A biologist is breeding fruit flies to include a specific genetic mutation that will be useful in understanding memory in humans. To check whether a fly has the mutation, a DNA sequence is analyzed in a way that kills the fly, so the biologist wants to test only a sample of the flies to estimate the proportion of flies that have the mutation.

The biologist selects 40 flies to sequence at random and finds that 9 of them have the genetic mutation.

1. Based on this sample, estimate the proportion of flies in this group that has the genetic mutation.
2. The scientist is worried that having only one sample may not be reliable for estimating the proportion of flies with the mutation, but does not want to sacrifice more flies to get a larger sample. The proportion from the sample is a good estimate for the population proportion, but it is difficult to understand the possible variability from a single value. Andre has a suggestion for how to better understand the variability:
   1. Assume the sample is representative of the population of flies, and create a simulation that mimics what the scientist found. Andre gets 200 pieces of paper, marks 45 of them as “mutant,” and puts them all in a bag. Because Andre decided to use 200 pieces of paper, why should 45 of them be marked “mutant”? What are some other combinations of total number of pieces of paper and number of papers marked “mutant” that he could use?
   2. Andre then simulates the scientist’s sample by drawing a slip of paper from the bag and noting whether it is mutant or not, then replacing the paper into the bag and drawing another paper until he has a sample of 40. He repeats this process for 50 trials and creates a dot plot showing the proportion that are mutant from each trial. Use the dot plot to estimate a range of proportions that include about 95% of the proportions from the trials.

      

   3. Andre then finds the standard deviation of this sampling distribution model to be 0.06. What should he report as the margin of error associated with his point estimate?
   4. Does your estimate for the range of proportions fit with the values Andre should report?

Elena and Clare are each working on a project about how high school students are having trouble finding jobs. They each find the proportion of students without jobs from a random sample, use a computer to do 1,000 simulations with the proportion they found, and then report the results.

Elena says, “The proportion of high school students without jobs is about 0.70 with a margin of error of 0.280.”

Clare says, “The proportion of high school students without jobs is about 0.75 with a margin of error of 0.138.”

1. Both students reported the margin of error based on 2 standard deviations from their simulations. What are the mean of the sample and standard deviation of the sampling distribution each student found? For at least 1 student, show your reasoning.
2. Clare and Elena try to figure out why Clare had such a smaller range of values in her report.
   1. First, they consider the proportion they used in the simulations. Elena says, “My simulation used 0.7 as the proportion since I found that proportion in my original sample.” Clare says, “My simulation used 0.75 as the proportion since I found that proportion in my original sample.” The students used different proportions in their simulations. Do you think this is why Clare has a lesser margin of error? Explain your reasoning.
   2. They look for more differences in their samples and simulation and discover that Elena surveyed 10 people in her initial sample and Clare surveyed 40 people. Then Elena used samples of size 10 in her simulation and Clare used samples of size 40 in her simulation. Do you think this is why Clare has a lesser margin of error? Explain your reasoning.