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Problem Five (Hypothesis Testing) Suppose want to test if Yifu claims to have Extrasensory Perception (ESP) , and that if draw card drawn randomly and with replacem...

Question

Problem Five (Hypothesis Testing) Suppose want to test if Yifu claims to have Extrasensory Perception (ESP) , and that if draw card drawn randomly and with replacement from an ordinary deck of cards, he can guess the suit without seeing the cardam sceptical and propose to test his claim_ formulate the hypothesis that when he tells me the suit; he is just guessing and the probability of his getting it right is 25%Therefore perform 100 trials where shuffle the deck, draw card, ask him to guess the

Problem Five (Hypothesis Testing) Suppose want to test if Yifu claims to have Extrasensory Perception (ESP) , and that if draw card drawn randomly and with replacement from an ordinary deck of cards, he can guess the suit without seeing the card am sceptical and propose to test his claim_ formulate the hypothesis that when he tells me the suit; he is just guessing and the probability of his getting it right is 25% Therefore perform 100 trials where shuffle the deck, draw card, ask him to guess the suit, and then replace the card: decide that since ESP is about guessing the card"s suit correctly; will make this one-sided (right) test with LOS 0.05. We perform the test and Yifu identifies 32 of the cards correctly: Now at this point; realize that if use the normal distribution, it will be an estimate and am not sure how well it will work; because B(100,0.25) is not symmetric. So decide for maximum precision | Il use the binomial directly; but how to calculate an LOS that goes along with discrete distribution (think about how to calculate the top % of B(3,1/2)4)? So decide to reason backwards: I'Il calculate P(X > 32) and see if it is ess than my LOS (which would mean it has to be in the critical region). (A) Perform the test as specified. Reject if P(X > 32) < LOS and fail to reject otherwise (B) What is the smallest number of cards Yifu would have to identify to make me reject my hypothesis at the % LOS? Hint: use binom cdf



Answers

An investigator wishes to estimate the proportion of students at a certain university who have
violated the honor code. Having obtained a random sample of $n$ students, she realizes that asking
each, "Have you violated the honor code?" will probably result in some untruthful responses.
Consider the following scheme, called a randomized response technique. The investigator
makes up a deck of 100 cards, of which 50 are of Type I and 50 are of Type II.
Type I: Have you violated the honor code (yes or no)?
Type II: Is the last digit of your telephone number a $0,1,$ or 2 (yes or no)?
Each student in the random sample is asked to mix the deck, draw a card, and answer the resulting question truthfully. Because of the irrelevant question on Type II cards, a yes response
no longer stigmatizes the respondent, so we assume that responses are truthful. Let $p$ denote the
proportion of honor-code violators (i.e., the probability of a randomly selected student being a
violator), and let $\lambda=P($ yes response). Then $\lambda$ and $p$ are related by $\lambda=.5 p+(.5)(.3) .$
(a) Let $Y$ denote the number of yes responses, so $Y \sim \operatorname{Bin}(n, \lambda) .$ Thus $Y / n$ is an unbiased
estimator of $\lambda .$ Derive an estimator for $p$ based on $Y .$ If $n=80$ and $y=20,$ what is your
estimate? [Hint: Solve $\lambda=.5 p+.15$ for $p$ and then substitute $Y / n$ for $\lambda . ]$
(b) Use the fact that $E(Y / n)=\lambda$ to show that your estimator is unbiased for $p$ .
(c) If there were 70 Type I and 30 Type II cards, what would be your estimator for $p ?$

In this question to start off, we are given this relationship between Land A and P, then in part a. We are told that why is a binomial random variable based on parameters N and Lambda? Therefore, why divided by n is an unbiased estimator for Lambda and we are asked to derive an unbiased estimator for P based on why we can rearrange the equation at the top of the sheet to give the following. This means that an estimator for P is given by the following. So that is our estimator for P and now given and equals 80. And why equals 20? We want to find our estimate for P. So we just plug this into the formula for estimator and this comes out to 0.2 for part B. We want to show that our estimator is unbiased. So we really want to show that the expected value of our estimator is equal to P. So this is equal to the expected value of two y over em minus 0.3. That's just using this equation with why over and is equal to Lambda and then using the linearity of expectation this can be re expressed as the following. So this is two times Lambda and the expectation of 0.3 is 0.3 and this is equal to P since the expected value of our estimator is the parameter we're estimating, for it is an unbiased estimator. And then for part C, we are given a slightly different set up for the question which would result in this relationship between Lambda and P would now be 0.7 times P plus 0.3 times 0.3. And now we are asked, What are estimator for P would be the estimator for Lambda remains. Why over em since why is still a binomial random variable? The estimator for P is equal to the estimated Verlander minus 0.9 divided by 0.7. And that's done simply by solving for P in this equation and then simply re expressing this substituting. Why over in for for the estimated for Lambda we get why over in minus 0.9 over a 0.7. So this is now our estimator for P

We want to conduct a pair differences test at the alpha equals 5% level testing the claim that population mean X bar A is greater than population X barbie. We have data a be given here, we assume amounts to mr distribution as you can see on the right. I've already calculated the mean difference D bar 6.125 The sample size and eight and the sample standard deviation of differences SD and 8.7 We complete the five steps us to blow to solve this problem first, let's evaluate the requirements to use a student's T distribution of the hypotheses because of the distribution shape it is appropriate to use a student distribution your degree of freedom and minus 27. No hypothesis mute equals zero. Alternative media is greater than zero and alpha equals 00.5 for confidence nexus, complete the test at and the P value our test that is T equals D. Bar over SDR. Again this gives 2.14 U. T. Table. This gives us a P value between 0.5 point 025 That means we can include that P is less than equal to alpha. So we reject the null hypothesis H not which means that we have evidence and you D. Is greater than zero.

Problem. 20. The null hypothesis is that B one minus B two is equal to zero, and the alternative hypothesis is is that the one minus B two is bigger than zero, where be one proportion is X one over N ones for 30/50 she's open six and B two, which is equal to extra over in two, which is 22/50 which is Oh, point or four. The pooled proportion, which is X one plus x 2 30 plus 22 over in one plus in two she is 100. She's equal toe 1000.52 toe. Find the value off until the statistics, which is equal to the P one minus B two over a square root off 4.5 to 1 minus 4.5 52 square root off one over n. One plus one over and to which is equal to 1.6 eso. The probability is equal to the probability that that or the P value is equal, that the probability that that is bigger than 1.6, which is equal to the probability that set is smaller than negative 1.6, which is equal to open toe five, 48 and the P value is bigger than open toe five. So we say it to reject the null hypothesis is so there is no sufficient evidence to support support the claim.


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