To conduct a single ANOVA t-test by hand, you would follow several steps, including calculating the sum of squares, degrees of freedom, mean squares, and the F ratio. However, for the purpose of this explanation, let's focus on understanding the concept of rejecting or accepting the null and alternative hypotheses and provide some examples.
Null hypothesis (H0): There is no significant difference between the groups being compared.
Alternative hypothesis (Ha): There is a significant difference between the groups being compared.
Example 1: Suppose you want to compare the average scores of three different study techniques (Technique A, B, and C) for a test where the null hypothesis states that there is no significant difference.
1. Collect the data: Gather test scores from a sample of students for each study technique separately (e.g., 30 scores for Technique A, 35 scores for Technique B, and 28 scores for Technique C).
2. Calculate the mean for each group: Determine the average test score for each study technique (e.g., Technique A = 75, Technique B = 80, Technique C = 85).
3. Perform ANOVA: Calculate the sum of squares within groups, the sum of squares between groups, and the mean squares for each (using appropriate formulas). Then calculate the F ratio, which is the ratio of the mean squares between groups to the mean squares within groups.
4. Reject or accept the null hypothesis: Based on the calculated F ratio, compare it to the critical F value from the F-distribution table at a given significance level (e.g., 0.05). If the calculated F ratio is greater than the critical F value, you reject the null hypothesis. Otherwise, you accept it.
Example 2: Suppose an industrial engineer wants to test the impact of different machine settings (Settings A, B, and C) on production time where the null hypothesis states that there is no significant difference.
1. Collect the data: Measure production times for a sample of products produced using each machine setting separately (e.g., 50 production times for Setting A, 45 for Setting B, and 55 for Setting C).
2. Calculate the mean for each group: Determine the average production time for each machine setting (e.g., Setting A = 80 seconds, Setting B = 85 seconds, Setting C = 90 seconds).
3. Perform ANOVA: Apply the same steps as outlined in Example 1 to calculate the F ratio based on the sum of squares within and between groups.
4. Reject or accept the null hypothesis: Again, compare the calculated F ratio with the critical F value at a chosen significance level. If the calculated F ratio exceeds the critical F value, you reject the null hypothesis. Otherwise, you accept it.
In both examples, if the null hypothesis is rejected, it means that there is convincing evidence to support the alternative hypothesis – that there is a significant difference between the groups being compared. Conversely, if the null hypothesis is accepted, there is not enough evidence to suggest a significant difference between the groups.
Single anova t test by hand.
Give some examples in details why null and alternatives hypothesis rejected and accepted.
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