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You may need to use the appropriate technology to answer this question. Information regarding the ACT scores of samples of students in three different majors is given below. Sample Size Average Major Management Finance Accounting 12 Within Treatments Total 26 Sample Variance (a) Compute the overall sample mean . 29.5 x 17 9 23 6 11 25. (b) Set up the ANOVA table for this problem including the test statistic. (Round your mean squares to four decimal places and your F statistic to two decimal places.) Source of Variation Sum of Squares Degrees of Freedom Mean Square Between Treatments 11 (c) Using a = 0.01, determine the critical value of F. (Round your answer to two decimal places.) F

You may need to use the appropriate technology to answer this question. Information regarding the ACT scores of samples of students in three different majors is given below. Sample Size Average Major Management Finance Accounting 12 Within Treatments Total 26 Sample Variance (a) Compute the overall sample mean . 29.5 x 17 9 23 6 11 25. (b) Set up the ANOVA table for this problem including the test statistic. (Round your mean squares to four decimal places and your F statistic to two decimal places.) Source of Variation Sum of Squares Degrees of Freedom Mean Square Between Treatments 11 (c) Using a = 0.01, determine the critical value of F. (Round your answer to two decimal places.) F

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Understanding ACT Scores Across Different Majors** The data provided showcases the ACT scores of samples from students in three different majors: Management, Finance, and Accounting. This illustration is used to analyze variance among groups using ANOVA (Analysis of Variance). Below is a table summarizing the sample data: | Major | Management | Finance | Accounting | |----------------|------------|---------|------------| | Sample Size | 12 | 9 | 11 | | Average | 26 | 23 | 25 | | Sample Variance| 17 | 6 | 11 | **ANOVA Table Setup** The following ANOVA table helps us determine the statistical differences between the score means of the different majors. The components to fill in include the sum of squares, degrees of freedom, and the mean square, which are crucial for calculating the F statistic. | Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F | |---------------------|----------------|--------------------|-------------|-------| | Between Treatments | | | | | | Within Treatments | | | | | | Total | | | | | **Tasks and Analysis** a) **Compute the Overall Sample Mean:** - The overall sample mean is calculated to be \( \bar{x} = 29.5 \). b) **ANOVA Table Setup:** - Populate the ANOVA table based on sample data and statistical formulas to assess variance. c) **Determine the Critical Value of F:** - Using a significance level (\( \alpha = 0.01 \)), determine the critical value for the F-distribution and round your answer to two decimal places. d) **Hypothesis Testing:** - Using the critical value approach, examine whether there is a significant difference in the means of the three populations: - Option 1: We should not reject \( H_0 \) and conclude there is no significant difference among the means. - Option 2: We should reject \( H_0 \) and conclude there is a significant difference among the means. These analyses will help determine if the differences in average ACT scores between majors are statistically significant, providing insights into academic performance across fields.
Transcribed Image Text:**Understanding ACT Scores Across Different Majors** The data provided showcases the ACT scores of samples from students in three different majors: Management, Finance, and Accounting. This illustration is used to analyze variance among groups using ANOVA (Analysis of Variance). Below is a table summarizing the sample data: | Major | Management | Finance | Accounting | |----------------|------------|---------|------------| | Sample Size | 12 | 9 | 11 | | Average | 26 | 23 | 25 | | Sample Variance| 17 | 6 | 11 | **ANOVA Table Setup** The following ANOVA table helps us determine the statistical differences between the score means of the different majors. The components to fill in include the sum of squares, degrees of freedom, and the mean square, which are crucial for calculating the F statistic. | Source of Variation | Sum of Squares | Degrees of Freedom | Mean Square | F | |---------------------|----------------|--------------------|-------------|-------| | Between Treatments | | | | | | Within Treatments | | | | | | Total | | | | | **Tasks and Analysis** a) **Compute the Overall Sample Mean:** - The overall sample mean is calculated to be \( \bar{x} = 29.5 \). b) **ANOVA Table Setup:** - Populate the ANOVA table based on sample data and statistical formulas to assess variance. c) **Determine the Critical Value of F:** - Using a significance level (\( \alpha = 0.01 \)), determine the critical value for the F-distribution and round your answer to two decimal places. d) **Hypothesis Testing:** - Using the critical value approach, examine whether there is a significant difference in the means of the three populations: - Option 1: We should not reject \( H_0 \) and conclude there is no significant difference among the means. - Option 2: We should reject \( H_0 \) and conclude there is a significant difference among the means. These analyses will help determine if the differences in average ACT scores between majors are statistically significant, providing insights into academic performance across fields.
### Educational Content: ANOVA Calculation and Interpretation #### (a) Compute the Overall Sample Mean \(\bar{X}\) - The provided value is **29.5**, but it seems to be marked incorrect, indicating a need for further calculation or verification. #### (b) Set Up the ANOVA Table To analyze the variance among different groups, we set up an ANOVA table. Below are the headers and spaces that generally need to be filled out for a standard ANOVA table. - **Source of Variation:** - Between Treatments - Within Treatments - Total - **Columns to Fill:** - Sum of Squares - Degrees of Freedom - Mean Square - F statistic (computed to two decimal places) #### (c) Determine the Critical Value of \(F\) Using \(\alpha = 0.01\) - This involves using statistical tables or software to find the critical \(F\) value, relevant to the degrees of freedom, at a significance level (\(\alpha\)) of 0.01. The answer should be rounded to two decimal places. #### (d) Using the Critical Value Approach Evaluate the hypothesis: - "We should not reject \(H_0\) and therefore cannot conclude that there is a significant difference among the means of the three populations." - The correct selection, indicated by a checkmark, suggests the null hypothesis is retained, meaning no significant difference is found with the given data. #### (e) Determine the p-value and Use It for the Test - The test involves comparing the p-value against a specified significance level. - Interpretation Example: - "Since this [p-value] is greater than the significance level, we should not reject \(H_0\) and conclude that there is not sufficient evidence in the data to suggest a significant difference among the mean ACT scores of the three majors." Each step involves rigorous statistical testing, and careful attention is required to fill out the ANOVA table, determine critical values, and interpret the p-value correctly. This process helps in identifying significant differences in group means and making informed conclusions based on the data.
Transcribed Image Text:### Educational Content: ANOVA Calculation and Interpretation #### (a) Compute the Overall Sample Mean \(\bar{X}\) - The provided value is **29.5**, but it seems to be marked incorrect, indicating a need for further calculation or verification. #### (b) Set Up the ANOVA Table To analyze the variance among different groups, we set up an ANOVA table. Below are the headers and spaces that generally need to be filled out for a standard ANOVA table. - **Source of Variation:** - Between Treatments - Within Treatments - Total - **Columns to Fill:** - Sum of Squares - Degrees of Freedom - Mean Square - F statistic (computed to two decimal places) #### (c) Determine the Critical Value of \(F\) Using \(\alpha = 0.01\) - This involves using statistical tables or software to find the critical \(F\) value, relevant to the degrees of freedom, at a significance level (\(\alpha\)) of 0.01. The answer should be rounded to two decimal places. #### (d) Using the Critical Value Approach Evaluate the hypothesis: - "We should not reject \(H_0\) and therefore cannot conclude that there is a significant difference among the means of the three populations." - The correct selection, indicated by a checkmark, suggests the null hypothesis is retained, meaning no significant difference is found with the given data. #### (e) Determine the p-value and Use It for the Test - The test involves comparing the p-value against a specified significance level. - Interpretation Example: - "Since this [p-value] is greater than the significance level, we should not reject \(H_0\) and conclude that there is not sufficient evidence in the data to suggest a significant difference among the mean ACT scores of the three majors." Each step involves rigorous statistical testing, and careful attention is required to fill out the ANOVA table, determine critical values, and interpret the p-value correctly. This process helps in identifying significant differences in group means and making informed conclusions based on the data.
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