Question
36 pointx MBalkS,at7 22Eo8My Notesel Yot teucnelAll euros have a national image on the "heads" side and a common design on the "tails" side. Spinning coin, unlike tossing it, may not give heads and tails with equal probabilities_ Polish students spun the Belgian euro 230 times, with its portly king, Albert, displayed on the heads side. The result was 140 heads. How significant is this evidence against equal probabilities? Follow the four-step process. (Round your test statist
36 pointx MBalkS,at7 22Eo8 My Notes el Yot teucnel All euros have a national image on the "heads" side and a common design on the "tails" side. Spinning coin, unlike tossing it, may not give heads and tails with equal probabilities_ Polish students spun the Belgian euro 230 times, with its portly king, Albert, displayed on the heads side. The result was 140 heads. How significant is this evidence against equal probabilities? Follow the four-step process. (Round your test statistic to two decimal places and your P-value to four decimal places. Assume a 95% confidence level:) P-value Conclusion: There is significant evidence that the proportion of times a Belgian Euro coin spins heads Is not 0.50, There is not enough evidence to conclude that the proportion of times a Belgian Euro coin spins heads is not 0.50. You may need to use the appropriate Appendix Table to answer this question_
Answers
Karl Pearson once tossed a coin 24,000 times and recorded 12,012 heads. a. Calculate the point estimate for $p=P($ head ) based on Pearson's results. b. Determine the standard error of proportion. c. Determine the $95 \%$ confidence interval estimate for $p=P(\text { head })$. d. It must have taken Mr. Pearson many hours to toss a coin 24,000 times. You can simulate 24,000 coin tosses using the computer and calculator commands that follow. (Note: A Bernoulli experiment is like a "single" trial binomial experiment. That is, one toss of a coin is one Bernoulli experiment with $p=0.5;$ and 24,000 tosses of a coin either is a binomial experiment with $n=24,000$ or is 24,000 Bernoulli experiments. Code: $0=$ tail, $1=$ head. The sum of the 1 s will be the number of heads in the 24,000 tosses.) e. How do your simulated results compare with Pearson's? f. Use the commands (part d) and generate another set of 24,000 coin tosses. Compare these results to those obtained by Pearson. Also, compare the two simulated samples to each other. Explain what you can conclude from these results.
1 26 when the euro coin was introduced in two dozen to two. Math professors have their statistics. Students tests whether the Belgian one euro coin was a fair coin. They spun the coin rather than talks in it and found that out of 250 spins, 140 showed a head event H, while 110 showed a tail event teeth. On that basis, they claim that is not fair point for a based only given data, find their probability of H and find the probability of tea for heads and tails. So in this situation we had a total of 250 spins, told us that a total of 140 ended up being heads, and that reduces to approximately 8.56 A total of 110 were tails out of the 2 50 and that his point 44 and B use a tree to find the probabilities of each possible outcome for the experiment of tossing the coin twice. So in our tree diagram, Tulse number one has two possibilities that's either gonna land on heads or tails, and according to our data above. There's a 56% chance of heads and a 44% chance of tales because the second flip was independent, the first, and we have the same two outcomes heads or tails. They're still at 56% chance of a heads and 44% chance of tales. And that is true on both sets of branches, and they're wanting us to show the possibility of each outcome. So if we take and go down the branches and is as weed run into heads and then another heads, for example, we can multiply the two values to figure out the possibility of getting the heads on the first horse and the heads on the second toast. So let's do that. We can multiply 0.56 in 0.56 and that ends up being 0.3136 four heads and then a tales. His 0.56 times 0.44 which gives us points. This gives us a point 2464 for the probability of getting details and then heads. It's a point for 4.56 that is also 0.2464 and then the probability of a tales than another tails 0.44 Time 0.4 gives us 0.19 36 Question C used to treat if on the probability of attaining exactly one head and two tosses the coin. Well, if you look at the outcomes, we've obtained one head in this situation and in this situation. So the probability of one head would be those two numbers at together 20.2464 plus 0.2464 which is 0.4928 deep. Use the tree to find the probability of obtaining at least one head. So any time you see at least one you want to think of the compliment of at least one is none. So no heads is what we're looking for. So we're gonna find the probability of no heads, which is the same exact thing is two tails because that's the only situation in which there are no heads. We know the probability of two tails is 1936 so we're gonna take that away from one. And the probability of at least one head is 10.80 64
This is yet another experiment using the Montecarlo mental, um, where you will have to use the definition. Probably two affinity. Vandy's the number off the successful outcomes divided by the total sum will stress and the present situation we're supposed to use. I can't later. They're going to use it for spread. She doing chocolate A t i 84 plus. Then you said it is ah instructed. And then you are going to use it to toss. Ah, coin Andy is supposed to dump. It is out of several is the head and it is out off wine is a tale. The new drink. What we How many zeros they don't do? Gentlemen, is there anyone? Do you get me? Ah, wanted one. Do you get they then small fun. And then after that, you can just do I count in order for you to answer the following questions. Business may experiment. Number one God, the probability that exactly you get exactly foreheads. Remember, you're toasting left ends 12345 And then you make observations. So how many times did you get foreheads in a cross? And it causes a mobile five horses. So do this repeatedly for 50 times. So you will do realize that, um, your answer will be fluctuating between will be fluctuating around 0.156 to 5 off which it is the exact under. If we're going to use the other method. Look, the experiment method differently. Just going to use Ah, the the definition of probability. Then the probability in the second Pierce, who also deflect rating I have four or down off the opening 3125 to be somewhere closer to back so you can do this and enjoy it.
I This again is one experiment that you have to kill out. Specially interesting experiment, uh, which we drive from Mom under Carlos Metal finding ability. And, um, in this case, you are going to use a simple definition of probability which says the probability off in event is it close to the number of successful outcomes off the studio Sambo space. So that's what German Jews. And you can use your calculator your t i a n four plus, and then you you sent it to the correct amount. And in this guess you want to be waking on. Ah, two days that are being award and we are going to repeat the experiment 50 times. Uh then after 50 times, you have to go to their doubts. And, um, when there's out that you have noted, the first part of the person is that you're supposed to find the probability that this some is nine or more. So you just count how many name or a number which is greater than nine is a sound? What do you find? And in the second case, you also supposed to find the probability that the sum is less than these are in session. So which is you? Some off true Islam or three Summer four is on the show. Life's on six, okay, and seven is not included there. But in the offense tickets where you wanted to some off 99 is included. Or more so the sum is always up to talk toward his eyes, so it will be 9 10 11 and 12 that you'll be checking off. You come all the substance Slocum's. Then you divide by by 50 petitions that you give that and you actually see that there is also always fluctuating for the first person. They will be fluctuating on, uh, between 5 18 and in the second case around faithful 12.
All right, This is a practical question that you have to do. It's an experiment. Um, we know that one way to solve probability problem is to repeated the experiment many times, uh, keeping sank off the results. Then the probability can be approximated using the basic definition off a little in effect on. And we can say that probably took it. Vandy's e. I mean, the outcome off a defender Success world. Come on. The he so that our village off e then would be number for ah successfully trance divided by the symbol space. And then we get it Eso What we use when we're during those child is what it's called them on the perimeter or finding come a beauty The Monday Carlo just shows you in a you are approximate through an experiment how to get dissident probability. So it's getting a probability theory and experiment. So if we simulates a coin little Tosa Memoria di on a t. I ate for glass cooperator. You can change the setting to fix this More modern don't do you displayed and and uh and oh, aren't ah then there. That's six plus point right, respectively, for a coin Ah, to us. Interpret is Enel is and and one is a tale in the in the case, the e l I G r Ky can be pressed repeatedly. Therefore, model simulations source. We are now supposing that three dice and old shoes he used them under Carlo method with 100 repetitions to approximate the following probabilities. And the 1st 1 is the probability that the sum is less is five or less in the second case is the probability is a one on our six is world Now If you cut out the experiment on bias Lee, you can see that the results for a well flat rate around several points There are 463 and the results will be we'll also flattering around 0.2 man 63 So you just experimented and then you see and this is quite interesting