Single anova t test by hand.
Give some examples in details why null and alternatives hypothesis rejected and accepted.
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.
To perform a single ANOVA t-test by hand, you will need the following information:
- The sample means of each group
- The sample sizes of each group
- The sample variances of each group
Here are the step-by-step instructions to conduct a single ANOVA t-test by hand:
Step 1: State the null and alternative hypotheses.
- The null hypothesis (H0): There is no significant difference between the means of the groups.
- The alternative hypothesis (Ha): There is a significant difference between the means of the groups.
Step 2: Calculate the total sum of squares (SSTO).
- SSTO measures the total variability in the data and is calculated using the formula:
SSTO = Σ (Xi - X)², where Xi is the individual value and X is the overall mean.
Step 3: Calculate the between-group sum of squares (SSB).
- SSB indicates the variability between the groups and is calculated using the formula:
SSB = Σ (Ni * (X_i - X)²), where Ni is the sample size of each group, Xi is the mean of each group, and X is the overall mean.
Step 4: Calculate the within-group sum of squares (SSW).
- SSW reflects the variability within each group and is calculated using the formula:
SSW = Σ ((Xi - X_i)²), where X_i is the individual value within each group.
Step 5: Calculate the mean square between (MSB).
- MSB is calculated by dividing the SSB by the degrees of freedom between (dfB), which is equal to the number of groups minus one (k - 1):
MSB = SSB / dfB.
Step 6: Calculate the mean square within (MSW).
- MSW is calculated by dividing the SSW by the degrees of freedom within (dfW), which is equal to the total sample size minus the number of groups (N - k):
MSW = SSW / dfW.
Step 7: Calculate the F statistic.
- The F statistic is the ratio of MSB to MSW:
F = MSB / MSW.
Step 8: Look up the critical F value.
- Determine the critical F value from the F-distribution table at a specified significance level (α) and degrees of freedom values (dfB and dfW).
Step 9: Compare the calculated F value with the critical F value.
- If the calculated F value is greater than the critical F value, reject the null hypothesis. There is evidence to suggest a significant difference exists between the means of the groups.
- If the calculated F value is less than or equal to the critical F value, fail to reject the null hypothesis. There is insufficient evidence to suggest a significant difference between the means of the groups.
To provide examples of rejecting and accepting the null and alternative hypotheses, let's consider two scenarios:
Example 1:
- Null hypothesis (H0): The mean duration of sleep is the same for three different age groups.
- Alternative hypothesis (Ha): The mean duration of sleep differs across three different age groups.
After conducting the ANOVA t-test, if the calculated F value is greater than the critical F value, let's say 3.85, we would reject the null hypothesis. This would indicate that there is evidence to suggest a significant difference exists between the mean durations of sleep across the age groups.
Example 2:
- Null hypothesis (H0): The mean scores on a math test are equal for two different teaching methods.
- Alternative hypothesis (Ha): The mean scores on a math test differ between two different teaching methods.
After conducting the ANOVA t-test, if the calculated F value is less than or equal to the critical F value, let's say 2.26, we would fail to reject the null hypothesis. This would indicate that there is insufficient evidence to suggest a significant difference between the mean scores on the math test using the two teaching methods.
In conclusion, the decision to reject or accept the null and alternative hypotheses in a single ANOVA t-test depends on comparing the calculated F statistic with the critical F value.
To perform a single-factor Analysis of Variance (ANOVA) test by hand, you need to follow these steps:
1. Calculate the sample means: Find the mean of each group or condition.
2. Calculate the overall mean: Find the mean of all the data combined.
3. Calculate the Sum of Squares Total (SST): This represents the variation in the data overall.
SST = Σ (Xi - Xoverall)²
Here, Xi represents each individual score, and Xoverall represents the overall mean.
4. Calculate the Sum of Squares Between (SSB): This represents the variation between the groups.
SSB = Σ (Xgroup - Xoverall)²
Here, Xgroup represents the mean of each group.
5. Calculate the Sum of Squares Within (SSW): This represents the variation within the groups.
SSW = Σ (Xi - Xgroup)²
Here, Xi represents each individual score, and Xgroup represents the mean of each group.
6. Degrees of Freedom (df) calculations:
- Total degrees of freedom: (N - 1), where N is the total number of scores.
- Between-group degrees of freedom: (k - 1), where k is the number of groups.
- Within-group degrees of freedom: (N - k).
7. Calculate the Mean Squares (MS):
- MS Between = SSB / (k - 1)
- MS Within = SSW / (N - k)
8. Calculate the F-statistic:
F = MS Between / MS Within
9. Determine the critical value: Use a table or statistical software to find the critical F-value for the desired significance level (e.g., α = 0.05).
10. Compare the calculated F-value with the critical F-value:
- If the calculated F-value is greater than the critical F-value, reject the null hypothesis.
- If the calculated F-value is less than or equal to the critical F-value, fail to reject the null hypothesis.
11. State the conclusion: Based on the above comparisons, state whether the null hypothesis is accepted or rejected.
Examples:
1. Null Hypothesis Rejected:
- Null Hypothesis (H0): The means of all groups are equal.
- Alternative Hypothesis (Ha): At least one group mean is different.
- Conclusion: If the calculated F-value is greater than the critical F-value, we reject the null hypothesis. This indicates that there is sufficient evidence to suggest that at least one group mean is different.
2. Null Hypothesis Accepted:
- Null Hypothesis (H0): The means of all groups are equal.
- Alternative Hypothesis (Ha): At least one group mean is different.
- Conclusion: If the calculated F-value is less than or equal to the critical F-value, we fail to reject the null hypothesis. This suggests that there is insufficient evidence to conclude that any of the group means are significantly different.
It is important to note that performing ANOVA by hand can be time-consuming and prone to errors. It is advisable to use statistical software for accurate and efficient analysis.