Problem: A recent report from Pew Research Center (September 14, 2018) discussed key finding about U.S. immigrants. One result was that Mexico is the top origin country of the U.S. immigrant population. Specifically, in 2016, 26% of all immigrants living in the U.S. were from Mexico. Assume that a random sample of nine immigrants living in the U.S. is obtained. Let the random variable x equal the number of immigrants out of the sample of nine that are that are from Mexico. Generate the binomial probability distribution for x. * Source: http://www.pewresearch.org/fact-tank/2018/09/14/key-findings-about-u-simmigrants/ Part I Theoretical Distribution Use STATDISK to generate the binomial probability distribution (see example on p. 20 of the Chapter 6 handouts). Snip or screen print your output distribution for printing. Make sure you get the whole output box, including the input information.
Part II Experimental Distribution
Answer Sheet Assignment 6 Part I Theoretical Distribution Fill in all of the blanks: n = 9 s = success = (describe in words) = immigrant who is Mexican p = 0.26 x = 0 through 9 (possible number of immigrants from Mexico selected out of 9) 1. Explain why this is a binomial experiment, making sure to address each of the four criteria (given on p. 12 of the Chapter 6 Handouts). 1. There was a fixed number of trials, 9. 2. Each trial was independent. 3. There were two outcomes: From Mexico, or not. 4. Probability of success was the same for each trial, p = 0.26. 2. What is the theoretical probability that exactly 2 of the 9 selected U.S. immigrants are from Mexico? (to three significant figures) From STATDISK, P(2) = 0.296 3. What is the theoretical probability that less than 2 of the 9 selected U.S. immigrants are from Mexico? (to three significant figures) From STATDISK, P(1 or fewer) = 0.277 4. What is the theoretical probability that at least 3 of the 9 selected U.S. immigrants are from Mexico? (to three significant figures) From STATDISK, P(3 or greater) = 0.427
5. Would it be unusual if 5 or more of the 9 selected U.S. immigrants are from Mexico? Explain why or why not, using your results from the theoretical probability distribution. Use the probabilities method on p. 24 of the Chapter 6 Handouts, and you must quote a specific probability here. From STATDISK, P(5 or more) = 0.0571. No, that would NOT be unusual, because the probability is > 0.05. 6. Make a probability histogram for the theoretical probabilities on the following grid. 7. Describe the shape of the probability distribution, and explain briefly why it has that shape. Specifically, look at p. 22 of the Chapter 6 Handouts, and address the values of both n and p. Calculate the value of np(1 p). It is right-skewed because p (0.26) < 0.5, and the trial size (n = 9) is smallish. Specifically, np(1 p) = (9)(0.26)(0.74) = 1.73 which is less than 10.
8. Show me your hand calculations (using the binomial distribution formulas) for the mean and standard deviation of the theoretical distribution. Mean: : = np = (9)(0.26) = 2.34 Std. Dev.: = np( 1 p) = (9)(0. 26)(0. 74) = 1.316 Part II Experimental Distribution 1. What was the experimental probability that exactly 2 of the 9 selected U.S. immigrants are from Mexico? (to three significant figures) From EXCEL, P(2) = 0.48 2. What was the experimental probability that less than 2 of the 9 selected U.S. immigrants are from Mexico? (to three significant figures) From EXCEL, P(less than 2) = P(0) + P(1) = 0.267 3. What was the experimental probability that at least 3 of the 9 selected U.S. immigrants are from Mexico? (to three significant figures) From EXCEL, P(3 or more) = P(3) + P(4) + + P(9) = 0.253 4. What was the mean and the standard deviation of the experimental probability distribution? Mean = 2.04 (2.0) Standard Deviation = 1.171 (1.2)
5. Make a probability histogram for the experimental probabilities on the following grid. 6. Comment on/discuss the differences and/or similarities between the experimental and theoretical results for this procedure. Specifically address the differences in the probability distributions (shape), the means and the standard deviations. The probability distributions are both right-skewed. There are definitely some differences though (more than usual!). This could be due to measurement error or random chance samples vary! P(2) is much higher for the experimental results compared to theoretical. The means were fairly close (2.0 experimental vs. 2.3 theoretical) and the standard deviations were fairly close (1.2 experimental vs. 1.3 theoretical). Overall about what you would expect, given that we only ran the experiment a limited number of times (75 runs). If we repeated the experiment many more times, the results would gradually approach the theoretical probabilities. Note that the theoretical distribution does not have a specified number of trials, it is simply what the mathematics predicts should happen in the long run.
7. Immigration has been a huge topic in the U.S. news over the last several years. The specific topic of this assignment deals with a Pew Research report that presents key findings about U.S. immigrants. This is a free form question. What question would YOU like to answer, specifically relative to immigrants in the United States? Basically, write your own question, and answer it. Research some topic about immigrants in the U.S. Briefly tell me something(s) that you learned, including some applicable statistics about the information, and any comments that you have. You must cite a credible source(s) for your information. Note: feel free to type this up if you want, and attach. You are welcome to include your personal opinion, but I also want to see some results of your research. List your source: