Perform a one-way ANOVA analysis, testing at the 0.05 level. Also, calculate the mean number of sick days for each group. Describe your results. .
Group Sick days per year
Placebo: 7, 4, 6, 8, 6, 6, 2, 9
Flu Shot 1: 5, 3, 3, 5, 4, 7, 3, 3
Flu Shot 2: 2, 4, 1, 2, 2, 1, 2, 5
To perform a one-way ANOVA analysis and calculate the mean number of sick days for each group, you can follow these steps:
Step 1: Set up the hypothesis:
- Null hypothesis (H0): There is no significant difference in the mean number of sick days among the groups.
- Alternative hypothesis (Ha): There is a significant difference in the mean number of sick days among the groups.
Step 2: Calculate the mean number of sick days for each group:
- Placebo group: 7, 4, 6, 8, 6, 6, 2, 9. Calculate the mean by adding the sick days and dividing by the number of observations (8).
- Flu Shot 1 group: 5, 3, 3, 5, 4, 7, 3, 3. Calculate the mean by adding the sick days and dividing by the number of observations (8).
- Flu Shot 2 group: 2, 4, 1, 2, 2, 1, 2, 5. Calculate the mean by adding the sick days and dividing by the number of observations (8).
Step 3: Calculate the sum of squares (within groups) and degrees of freedom:
- Sum of squares (within groups) is the sum of the squared deviations from each observation to the mean of the respective group.
- Degrees of freedom (within groups) can be calculated by subtracting the number of groups from the total number of observations.
Step 4: Calculate the sum of squares (between groups) and degrees of freedom:
- Sum of squares (between groups) is the sum of the squared deviations of each group mean from the overall mean, weighted by the number of observations in each group.
- Degrees of freedom (between groups) can be calculated by subtracting 1 from the number of groups.
Step 5: Compute the F-statistic and the p-value:
- The F-statistic is the ratio of the mean sum of squares (between groups) to the mean sum of squares (within groups).
- The p-value is the probability of observing a test statistic as extreme as the one computed, assuming the null hypothesis is true.
Step 6: Compare the computed F-statistic with the critical value of the F-distribution:
- Determine the critical value of the F-distribution corresponding to the desired significance level (in this case, 0.05) and the degrees of freedom for both the numerator (between groups) and denominator (within groups).
- If the computed F-value is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Once you have performed the above steps, you can describe your results based on the outcome of the hypothesis test and provide interpretations.