 # Statistics Problems

Subject Mathematics General Statistics

## Question

Problem 1.
You are working for a school district that is experimenting with a new form of teacher evaluation based on student performance as well as classroom observation. To evaluate whether student evaluations are effective in improving teaching in ways that promote student learning, the Superintendent randomly assigned elementary schools to either (a) an experimental group that used the new evaluation system or (b) a control group that retained the old evaluation (based on classroom observation and self-reporting of accomplishments). After two years, the district collected data on student performance from the experimental and control schools. Your assignment is to determine whether there is evidence that the new teacher evaluations helped to improve student performance.
1. Calculate the average improvement in student math test scores for the control group and experimental group of schools. What do you observe?
2. You want to be sure that your conclusion is valid. So, you will need to conduct a hypothesis test. What should the null and alternative hypothesis for this test be?
3. Use the “t-test: Two sample assuming equal variance” function in the data analysis function (you may need to install this add-in into Excel). Are you able to reject the null hypothesis, just looking at the post-test year results?
4. The data also distinguish whether schools (both in the control group and the experimental groups) are low-SES schools meaning that more than 50% of the students qualify for free or reduced cost lunches. A “1” indicates a high SES school, while a “0” indicates a low SES school. Controlling for whether a school is high or low SES compare the average student improvement in math scores between the control and experimental groups.
5. Conduct two additional hypothesis tests to check whether these two differences are statistically significant.
6. What do you conclude about the efficacy of teacher evaluations in improving student performance? Why?

Problem 2.
You are working for the chief executive of a large government agency. Last year the agency introduced a performance management system where the work of project managers and line managers was evaluated bimonthly according to a series of key performance metrics. At the end of the meetings managers were given action memos on the areas of their work that needed improvement, and their evaluation at the end of the year was based on how well they responded to these action memos. As part of the evaluation three groups of employees were surveyed:      1) senior managers who ran the performance management system but were not evaluated by it; 2) project managers, and 3) line managers. They were each asked whether they thought the individual year-end evaluations were fair or not. They could either respond “Yes” (coded as 1 in the data) or “No” (coded as 0).
1. Calculate the proportion of each category of worker (e.g., executive managers, project managers, and line managers) that thought the individual evaluations were fair. [As a check 66.7% of the executive managers think the process is fair]
2. Based on a comparison of these proportions, how would you characterize the level of support on the part of staff employees compared to managers?
3. Use the chi-square test to determine whether the observed differences are statistically significant. In Excel 2010 or newer, you can use the function =chisq.test. In Excel 2007 or 2003, you can use the function =chitest.   What is the p-value for the chi-square test?
4. Are you able to reject the null hypothesis that all three types of employees have similar perspectives on the incentive pay system? Explain

Problem 3.
You have been asked to evaluate a reading mentoring program that has been in operation for six months. The program managers provide you with data on 40 program participants. The data show each students increase (or decrease) in reading scores over the last six months and the hours of mentoring they have attended
1. Graph the relationship between hours-of-mentoring and test-score-improvement. In Excel, highlight both columns of data, including the column names, and insert a scatterplot.
2. Calculate the correlation between mentoring hours and change in reading score. You may use the correl function in Excel for this.
3. What can you say about the success of the program based on this correlation? Does it demonstrate that mentoring increases scores?

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