A researcher wants to investigate the effect of education level on women's incomes. A simple random sample of working women aged between 30 and 50 was selected from each of three education categories. The women's annual incomes (in thousands of dollars) were recorded. The results are as follows:
| | High School |
|------------|-------------|
| | 49 |
| | 31 |
| | 35 |
| | 42 |
| | 33 |
| | 37 |
| | 48 |
| | 45 |
| | College |
|---------|---------|
| | 36 |
| | 85 |
| | 28 |
| | 32 |
| | 44 |
| | 63 |
| | 46 |
| | 42 |
| | Advanced Degree |
|-----------------|-----------------|
| | 58 |
| | 63 |
| | 52 |
| | 31 |
| | 30 |
| | 99 |
| | 112 |
| | 96 |
Test the claim that the three different samples come from populations with the same mean. Use a significance level of 0.01. Assume that incomes are normally distributed.
1. Complete the populations, null and researcher's hypothesis
Population 1:
The set of all $\square$ incomes from women with $\square$ a high school diploma
Population 2:
The set of all $\square$ incomes from women with $\square$ a college degree
Population 3:
The set of all $\square$ incomes from women with $\square$ an advanced degree
$H _0$ : $\square$
$\square$
$\square$
$\square$
$\square$
$H _r$ : $\square$
$\square$
$\square$
$\square$
$\square$
2. Which test would you use to verify the requirements? $\square$ X is normally distributed
Has the requirements of the test met? $\square$ Yes
What is the sampling distribution? $\square$ F distribution
3. Determine the cutoff score, (critical value) for significance. $\square$ 6.3589
4. Fill in the blank for the necessary statistics to test the claim. Round $x$ and $s^2$ to twice as many as one more decimal place as your data.
| | High School | College | Advanced Degree |
|------|-------------|---------|-----------------|
| n | 6 | $6$ | 6 |
| $x$ | 42.17 | 40.33 | 68.67 |
| $s^2$ | 46.17 | 134.67 | 893.47 |
5. Determine the standardized sample score (from the sample taken). Round your answer to four decimal places. Include trailing zeros as needed. $\square$ 4.2122
6. Determine the p-value from the standardized sample score. Round your answer to four decimal places. Include trailing zeros as needed. $\square$ 0.0353
7. Make your decision to reject or fail to reject the null hypothesis. $\square$ fail to reject
Answer (2)
Define populations for each education level.
State null hypothesis ( H 0 : μ 1 = μ 2 = μ 3 ) and alternative hypothesis (at least one μ i is different).
Calculate the F-statistic: F = 3.5893 .
Since p-value (0.0456) > significance level (0.01), fail to reject the null hypothesis. fail to reject
Explanation
Define Populations and Hypotheses Let's start by defining the populations and hypotheses for this ANOVA problem. We want to determine if there's a significant difference in the mean incomes of women based on their education levels (High School, College, and Advanced Degree).
Define Populations Population 1: The set of all incomes from women with a high school diploma. Population 2: The set of all incomes from women with a college degree. Population 3: The set of all incomes from women with an advanced degree.
State Null and Alternative Hypotheses The null hypothesis ( H 0 ) states that the population means are equal: H 0 : μ 1 = μ 2 = μ 3
The alternative (researcher's) hypothesis ( H r ) states that at least one population mean is different: H r : At least one μ i is different where μ 1 , μ 2 , and μ 3 represent the mean incomes for the High School, College, and Advanced Degree populations, respectively.
Verify Requirements and Sampling Distribution The problem states that incomes are normally distributed, which is a requirement for ANOVA. We would typically also need to check for equal variances across the groups. The sampling distribution for this test is the F distribution.
Determine the Cutoff Score The critical value (cutoff score) for a significance level of α = 0.01 is given as 6.3589.
Calculate Sample Statistics Now, let's verify the sample statistics. The problem provides some values, but they appear to be based on n = 6 instead of the actual sample size of n = 8 . Based on the calculations performed:
High School: n = 8 , x ˉ = 40.00 , s 2 = 48.29 College: n = 8 , x ˉ = 47.00 , s 2 = 348.86 Advanced Degree: n = 8 , x ˉ = 67.62 , s 2 = 981.98
Determine the Standardized Sample Score The F-statistic is calculated as F = MS W MSB , where MSB is the mean square between groups and MSW is the mean square within groups. From the calculations, F = 3.5893 .
Determine the p-value The p-value associated with the F-statistic is calculated to be 0.0456.
Make a Decision Finally, we make a decision. Since the p-value (0.0456) is greater than the significance level (0.01), we fail to reject the null hypothesis. Also, the calculated F-statistic (3.5893) is less than the critical value (6.3589), which leads to the same conclusion.
Conclusion Therefore, we fail to reject the null hypothesis. There is not enough evidence to conclude that there is a significant difference in the mean incomes of women with different education levels at a significance level of 0.01.
Examples
Consider a company trying to determine if different training programs lead to different performance levels among employees. ANOVA can help assess whether the average performance scores differ significantly across these training groups. Similarly, in agriculture, ANOVA can be used to compare the yields of different crop varieties to see if there's a significant difference in their average yields. These applications demonstrate how ANOVA helps in comparing means across multiple groups to identify significant differences.
The analysis indicates that we cannot reject the null hypothesis that the means of women’s incomes across different education levels are equal, based on our significance level of 0.01. The calculated p-value exceeds the alpha level, suggesting no significant differences. Thus, we conclude there is insufficient evidence to claim that education level affects women's incomes significantly.
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