Assignment 3 1 Proportions and Di'erences The Louisiana Department of Education [LDE} is concerned with the scores of the standardized math and scince exams among the 627 high schools in the state. The study sampled n. = 442 schools and resulted in the variable statistics fig; = % = [L43 and mh = % = 1146, showing the percentage of students that pass the exam in each discipline, respectively. After analysis, the covariance was determined to be CV[sci,math] = ELIE. l. 1What is the sample mean and variance of sci math [i.e., pm 31mm}? [3 pts} 2. Initially, it appears that the math exam is harder as science scores are above math by 2% in our sample. TWhat is the probability that the true science score exceeds the math score by more than 5% {that is, \"(padmath 2} 0.05} or Frii punt}; } 0.85)}? [6 pts] 3. The LDE claims that \"Ideally, the two standardized tests should be equally difcult. We should expect the percentage of students that pass the math exam to be equal to the per- centage of students that pass the science exam (pm. = pmathj.\" Test this claim. (3 pts) 2 Comparing Proportions A cosmetics manufacturer has hired a team of McNeese MBA students to statistically test the success of two types of nail polish removal methods on n = 100 people. Each subject (person) has the exact type of nail polish on each testable finger. For each subject i, define X," = 1 if method M is successful (that is, reducing the target amount of polish on the applied nail within a given timeframe) and zero otherwise. As this research is for statistical clarity more than method content, for brevity we say this can be nail polish remover of any type (acetone-based vs. acetone-free, remover strips vs. soaking, etc.). We will refer to the method ONE approach as "M1" and the method TWO approach as "M2" for comparability. We can apply MI to any right-hand finger, and M2 to the corresponding finger on the left hand, simultaneously. 4. Open the Excel file Data File.xlex. To determine the joint distributions for each person, you can use the Excel "IF" function. In cell D2, type "=IF(AND(B2=0,C2=0),1,0)". Similarly, type the following functions in each of their respective cells: . Type "=IF(AND(B2=1,C2=0),1,0)" in cell E2 . Type "=IF(AND(B2=0,C2=1),1,0)" in cell F2 . Type "=IF(AND(B2=1,C2=1),1,0)" in cell G2 Using the fill box (the small box at the bottom of each cell), drag each formula down to row 101 (that is, person 100). Now, in cell B102, type "=AVERAGE(B2:B101)" to get PM1. By clicking the fill box at the bottom of cell B102 and dragging across to cell G102, we have the following concepts: . The marginal probability PMI = Pr(XMI = 1) in cell B102 . The marginal probability PM2 = Pr(XM? =1) in cell C102 The joint probability Pr(XMI = On XM2 = 0) in cell D102 . The joint probability Pr(XMI = 1n XM= = 0) in cell E102 . The joint probability Pr(XMI = On XM2 =1) in cell F102 . The joint probability Pr(XMI = 1n XM2 =1) in cell G102 Using the above values, complete the joint distribution table below. (8 pts) XM2 0 1 Pr(XMI) XMI 0 Pr(XM2)5. Now, calculate the variance of methods ONE and TWO (remember, they are both 0-1 vari- ables, so you simply need to use the variance of a Bernoulli distribution for each). (8 pts) 6. Determine the confidence intervals for each method (PMI and PM2). (8 pts) 7. Do the confidence intervals from question 6 overlap? Assuming this is an accurate statistical technique, what does it tell you about the methods? (6 pts) 8. Based on historical data, management has determined the company's original method ("MO") is 59% effective (successful) in removing nail polish. Test the null hypothesis that method ONE is just as effective as the original method at the 5% level. (8 pts)9. Determine the sample covariance between methods ONE and TWO. (6 pts) 10. Follow the the procedure below. . Step 1 (adding in the Analysis ToolPak): FILE (top left of screen)->Options->Add- Ins->Manage: Excel Add-ins-Go-Analysis ToolPak (check the box) . Step 2 (getting the Data Analysis kit): DATA (tab at the top of screen) -Data Analysis-+Covariance-OK . Step 3 (cacluating the covariance): Follow the image below. Covariance 7 X Input SBS1:SC$101] OK Input Range: Cancel Grouped By: Columns Rows Help Labels in First Row Output options Output Range: New Worksheet Ply O New Workbook Complete the variance/covariance table below. M1 M2 M1 M2 What did you find? How does your answers from questions 5 and 9 fit in the table? (6 pts)11. What is the mean and variance of the difference, XMI - XM? (or PMI - PM2)? (8 pts) 12. Construct a 95% confidence interval for the difference, PM1 - PM2- (6 pts) 13. Suppose we wanted to test the null hypothesis, Ho : PM1 = PM2 at the 5% level. Another way to write the null hypothesis is Ho : PM1 - PM2 = 0. In plain English, briefly explain: What does this null hypothesis say? (8 pts) 14. After testing the null at the 5% level, what does that tell us about the two methods? Have we determined that one is more effective than the other? (6 pts)