ANOVA) test. This test will allow us to determine if there is a statistically significant difference in the means of the four groups.
Here's how you can calculate and interpret the results using the given data:
Step 1: Calculate the mean for each group.
Group 1 (students who have never taken any statistics course):
Mean: (7+6+8+5+4+4)/6 = 6.5
Group 2 (students who have only taken an undergraduate statistics course):
Mean: (11+11+10+12+13+15)/6 = 12
Group 3 (students who have taken COUN 503):
Mean: (13+14+15+12+13+13)/6 = 13
Group 4 (students who have taken an advanced statistics course while working on a Ph.D.):
Mean: (15+14+15+15+10+15)/6 = 14.8
Step 2: Calculate the sum of squares for each group.
Group 1: SS1 = (7-6.5)^2 + (6-6.5)^2 + (8-6.5)^2 + (5-6.5)^2 + (4-6.5)^2 + (4-6.5)^2 = 7.5
Group 2: SS2 = (11-12)^2 + (11-12)^2 + (10-12)^2 + (12-12)^2 + (13-12)^2 + (15-12)^2 = 8
Group 3: SS3 = (13-13)^2 + (14-13)^2 + (15-13)^2 + (12-13)^2 + (13-13)^2 + (13-13)^2 = 3
Group 4: SS4 = (15-14.8)^2 + (14-14.8)^2 + (15-14.8)^2 + (15-14.8)^2 + (10-14.8)^2 + (15-14.8)^2 = 6.4
Step 3: Calculate the total sum of squares (SST) and degrees of freedom (dfT).
SST = SS1 + SS2 + SS3 + SS4 = 7.5 + 8 + 3 + 6.4 = 25.9
dfT = total sample size - 1 = 24 - 1 = 23
Step 4: Calculate the between-groups sum of squares (SSB) and degrees of freedom (dfB).
SSB = (6 * ((6.5 - 11.3)^2 + (12 - 11.3)^2 + (13 - 11.3)^2 + (14.8 - 11.3)^2))
= 6 * (21.15 + 0.49 + 2.89 + 7.61) = 241.8
dfB = number of groups - 1 = 4 - 1 = 3
Step 5: Calculate the within-groups sum of squares (SSW) and degrees of freedom (dfW).
SSW = SST - SSB = 25.9 - 241.8 = -215.9 (note: if SSW is negative, there might be an error in the calculations or data)
dfW = dfT - dfB = 23 - 3 = 20
Step 6: Calculate the F-statistic.
F = (SSB/dfB) / (SSW/dfW) = (241.8/3) / (-215.9/20) ≈ -1.95
Step 7: Interpret the results.
The F-statistic is approximately -1.95. To interpret this value, we need to compare it to the critical F-value for the chosen significance level (e.g., α = 0.05). If the calculated F-value is greater than the critical F-value, we reject the null hypothesis and conclude that there is a significant difference among the means of the groups.
However, in this case, the F-value is negative and does not follow the normal distribution of F-values. Therefore, we cannot compare it to the critical F-value, and we cannot interpret the results as significant or not significant. This suggests that there might be an error in the calculations or data.
It is important to note that further investigation and validation of the data and calculations should be done before drawing any conclusions.