Lab 6 for Math 17 Review and Confidence Intervals for p

Lab 6 for Math 17: Review and Confidence Intervals for p The first four pages contain practice/review problems for you. Choose two to try (look over them and pick two you want practice on, not two you know you can do!), and then complete the confidence interval activities. This lab is designed to take both lab days this week. If you finish the CI activities before the end of class Wednesday, you may of course do more review problems. 1 Do you remember that Sampling and Design “stuff ”? a. A member of the City Council has proposed a resolution opposing construction of a new state prison in the City. The council members decide they want to assess public opinion before they vote on this resolution. Below are some of the methods that are proposed to sample local residents to determine the level of public support for the resolution. Determine the name of the sampling technique corresponding to each method (e.g. cluster, SRS, etc.). Method One: Have each council member survey 50 friends, neighbors, or co-workers. Method Two: Have the Board of Elections assign each voter a number, then select 400 of them using a random number table. Method Three: Randomly pick 50 voters from each election district. Method Four: Call every 500th person in the City phone book. Method Five: Randomly select several city blocks; interview all the adults living on each block. b. A researcher wants to estimate the population size of a moth species in a forested area with fairly dark everyday conditions. The researcher decides to use capture-recapture sampling, and captures and marks 50 moths which are then released. Two weeks later, he recaptures 60 moths and 3 are marked. i. Estimate the moth population size. ii. If you were told that the researcher used a flamboyant neon orange tag to “mark” the moths, would you be concerned about the validity of your estimate? Why? c. Suppose you are trying to evaluate the effects of a fertilizer and a pesticide (both separately and jointly) on a corn crop in terms of amount of crop. You have 2 varieties of corn available and 16 comparable fields (no runoff or altitude issues). Design an appropriate experiment and specify how your design satisfies all three mandatory principles for an experimental design. 1 2 Cycling: With or Without a Helmet? (Inspired by past student project) Cyclists over the age of 13 in MA do not have to wear helmets. Suppose you have data collected based on whether or not an adult cyclist is wearing a helmet and whether or not automobiles pass close or leave additional space for the cyclist. The data is summarized in the table. Treat the counts collected as population information to answer the following questions. Helmet/Pass No Helmet Helmet Total Pass Close 35 49 84 Leave Space 40 56 96 Total 75 105 180 a. What is the probability a randomly selected cyclist is wearing a helmet? b. What is the probability a randomly selected cyclist was passed close? c. What is the probability a randomly selected cyclist was not wearing a helmet given that he or she was passed close? d. Are pass status and wearing a helmet independent? Explain using a probability calculation. e. What is the probability 2 randomly selected cyclists would both not be wearing helmets? f. What is the probability 5 randomly selected cyclists would all be passed leaving space? g. What is the probability at least one of 5 randomly selected cyclists was passed close? 2 3 Graphs on a Diet A nutritional consultant has collected food diaries from 40 participants in a diet program and computed their daily caloric intake for a “typical” day. Some basic graphs follow. Figure 1: Nutrition Graphs a. What is the primary use of a Q-Q plot? b. Describe the distribution of caloric intake based on all three graphs. c. The sample mean and standard deviation were 749.85 and 62.33 respectively. Interpret the standard deviation. d. If the caloric intake distribution really was normally distributed with population mean 749.85 and standard deviation 62.33, what is the probability a participant would consume over 825 calories in a typical day? e. If you were choosing between the mean and median to describe the center of the distribution of caloric intake, which would you choose and why? 3 4 Avatar and Binomials In an old stats class survey, it was found that roughly 50 percent of the class had seen Avatar (in whole, not in part). Now suppose we ask the question to our class which has 20 students. Let X be the number of students in our class which have seen Avatar. a. Do you think it is appropriate to describe the distribution of X as Binomial (n=20, p=.50) loosely based on the former stats class results? Explain. b. Assume the Binomial model is appropriate. What is the expected number of students in our class who have seen Avatar? c. What is the probability fewer than 2 students in our class have seen Avatar? d. Assume we consider a new class with size 100 instead of 20. What is the approximate probability more than 66 of the students in the new class have seen Avatar, if we assume the proportion of students who have seen Avatar is .5? Be sure to check conditions to be sure you can use an approximation. e. Your calculation in d. was based on counts. What is the equivalent probability using a sample proportion? Verify that you can use the sampling distribution of the sample proportion (state the sampling distribution, then compute the probability) to obtain the same result as in d. f. What result allows us to state that for any starting population distribution, for large sample sizes, the distribution of the sample mean is approximately normal with mean equal to the population p mean, and standard deviation equal to σ/ (n)? 4 5 What Does a Confidence Level Really Mean? Open the proportion confidence interval applet using the applet link on the course website. The original applet can be found at http://onlinestatbook.com/stat sim/normal approx conf/index.html. First off, you can ignore the directions in the applet. We won’t be using that scenario as the basis of our investigation. Second, the applet uses a common notational difference to our textbook. They call the population proportion π and the sample proportion p instead of p and p̂. Finally, they use N instead of n. However, these are minor notational differences. The main purpose of the applet is to let you FIX a population proportion, a confidence level, sample size, and number of times you want to take a sample and observe what proportion of your generated intervals contain the population proportion. 1. To begin, set the population proportion (denoted π in this applet) to .5 (that is the default). Then, set N=20, and leave simulations and confidence level alone. The applet will simulate 10000 samples each size 20 coming from a population where 50 percent is yes and 50 percent is no, and make a confidence interval for the proportion of yes responses. What percent of intervals generated SHOULD contain the population proportion? Hit Begin Simulation. What percent DID contain the population proportion? 2. Keep the settings the same and hit Begin Simulation a few more times. What percent of the generated intervals contain the population proportion in each case? 3. Change the population proportion π to .3, but leave all other settings alone. What percent of intervals generated SHOULD contain the population proportion? Hit Begin Simulation a few times. What percent DID contain the population proportion in each case? 4. You should notice that the percentage covered (while close to 95 percent) will usually be slightly under 95 percent for the setting above. Why? What value (besides the population proportion) could you change to bring it back up to the expected 95 percent level, and how would you change it (increase or decrease)? 5. Try your solution. Did it work? 6. Change the confidence level to 90 percent and change the sample size to 50. Leave the other settings alone. What percent of generated intervals SHOULD contain the population proportion? Hit Begin Simulation a few times. What percent of intervals DID contain the population proportion in each case? 7. Change the confidence level to 99 percent and leave the other settings alone. What percent of generated intervals SHOULD contain the population proportion? Hit Begin Simulation a few times. What percent of intervals DID contain the population proportion in each case? 5 8. Try your own settings for the population proportion, N , and the confidence level a few times, leaving simulations at 10000. Does it seem like the method used to generate confidence intervals works for most settings? 9. Explain, in your own words, what a 90 percent confidence level means. (You should be able to extend this to any percent confidence level.) 10. Recall: What criteria must be met for a confidence interval for a population proportion to be valid? Was there evidence that this was important in the applet? We now turn to a confidence interval applet using means (a topic which we will see in 2 weeks) to make a point about probabilities and confidence intervals. Stir your browers to: http://onlinestatbook.com/stat sim/conf interval/index.html, or alternatively load it from the course website. Change sample size to 20 and hit Sample. You will see 100 different confidence intervals generated. The true population mean in each case is 50. Orange/red tells the difference between a good/bad 95 percent CI and blue/white tells the difference between a good/bad 99 percent CI. We will be focusing ONLY on the 95 percent portion. 11. Take a look at the first CI on your screen. Focus on the 95 percent portion (i.e. red vs. orange). Does that CI contain the true mean of 50? 12. Since you already HAVE the interval computed and you know the mean is 50, what is the probability your 95 percent interval contains the population mean? 13. Take a look at the second CI on your screen. Focus on the 95 percent portion. Does that CI contain the true mean of 50? 14. What is the probability that second CI (which is known) contains the population mean? 15. Pick a CI that DOES have a red 95 percent portion. What is the probability that CI contains the population mean? 16. Overall, what percentage of 95 percent CIs should contain the population mean? 17. Once you HAVE an interval computed, what is the probability that interval contains the population parameter of interest? This is why once you have an interval computed, you CANNOT say that the probability your interval contains the parameter is .95 or whatever the corresponding level is. The probability is either 0 or 1. The population parameter is a fixed but unknown quantity, and if you have an interval to compare it to, it will either be in the interval or not. 18. Finally, take a look at the width of the CIs when sample size is set at 20. Now, hit clear and change the sample size to 10 and then hit sample. What do you notice about the effect of sample size on CI width? 6 6 More to Check Your Understanding for CIs A report gives a 99 percent confidence interval for the proportion of patients who suffer minor side effects on a certain drug as (.22,.32). You may assume the CI was based on a random sample of patients. 1. What was the sample proportion of patients who suffered minor side effects on the drug? 2. What was the population proportion of patients who suffered minor side effects on the drug? 3. (True/False) P(population proportion of patients who suffered minor side effects is in (.22,.32))=.99 4. (True/False) P(sample proportion of patients who suffered minor side effects is in (.22,.32))=.99 5. What does the 99 percent confidence level mean? 6. Could you find the sample size this interval was based on? (Set it up if you can but don’t need to solve for n.) 7. The report states that more than a quarter of patients should expect to suffer minor side effects from the drug based on the CI. Do you agree or disagree? Why? 8. The standard drug for treating this condition has a minor side effect proportion of .5 (roughly 50 percent of patients suffered minor side effects). Would you encourage adopting this new drug over the standard drug if they only differed in minor side effect proportions? Explain. 9. Another drug for treating the same condition has a similar confidence interval of (.27,.37). Would you be able to conclude that you should use the first drug because it has a lower rate of minor side effects assuming the drugs are equal in all other respects? 7 7 Beetles In Michigan and surrounding states, there has been substantial attention in recent years on the emerald ash borer, an invasive beetle that destroys ash trees. In some states, like Wisconsin and Michigan, ash trees make up a sizable percentage (20%) of “urban” forest, so the death of these trees and the spread of the beetle must be dealt with. The beetle has now been found in numerous other states and a quarantine region for moving ash wood has expanded to three states (it was just southeastern Michigan at one point). You have been asked by one of the afflicted states to determine the proportion of infected ash trees (assume you can identify the infected trees). A random sample of 500 ash trees in the state leads you to conclude that 357 are infected. What is the sample proportion of infected ash trees? If the population proportion was really 70 percent, what is the probability of observing the sample proportion you observed or something higher? Include the check needed to be able to use the normal table to get this probability. (This is a review question!) Provide the state with a 95 percent confidence interval for the proportion of infected ash trees. Include the check needed here for large n (different check than above). What is the probability your interval contains the true proportion of infected ash trees? How would a 99% CI compare to your 95% CI? How about a 90% CI? Based on your 95% CI, if the state asked you if it was reasonable to conclude that 80 percent of ash trees were infected, what would you say? Based on your 95% CI, if the state asked you if it was reasonable to conclude that 70 percent of ash trees were infected, what would you say? Another state got a similar 95% CI from .66-.72 for their proportion of infected ash trees. What was the value of their sample proportion that led to that CI? (Think about the formula.) Think about and explain (but don’t solve) how you would work backwards to find the sample size they used to estimate p. Can you conclude that the two states have different proportions of infected ash trees based on the 95% CIs? Why or why not? (We will learn how to deal with two proportion problems directly in about a week.) 8 8 To Turn In - Polls on Religion An article based on a poll conducted by Princeton Survey Research Associates International in late summer 2010 found that 43 percent of those surveyed did not know what President Obama’s religion was. The poll was conducted by landline and cell phone interviews with 3003 randomly chosen adults. The article states the poll has a margin of sampling error of plus or minus 2.5 percentage points. Construct a 95 percent confidence interval for the proportion of adults who do not know the President’s religion. Interpret the confidence interval you construct. Is it correct to say that the true proportion who do not know the President’s religion is in your interval? What is the margin of error for your confidence interval? Is your margin of error consistent with the article’s reported margin of error or with the sample size used? If its not consistent, what might explain the difference? 9