Lab 3 for Math 17 Normality, Standardization, and Capture-Recapture

Lab 3 for Math 17: Normality, Standardization, and Capture/Recapture 1 Capture-Recapture Sampling 1. This is a repeated sampling process, requiring at least 2 samples with usually a pre-determined length of time between samples. 2. You take the first sample (which may or may not have fixed size), and you somehow mark each subject in the sample, then release the subject (fish, bird, etc.). Marking should be done in such a way that it does not affect survivability of the subjects in any way. 3. Later, you take a second sample, and count the number of marked subjects in the second sample (i.e. the number in the second sample that were also in your first sample). 4. Claim: the proportion marked in your second sample should be roughly equal to the proportion of the whole population that you marked as the first sample assuming both samples are SRS and there wasn’t too much disturbance to the population between the first and second samples (e.g. it is a complication if marked subjects die, but adjustments can be made in practice). So, there are four quantities of interest: N = estimate of population size, M = number marked in first sample, C = number captured in second sample, and R= number in second sample which are marked, and the presumed relationship is M/N = R/C or C/N = R/M . Solve for N in terms of the other three quantities: Let’s try it. Say your first sample consists of 200 fish which you tag and release. A year later, you take another sample and catch 120 fish, of which 12 are marked. What is your estimate of the population size? How about if your original sample was 60, and 2 weeks later you catch 100 of whom 5 are marked? What would your population estimate be? Do you think the first or second sample is usually larger? Why? Are there any possible issues with one sample being a lot bigger than the other? 2 Trying Out Capture-Recapture Sampling We are going to examine Capture-Recapture sampling for ourselves to see how it works and how reliable it might be by trying to estimate the population size of cheddar goldfish. Each group (6 groups) will receive the following materials: cheddar population of goldfish, marked replacements (the colored goldfish), sampling implements (dixie cups and brown bags), and napkins. Groups one, two, and three will only mark one cup of goldfish each, while groups four, five, and six will mark two cups worth each. All other directions are below. You may eat unused goldfish, but don’t accidentally eat a population member! You WILL have to count the entire population size AFTER we do the sampling, so don’t start eating the population early. Your group must decide what to do 1 about partial goldfish. The questions will guide you to record R,C, and M , at each relevant step. To start: Empty your cheddar goldfish population into the brown bag. Also, write down your group number so you “mark” the correct amount. Group number: 1. In your group, estimate the population size (you may not use serving info from the bag). Our estimate of the goldfish population is: 2. Based on your group number, capture the specified “amount” of goldfish from the bag, and count how many that actually was. Then, replace those goldfish with the marked goldfish. How many goldfish did you mark? You may place captured goldish that were replaced back in the original bag to snack on. 3. Shuffle up the population in the sampling bag (you can determine how best to do this). 4. Sample one cup worth of goldfish. How many did you catch? How many are marked? What is your estimate of the population size from this sampling? Return those goldfish to the population. 5. Sample two cups worth of goldfish. How many did you catch? How many are marked? What is your estimate of the population size from this sampling? Return those goldfish to the population. 6. Sample three cups worth of goldfish. How many did you catch? How many are marked? What is your estimate of the population size from this sampling? Return those goldfish to the population. 7. Determine the actual size of the population of goldfish you had. How accurate were your results using capture-recapture sampling? Class discussion notes: Ideally, we would have repeated some of the sampling so you all could see the variability in the estimates, but due to time constraints we could not do that. However, capture-recapture sampling is a technique that has been extensively studied, with various modifications available depending on what you are studying, and has been used to estimate population sizes (and related quantities) for everything from zebras and tigers to human drug addicts. 2 3 Normality and Standardization - To Turn In 1. Normal distributions are characterized by these two values (names and notation): 2. Saying a distribution is standard normal means what? 3. Circle any of the following that are not valid normal distributions. (Be sure you can explain why.) a. N(0,6) b. N(50,.0001) c. N(-30,-10) d. N(-200,60) 4. True/False. The empirical rule is valid for all distributions for a quantitative variable. 5. Suppose a test results in normally distributed scores with a mean of 75 and standard deviation of 12. Compute a z-score for Betty’s score of 84. 6. Using the normal model, what is the probability a student scored 84 or higher? (if you are interested in learning how Rcmdr can compute this so you can check your answer from the table, just ask) 7. The top 25 percent of students scored above what value? 8. Another test taken by 16 students results in a sample mean of 40 and sample standard deviation of 5. Give appropriate notation for each statistic related to the test. 9. Interpret the sample standard deviation from question 8. 10. Assume the new test also has normally distributed scores and treat the mean and standard deviation provided as µ and σ. What score on the second test is equivalent to Betty’s 84 on the first test? 3