Population Mean and Proportions Analysis: A Survey on Sub-populations - Prof. David K. Nea, Study Guides, Projects, Research of Statistics

Download Population Mean and Proportions Analysis: A Survey on Sub-populations - Prof. David K. Nea and more Study Guides, Projects, Research Statistics in PDF only on Docsity! Dr. Neal, Spring 2009 MATH 203 Survey on Population Mean The first part of this project is on the analysis of a population mean. You will obtain data on a specific measurement X by performing a random survey on a targeted population Ω that is divided into two disjoint sub-populations Ω1 and Ω2 . The project involves computing statistics, finding confidence intervals, and performing hypothesis tests on the overall mean and on the difference 1 – 2 of the sub-population means. State the Measurement and Populations First, you will analyze the mean of some measurement X on a specific population and compare the differences within the two sub-populations. 1. Specify the measurement X that you wish to analyze and the population Ω that you will target in your survey. State the two disjoint sub-populations Ω1 and Ω2 that you will consider. (After defining your sub-populations, refer to them throughout by name, not simply as Ω1 and Ω2 .) 2. Give the approximate size N of your population Ω and the sizes N1 and N2 of the sub-populations Ω1 and Ω2 . Then specify the proportionate sizes of the sub- populations. (For example, Ω1 is 60% of the population and Ω2 is 40%.) Initial Estimates Before conducting the survey, you must make some initial estimates about the mean of your measurement. Later, you will use these estimates for your hypothesis tests. 1. Give estimates for the means 1 and 2 of the measurement X on your two sub- populations Ω1 and Ω2 . Explain the reasoning you used for your estimates. (The difference of your estimates then becomes your estimate M for the difference in means 1 – 2 in Question 3 under Hypothesis Tests.) 2. Using your estimates in 1, calculate a weighted estimate of the overall population mean . (This estimate is to be used in Question 1 under Hypothesis Tests.) 3. How do you personally compare with your estimated average? That is, what is your measurement for X and you are above, below, or close to your estimated average? Determining Sample Size 1. Give an estimate for the possible range [c , d ] of your measurements. Give a logical explanation for your chosen bounds. 2. For your measurement, choose a desired margin of error e . This value should depend on the size of your measurements and should be about 5% of the size of your estimate for the overall population mean . 3. For both Ω1 and Ω2 , find the sample sizes n1 and n2 required to obtain 95% confidence intervals that have no larger than your desired margin of error e . (Over) Dr. Neal, Spring 2009 Recall: Using N1 for the size of Ω1 , the required sample size n1 for Ω1 is found by n1 ≥ N1 1.96 × U e       2 N1 − 1 + 1.96 × U e       2 , where U = d − c 2 (Use N2 for Ω2 .) Conducting the Survey Next you must conduct a random survey on the targeted population to obtain sample measurements. For scientific purposes, there usually should be at least n1 respondents from sub-population Ω1 and at least n2 respondents from sub-population Ω2 . However, for instructional purposes here, you may limit yourself to around 85 measurements from the entire population with at least 30 from each sub-population. But you should use sample sizes that are in proportion to the sizes of the sub- populations. For example, if Ω1 is 60% of the population, then use a 60 : 40 breakdown such as n1 = 60 and n2 = 40, or perhaps n1 = 51 and n2 = 34. 1. Explain the sample sizes n1 and n2 that you will use. Then explain whether or not you will use the small population correction factors for each of Ω , Ω1 , and Ω2 . (For Ω1 , compute (N1 − n1)/ (N1 − 1) . If it is close to 1, then you do not need the correction factor for Ω1 . Use the analogous formulas for Ω and Ω2 .) 2. Take a random survey of people specifically within your target population Ω and record their measurements. Be sure to determine to which sub-population each respondent belongs. 3. State how the survey was conducted and how randomness was ensured. 4. Include the raw data of all responses collected above as an appendix. Note: Because you will be using the same populations for both survey questions, you should collect the data for both at the same time. Data for your responses can be easily sorted in a chart like the one below. (Be sure to obtain an exact measurement from each respondent so that the statistics can be calculated accurately.) Response In Pop ? 1 2 Meas. X Yes Not Yes Person 1 √ √ 20 √ Person 2 √ √ 14 √ Person 3 √ √ 12 √ . . . . . . . . . . . . . . Dr. Neal, Spring 2009 Survey of Population Proportions This second part of the project is on the analysis of a population proportion. Now you will analyze the “Yes/No” data from your survey on your targeted population and its two sub-populations. This part of the project involves constructing confidence intervals and performing hypothesis tests for the overall population proportion p and for the difference p1 – p2 of the sub-population proportions. State the Question and Populations 1. State your (Yes/No) question and which response you want to measure. This will be called a favorable response. 2. State again the population Ω and the two sub-populations Ω1 and Ω2 that you are considering. Henceforth, refer to them by name and not just as Ω1 and Ω2 . (These must be the same populations used in the first part of the project.) Initial Estimates Before conducting the survey, make some initial estimates of the proportions. You will use these estimates later for your hypothesis tests. 1. Give estimates of the true proportions p1 and p2 within each sub-population Ω1 and Ω2 that you think will respond favorably. Give reasons for your estimates. (The difference of your estimates then becomes your estimate for the difference in proportions p1 − p2 for Question 3 in Hypothesis Tests.) 2. Using your estimates in Question 1, calculate a weighted estimate P for the overall population proportion p . (This estimate is to be used in Question 1 under Hypothesis Tests.) 3. How would you personally respond to your question? Do you think that you would be with the majority or minority with your response? Determining Sample Size 1. For your proportion, choose a desired margin of error e from 0.02 to 0.04. 2. For both Ω1 and Ω2 , find the sample sizes n1 and n2 required to obtain 95% confidence intervals that have no larger than your desired margin of error e . Recall: Using N1 for the size of Ω1 , the required sample size n1 for Ω1 is found by n1 ≥ N1 × 1.96 × 0.5 e       2 ( N1 −1) + 1.96 × 0.5 e       2 (Use N2 for Ω2 .) Dr. Neal, Spring 2009 Conducting the Survey Now you must analyze the Yes/No data obtained while doing Survey 1. 1. Give the overall sample proportion p = m /n as a fraction (# yes / # surveyed) and as a percentage. How does p compare with your initial estimate of p ? 2. For both Ω1 and Ω2 , give the respective sample proportions p 1 = m1/n1 and p 2 = m2 /n2 in fraction form and as percentages. Do these values compare favorably with your estimates? 3. Does it seem that one sub-population is much more likely to respond favorably ? 4. Based on the values of p 1 and p 2 and possible margin of error, do you think it is possible for the true sub-population proportions p1 and p2 to be equal? Explain. Two-Way Table We next wish to display the data graphically with two-way tables: Y = “Yes” N = “Not Yes” Ω1 total in Ω1 Ω2 total in Ω2 Total “Yes” Total “Not Yes” total 1. Make two tables, the first of which shows the number of people in all of the categories, and the second of which shows the percentage of persons in all of the categories. (For the second, simply divide everything in the first table by the total in the bottom right.) 2. Compute the conditional percentages P(Ω1 | Y ) and P(Ω2 | Y ) and explain what these values mean. Dr. Neal, Spring 2009 Confidence Intervals Next, you will construct confidence intervals for the true population proportion p , and the true sub-population proportions p1 and p2 . 1. Compute 95% confidence intervals for p , p1 , and p2 , where p ≈ p ± 1.96 p (1− p ) n or p ≈ p ± 1.96 p (1− p ) n N − n N − 1 (for population size N ) Use similar formulas for p1 and p2 (with p 1 , n1 , N1 , and p 2 , n2 , N2 ). 2. Do you feel that the margins of error are small enough to pinpoint the true population proportions? Explain. 3. Based on the confidence intervals for p1 and p2 , is it possible for p1 and p2 to be equal; that is, is there any overlap in their confidence intervals? 4. Find a 95% confidence interval for the difference p1 − p2 and explain the interval in words. Based on this confidence interval, is it now possible for p1 and p2 to be equal? Hypothesis Tests Lastly, you will use the sample data to perform hypothesis tests on the initial personal estimates that you made. 1. Let P denote your personal estimate for p . (See Initial Estimates Question 2.) Use your data to test the null hypothesis H0 : p = P with a one-sided alternative (you can use the 1–PropZTest). Explain your conclusion in terms of the p -value. 2. Test the null hypothesis H0 : p1 = p2 with a one-sided alternative (2–PropZTest). Explain your conclusion in terms of the p -value. What do you infer now about the proportions being the same for each sub sub-population? 3. Now let P denote your personal estimate of the difference p1 − p2 . (See Initial Estimates Question 1.) Use your data to test the null hypotheses H0 : p1 − p2 = P with a one-sided alternative. Explain your conclusion in terms of the test statistic and rejection region. (Recall: In this case, if P ≠ 0, then we define the test statistic by z = ( p 1 − p 2 ) − P p 1(1 − p 1) n1 + p 2 (1− p 2 ) n2 which follows an approximate standard normal distribution Z ~ N(0, 1) for large sample sizes.) State any final conclusions and add any other comments that you like.