4) What do you mean by Analysis of Variance? Why it is preferred in comparison to ‘t’ test while determining the significance difference in the means.

Analysis of Variance (ANOVA) is a statistical test used to compare the means of two or more groups simultaneously. It is preferred over multiple t-tests when dealing with more than two groups because it:

• Is more efficient, requiring fewer calculations.
• Reduces the risk of Type I errors, which can inflate when multiple t-tests are conducted.
• Allows for examination of both between-group and within-group variance.

5) What is the relationship between S.D. and Mean sum of squares within the groups?

The mean sum of squares within groups (MSW) is an estimate of the common variance within each group. It is related to the standard deviation within each group. If the standard deviations within the groups are similar, then the square root of MSW is a pooled estimate of the common standard deviation.

6) Why it is necessary to fulfill the assumptions of ‘F’ test, before to apply analysis of variance.

ANOVA is based on certain assumptions that are necessary for the validity of its results:

• Normality: The data within each group should be approximately normally distributed.
• Homogeneity of variance: The variances of the different groups should be roughly equal.
• Independence: The observations within and between groups should be independent of each other.

Violation of these assumptions can lead to inaccurate or misleading conclusions.

7) Why the ‘F’ ratio test and ‘t’ ratio tests are complementary to each other.

The F and t tests are complementary in that:

• When F is significant: Post-hoc t-tests are often used to pinpoint which specific group means differ significantly.
• When t is not significant for two groups: F-tests might still reveal significant differences because ANOVA considers within-group variance.
• They are mathematically related: F = t2

8) What should be the various problems of psychology and education. Where the ANOVA can be used successfully.

Examples of ANOVA applications include:

• Comparing the effectiveness of different therapies: Assessing whether cognitive-behavioral therapy (CBT), psychodynamic therapy, or a control condition lead to different levels of anxiety reduction.
• Examining the impact of teaching methods: Determining if lecture-based, project-based, or blended learning approaches result in different levels of student comprehension.
• Investigating the effects of different drug dosages: Analyzing whether various doses of a medication have significantly different effects on symptom severity.

9) State the assumptions of ANOVA.

As mentioned earlier, the crucial assumptions of ANOVA are:

• Normality of data within each group.
• Homogeneity of variances across groups.
• Independence of observations.

10) What happens when these assumptions are violated?

Violating these assumptions can affect the accuracy and reliability of the F-test results, potentially leading to:

• Increased Type I errors: Rejecting the null hypothesis when it is actually true.
• Decreased statistical power: Failing to detect a true difference between group means.

11) Compare the ‘F Ratio’ test and ‘t Ratio’ test in terms of their relative merits and demerits.

Merits of F Test (ANOVA):

• Can compare multiple groups simultaneously.
• More efficient than multiple t-tests.
• Controls the Type I error rate better than multiple t-tests.

Demerits of F Test:

• Requires meeting certain assumptions (normality, homogeneity of variance, independence).
• Does not indicate which specific group means differ.

Merits of t Test:

• Simpler to calculate and interpret than ANOVA.
• Can be used for both independent and correlated samples.

Demerits of t Test:

• Only suitable for comparing two groups.
• Conducting multiple t-tests increases the Type I error rate.

12) What is the mathematical relationship between F and t.

The mathematical relationship is F = t^2. This means that for two groups, the F statistic from ANOVA is simply the square of the t statistic.

13) When the post ANOVA test of difference is applied?

Post-ANOVA tests of difference (post-hoc tests) are applied when the overall F test is statistically significant, meaning there is evidence of a difference between at least two group means. Post-hoc tests help identify which specific group means differ from each other.

14) How many degree of freedom are associated with the variation in the data for A comparison of four means for independent samples each containing 10 cases?

For a comparison of four means with 10 cases in each independent sample, the degrees of freedom are:

• Between Groups (df1): k - 1 = 4 - 1 = 3 (where k is the number of groups)
• Within Groups (df2): N - k = 40 - 4 = 36 (where N is the total sample size)

15) A comparison of three groups selected independently each containing 15 units.

For a comparison of three groups with 15 units in each independent sample, the degrees of freedom are:

• Between Groups (df1): k - 1 = 3 - 1 = 2
• Within Groups (df2): N - k = 45 - 3 = 42