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QUESTION 3 Students were given different drug treatment before revising for their examinations. some were given a memory drug, some a placebo and others no treatment. the exam scores (%) are shown for the three groups: MEMORY DRUG: 70,77,83,90,97. Mean: 83.40, variance: 112.40 PLACEBO: 37,43,50, 57, 63. mean:50.00, variance: 109.00 NO TREATMENT: 2, 10, 17, 23, 30 . Mean: 16.60, Variance: 112.30 GRAND MEAN: 50.00 GRAND VARIANCE: 892.14 (A) Carry out an analysis of variance to determine wether the treatments have different effects on the performance of the student (use alpha = 0.05)

Question

QUESTION 3 Students were given different drug treatment before revising for their examinations. some were given a memory drug, some a placebo and others no treatment. the exam scores (%) are shown for the three groups: MEMORY DRUG: 70,77,83,90,97. Mean: 83.40, variance: 112.40 PLACEBO: 37,43,50, 57, 63. mean:50.00, variance: 109.00 NO TREATMENT: 2, 10, 17, 23, 30 . Mean: 16.60, Variance: 112.30 GRAND MEAN: 50.00 GRAND VARIANCE: 892.14 (A) Carry out an analysis of variance to determine wether the treatments have different effects on the performance of the student (use alpha = 0.05)

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Solution

To carry out an analysis of variance (ANOVA), we need to calculate the sum of squares between groups (SSB), sum of squares within groups (SSW), and total sum of squares (SST).

  1. Calculate the SSB (Sum of Squares Between groups): This is the sum of the squared differences between each group mean and the grand mean, multiplied by the number of observations in each group.

    SSB = Σ [n*(mean of each group - grand mean)^2]

    For Memory Drug: 5*(83.4 - 50)^2 = 4445 For Placebo: 5*(50 - 50)^2 = 0 For No Treatment: 5*(16.6 - 50)^2 = 4445

    SSB = 4445 + 0 + 4445 = 8890

  2. Calculate the SSW (Sum of Squares Within groups): This is the sum of the variances of each group, multiplied by the number of observations in each group minus 1.

    SSW = Σ [(n-1)*variance of each group]

    For Memory Drug: 4112.4 = 449.6 For Placebo: 4109 = 436 For No Treatment: 4*112.3 = 449.2

    SSW = 449.6 + 436 + 449.2 = 1334.8

  3. Calculate the SST (Sum of Squares Total): This is the sum of SSB and SSW.

    SST = SSB + SSW = 8890 + 1334.8 = 10224.8

  4. Calculate the degrees of freedom for SSB (dfB), SSW (dfW), and SST (dfT).

    dfB = number of groups - 1 = 3 - 1 = 2 dfW = total number of observations - number of groups = 15 - 3 = 12 dfT = total number of observations - 1 = 15 - 1 = 14

  5. Calculate the mean squares for SSB (MSB) and SSW (MSW).

    MSB = SSB/dfB = 8890/2 = 4445 MSW = SSW/dfW = 1334.8/12 = 111.23

  6. Calculate the F statistic.

    F = MSB/MSW = 4445/111.23 = 39.94

  7. Compare the F statistic with the critical F value from the F distribution table for alpha = 0.05, dfB = 2, and dfW = 12. The critical F value is 3.89.

Since the calculated F statistic (39.94) is greater than the critical F value (3.89), we reject the null hypothesis that the treatments have the same effect on the performance of the students. Therefore, the treatments have different effects on the performance of the students.

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