Math 146, Summer 2017 Instructor Linda C. Stephenson (due Wednesday, August 9) The purpose of the assignment is to find confidence intervals to predict the proportion of a population. The population in question is the outcomes from rolling a single die (singular of dice). Specifically, what proportion of the rolls will be greater than 3, in other words an outcome of 4, 5 or 6 on a single roll of a die. Theoretically, what is the population proportion, p, of rolls that will be greater than 3? You will try to capture the true population proportion by rolling a die (simulated), and using the samples to come up with confidence intervals to predict the population proportion. What IS the true population proportion, p, that we are trying to capture in the confidence intervals? In other words, what is the proportion of rolls, theoretically, that will be greater than 3? A. Roll a die 40 times, and construct both an 75% and a 95% confidence interval estimate of the population proportion of rolls that come up greater than 3. B. Roll a die 500 times, and construct both an 75% and a 95% confidence interval estimate of the population proportion of rolls that come up greater than 3. Do your confidence intervals contain the true population proportion? Also, what differences do you see between the confidence intervals for n = 40 vs. n = 500? And what differences do you see between the confidence intervals for 75% vs. 95% confidence? Page 1 of 9
Math 146, Summer 2017 Instructor Linda C. Stephenson Instructions Parts A and B Confidence Intervals For Part A you need a simple random sample of 40 rolls of a single die, and for Part B a simple random sample of 500 rolls of a die. Use the same sample in either case to construct both of the confidence intervals (75 and 95%). 1. You re going to use the STATDISK Dice Generator to simulate rolling a die. Open STATDISK, and go to Data/Dice Generator. 2. Change the sample size to 40, the number of dice to 1, and hit the Generate button. Your output will appear on the right. The first column Roll is just listing the roll number from 1 through 40. The second column Total is the actual roll of the die. 3. Hit the Copy button at the bottom right, which will copy just the second column, the actual die rolls. 4. Click on the top cell in STATDISK (Row 1, Column 1), and hit the Paste button. Your 40 die rolls should now be pasted into Column 1. 5. You want to count up how many of the rolls were greater than 3, in other words how many 4, 5 or 6 rolls did you get. The easiest way to do that is to sort the data from biggest to smallest. In the tool bar, hit the Data Tools then Sort Data. Choose One column, using column 1, and change the order from Z to A. Then hit Sort. In other words, it will sort them with the 6 s and the top, and the 1 s at the bottom. Page 2 of 9
Math 146, Summer 2017 Instructor Linda C. Stephenson So now your output should look something like this: I highlighted row 21, because that tells me that I had 21 rolls that were greater than 3. In other words, my sample proportion was: p = 21 40 = 0.525 I want to see a snip of this output, so save it for printing! 6. Now, repeat the whole process, but this time change your sample size to 500. Sort them from Z to A again. Scroll down, and just snip a portion of the output (not all 500 lines!) so that I can see how many of your rolls were greater than 3. For example, I had 246 rolls out of 500 that were greater than 3: Again, save this output for printing! Page 3 of 9
Math 146, Summer 2017 Instructor Linda C. Stephenson 7. Now you have everything you need to calculate a confidence interval for the proportion, p, the true population proportion of rolls that are greater than 3. And of course, you already know the true population proportion (because you can calculate it theoretically), but the point is to see how well the confidence interval method works! Did you capture the true population proportion in your confidence intervals? So, calculate the confidence interval for each of your two sample sizes, for both a 75% and 95% confidence level. You will have a total of four different confidence levels when you are done. NOTE: Do not round off your results until the very end. Use ALL of the digits for your sample proportion. VERY IMPORTANT CAUTION!!! Remember that you are calculating the confidence interval around your POINT ESTIMATE p, not around the true proportion p! The whole point is to find out if your predicted confidence interval includes the true proportion. If you create the confidence interval by adding and subtracting the margin of error E from the true proportion p, then it will definitely contain the true proportion, right? Which is NOT the point! Use your point estimate! What You Need to Turn In 1. Print out from STATDISK showing the two snips of your sorted samples for both n = 40 and n = 500. See examples on p. 3. 2. Print out of the following Answer Sheets (five pages total), filled out completely. Page 4 of 9
Answer Sheet After your analysis of the population distribution, what can you conclude about using the methodology to obtain a confidence interval for the population proportion. Is it reasonable to use the procedure for both the sample size n = 40 and n = 500? Explain briefly, and specifically address the two requirements given in the procedure on p. 426 of the textbook, also given on p. 22 of the Chapter 9 Handouts. To be extremely clear, I am talking about these requirements: Note: the second requirement may not quite be met for n = 40, but it should be pretty close, it will depend on your results. The third requirement (which I lined out) doesn t really apply, because the population size here is theoretically infinite. Population Proportion List the theoretical population proportion of rolls that will be greater than 3. Population proportion: p = Page 5 of 9
Parts C and D Confidence Intervals Note that I want the confidence intervals shown in interval notation, like: (lower bound, upper bound). Round them to THREE DECIMAL PLACES! E = margin of error CI = Confidence Interval Point estimate, p n = 40 n = 500 z /2, 75% E (75% confidence) CI (75%) (, ) (, ) Did 75% CI include the true population proportion? (yes/no) z /2, 95% E (95% confidence) CI (95%) (, ) (, ) Did 95% CI include the true population proportion? (yes/no) Page 6 of 9
Graph your Intervals: Use the following scales to very carefully draw your four confidence intervals. Draw them like this: (-----------------------) For each sample size, put your two intervals right above the scale. Next to each interval, label the confidence level: 75%, 95%. n = 40: n = 500: p Note: scale units are proportion, in decimal form, which is basically unitless. Page 7 of 9
1. Discuss (and explain) the effect of the sample size (n = 40 vs. n = 500) on the confidence intervals. How does it change the confidence intervals, and why? 2. Discuss (and explain) the effect of the confidence level (75% vs. 95% confidence) on the confidence intervals. How does it change the confidence intervals, and why? Page 8 of 9
3. Did your results play out the way you expected them to? Or, was there anything unexpected about your analysis? (Remember, there might be, because samples vary!). For your answer, for one thing, specifically address the differences in the values of your point estimates for n = 40 and n = 500. Was one closer to the true proportion p than the other, and was it as expected? Also specifically address which of your confidence intervals were more likely to contain the true proportion, and which were less likely. Did it turn out that way? You don t have to write a novel, but give me a little explanation, not just a yes or no answer. Page 9 of 9