Worked Problem.....One-Way Analysis of Variance.....Psyc205/Soc205.....Vernoy

full2.gif (11401 bytes)
One-Way Analysis of Variance
(A worked Problem: Procedure explained in more in Chapter 13)

As a psychologist who works with people who have Down's syndrome, you design a study intended to determine which rewards are most effective for use in training your patients. You select four different, independent, groups of six patients and record the number of days it takes to teach them a particular task, with each group receiving one of four types of rewards: Reward 1, Reward 2, Reward 3, and Reward 4. the number of days are given in the following table.

Reward 1	Reward 2	Reward 3	Reward 4
3 5 6 2 1 2	6 7 9 7 11 6	9 10 15 12 11 10	12 13 15 18 15 13

Use the data above to conduct a one-way analysis of variance.

I. State your hypotheses

Null hypothesis: The type of reward does not make a difference in the number of days required for
Down's Syndrome patients to learn a task.

Research hypothesis: The type of reward makes a difference in the number of days required for
Down's syndrome patients to learn a task

II. After stating the hypotheses, always begin an analysis of variance problem by computing all required sums.

Reward 1	Reward 2	Reward 3	Reward 4
3 5 6 2 1 2	6 7 9 7 11 6	9 10 15 12 11 10	12 13 15 18 15 13

III. Compute SS_total

wp1an 6.gif (2132 bytes)

IV. Compute SS_bg

wp1an 7.gif (4018 bytes)

V. Compute SS_wg

VI. Compute df_total

df_total = N_total - 1 = 24 -1 = 23

VII. Compute df_bg

df_bg = k-1 = 4 - 1 = 3

VIII. Compute df_wg

df_wg = (n₁-1) + (n₂-1) + (n₃-1) + (n₄-1) = (6-1) + (6-1) + (6-1) + (6-1)
df_wg = 5 + 5 + 5 + 5 = 20

IX. Compute MS_bg

wp1an 9.gif (1342 bytes)

X. Compute MS_wg

wp1an 10.gif (1346 bytes)

XI. Compute F

wp1an 11.gif (1365 bytes)

XII. Find the critical value of the F ratio in Table F and determine the significance of F

    A. df = (3, 20)
    B. Critical Value at alpha = .05 is 3.10
    C. Because F > Critical Value we can reject the Null Hypothesis and accept the Research Hypothesis
    D. The probablity of an F ratio this large happening just by chance is less than .05 (p < .05).

XIII. Create the Source Table

Source	SS	df	MS	F	p
Between	413.500	3	137.833	32.685	<.05
Within	84.333	20	4.217
Total	497.833	23

XIV. Because F is greater than the critical value we must compute the HSD

    A. MS_wg = 4.217
    B. n = 6    (Hint: The number of scores in each group)
    C. q = 3.96    (Hint: To enter table Q use k = 4...k is the number of groups...and df_wg = 20)
    D. Compute HSD

wp1an 12.gif (1759 bytes)

XV. Compare all pairs of means

wp1an 13.gif (2997 bytes)

XVI. Conclusions

    A. Reward 2 is significantly greater than Reward 1
    B. Reward 3 is significantly greater than Reward 1
    C. Reward 4 is significantly greater than Reward 1
    D. Reward 3 is significantly greater than Reward 2
    E.   Reward 4 is significantly greater than Reward 2
    F.   Reward 3 and Reward 4 are not significantly different from one another.