A company director took a sample of salaries of 13 employees in a big school during a certain time. she divided the wages into three groups according to the different jobs: teachers, assistants and coordinators as displayed in the table below. She wanted to find out whether or not the means of the salaries classified by the three jobs are significantly different. Let the level of significance be
(a) 5 % and
(b) 1 %.
Workers 1 2 3 4 5
Teachers 74 65 72 69 -
Assistants 75 78 74 76 72
Coordinators 56 55 53 52 -
Can someone please help me to solve this question. I tried setting a hypothesis but I don't know if that is correct. Do I have to compare means?
st.statistics
With only 13 employees, it would be best to use X^2 (Chi-square) test.
Ho: no diference
X^2 = ∑ (O-E)^2/E, where O = observed frequency and E = expected frequency.
∑ = sum of all the cells.
E = (column total * row total)/grand total
df = n - 1, where n = number of cells
Look up value in X^2 table in the back of your textbook.
To test whether or not the means of the salaries classified by the three jobs are significantly different, you can perform a one-way analysis of variance (ANOVA) test. This test will compare the means of three or more groups to determine if there is a statistically significant difference between them.
Here are the steps to perform the ANOVA test:
Step 1: Set up the hypothesis:
- Null hypothesis (H0): The means of the salaries for the three jobs are equal.
- Alternative hypothesis (Ha): The means of the salaries for the three jobs are not equal.
Step 2: Calculate the means and variance for each group:
First, calculate the mean and variance for each group (teachers, assistants, and coordinators) using the given sample data. The variance will give you an idea of the variability within each group.
Step 3: Calculate the overall mean and variance:
Calculate the overall mean and variance of all the salaries combined. This will give you an idea of the overall variability in the entire sample.
Step 4: Calculate the F-statistic:
The F-statistic is used to test the difference in means between the groups. It compares the variability between the groups to the variability within the groups. The formula for calculating the F-statistic is:
F = (Between-group variance / (k - 1)) / (Within-group variance / (n - k))
where k is the number of groups and n is the total sample size.
Step 5: Determine the critical F-value:
Look up the critical F-value in an F-distribution table or use a statistical software to determine the critical F-value for your given level of significance (5% or 1%) and degrees of freedom for the numerator and denominator.
Step 6: Compare the calculated F-value and the critical F-value:
If the calculated F-value is greater than the critical F-value, you reject the null hypothesis and conclude that there is a significant difference between the means of the three jobs. Otherwise, if the calculated F-value is smaller than the critical F-value, you fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference between the means.
Remember to check the assumptions of the ANOVA test, such as normality and equal variance, before interpreting the results.
To summarize, you have to calculate the means and variances for each group, calculate the F-statistic, look up the critical F-value, and compare the calculated F-value with the critical F-value to determine if the means of the salaries classified by the three jobs are significantly different.