Malaika Wauters Math 11 - Lab 4

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May 30, 2024

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Malaika Wauters Math 11 Denise Rava Lab 4: Simulating Bernoulli Trials 1. How many heads did you get in each of the four experiments? First trial - 1 head Second trial - 3 heads Third trial - 3 heads Fourth trial - 2 heads 2. What is the probability of getting zero heads in six tosses? What is the probability of getting exactly two heads in six tosses? a. 0.03125 b. 0.3125 3. In your simulation, how many times out of 1000 were there zero heads in the six tosses? How many times were there two heads? Include a histogram of your simulation results in your write-up along with your answers to these questions. As always, remember to make sure the axes are appropriately labeled. a. As depicted in the histogram above and according to the simulation ran through Minitab, there were only 23 times where I summed 0 heads in total versus a high 293 times where 2 heads were a result of the toss.
4. Compare your answers to the previous two questions. In the 1000 simulations, did you get zero heads about as many times as you expected (remember that the number of times you expect to get zero heads is the number of simulations times the probability of getting zero heads in one simulation)? What about two heads? a. The expected values for the trials don’t exactly line up to be completely accurate with the data that I yielded yet itr is not too far off and it can be concluded that with even more than 1000 trials, the data that I should yield would more accurately resemble the expected value. I had only 23 trials in which there were to be 0 heads while it should be approximately 31.25 times. And again, in regards to discovering 2 heads as a result, it should have happened 312.5 times yet it only occurred 293 times. 5. First, simulate tossing a coin 10 times and counting the number of heads. Do 1000 repetitions of this procedure (so you will generate 1000 numbers, each a binomial random variable with n = 10 and p = 1/2). Present a histogram of the results. 6. Find the mean and standard deviation of the 1000 numbers that you got. (Remember you can get this using Stat --> Basic Statistics --> Display Descriptive Statistics .) Are these numbers close to the expected value and standard deviation of a binomial random variable with n = 10 and p = 1/2? a. Mean = 5.479 b. StDev = 1.641 c. By binomial distribution, E[ X ] = np and SD( X ) = ( np (1- p )) ½ So the expected value would be half of 10, 10(½) = 5 And the expected StDev would be the square root of 10(½ )(½ ) = 1.581 d. Evidently, the values that were yielded, also known as the actual observed values are very close to the expected values (5.479 - 5 = 0.479) and (1.641-1.581 = 0.06)
7. Now simulate tossing a coin 100 times. As before, do 1000 repetitions of this procedure, and present the results in a histogram. Based on the histogram, if you tossed a coin 100 times, would you be surprised if the number of heads were 5 more (or 5 fewer) than expected? Would you be surprised if the number of heads were 20 more (or 20 fewer) than expected? 8. Next simulate tossing coins 1000 and 10,000 times. As before, do 1000 repetitions of each procedure, and make two histograms. If you tossed a coin 10,000 times, would you be surprised if the number of heads were 5 more (or 5 fewer) than expected? Would you be surprised if the number of heads were 20 more (or 20 fewer) than expected? a. After flipping the coin 10,000 times I would expect to have plus or minus 5, or even perhaps plus or minus 20 heads that I initially predicted because of the increase in the standard deviation and spread of the data, although it would be slightly less likely to expect a variation of plus or minus 20 heads as a result because that is a bigger difference.
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