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1:Roll a 20-sided die and observe the number. The chance thatthe number is less than 5 is 20% and the chance that the numberis bigger than 10 is 55%. What is the c...

Question

1:Roll a 20-sided die and observe the number. The chance thatthe number is less than 5 is 20% and the chance that the numberis bigger than 10 is 55%. What is the chance that thenumber is less than 5 or bigger than 10?Enter your probability as a decimal value of the formX.XX2: Pick a student at random. Let B denote theevent that the student ate breakfast this morning; let M denote theevent that the student is male.One of the following choices is larger than the other two. Whichis it?a: P(B)

1:Roll a 20-sided die and observe the number. The chance that the number is less than 5 is 20% and the chance that the number is bigger than 10 is 55%. What is the chance that the number is less than 5 or bigger than 10? Enter your probability as a decimal value of the form X.XX 2: Pick a student at random. Let B denote the event that the student ate breakfast this morning; let M denote the event that the student is male. One of the following choices is larger than the other two. Which is it? a: P(B) b: P(B or M) c:P(B and M) 3: A quiz consists of 10 multiple-choice questions, each with 4 possible answers, only one of which is correct. A student who does not attend lectures on a regular basis has no clue what the answers are, and therefore uses an independent random guess to answer each of the 10 questions. What is the probability that the student gets at least one question right? Let's decide on some notations. Let L be the event that the student gets at least one of the questions right. We'll use R for getting a question right and W for getting it wrong (W is essentially "not R"). What is the value of P(L)? Enter your probability as a decimal rounded to 4 decimal places in the format X.XXXX.



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're given this relationship between lambda 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 -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 .3 is .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 our 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 -0.09, 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 -0.09 Over a 0.7. So this is now our 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

So we know that he has 10 Multiple choice questions and five alternatives. So the probability that he gets one right by guessing is 12 out of one out of five or point to. And we know I'm going to let our stand for the number that he gets correct As my random variable. So on part a we want to know what's the probability that he gets all 10 correct? And we know that in the binomial setting that I'm gonna have 10 shoes 10 and I'm going to have point to to the 10th and I'm going to have .8 to the zero. And when we do that, we end up getting 1.24 times 10 to the negative seventh power. So it's extremely unlikely for that to happen now. What's the chance that he doesn't get any of them right? So he gets all of them wrong? And likewise, we're going to have 10-0 and we're going to have point to to the zero. It and we're going to have .8 to the 10th power And that .8 Yeah. Let me just change my question here when we do this. We he has a About a 10.7% chance of that happening now. We want to find the probability that he gets at least one right and we know that's going to be one minus the probability that he gets them all wrong. So one minus that answer is going to end us. Because this is that answer is going to give us .89-6. And then in part D we want to find the probability that he gets at least half of them right? And we can find 01234 and take one minus that. Or find 56789 10. I am going to use my Binomial CDF button And I have 10 and I have point to as my success and The events that this does not include. R zero up through four. So I want to find this. And so when I take one minus Binomial Cd app With 10 trials point to is the probability of success and four. That gives me A zero, So he doesn't have a good chance of guessing and getting half of them right.

Question number 16. It's from a given in equally then the financial binomial probability Be off X equal key equal in CK the people key the one minus b power in minus K equal factorial in over factorial key Multiply boy factorial in minus key The people k dot one minus p or in minus key. One off, one out off. Five answers is correct. The probability the number off favorable outcomes divided by the number off possible outcomes be equal 1/5 equally point to below it at K equal thin. So probability off X equal 10 equal Factorial 10 over. Factorial 10 Multiply Boyton minus 10 story In that 0.2. Parton the one minus point to Burton minus 10 equal. Almost zero question number we given an equal. Then we would use the above formula. So 11 out off five answers is correct. The probability is a number off favorable outcome divided by the number off possible outcomes The equal 1/5 equal point to value it AT T equals zero probability Off X equals zero equal factorial 10 over. Factorial zero boy didn't minus zero factorial that 0.2 point zero that one minus point to for them minus zero almost equal 0.17 four equal 10 point 10.74% Question Number. C Result. Port P Prosperity of X equal zero equal 00.1074 equals 10.4 at a 10.74 person A p equal 4.2 addition rule for multiple exclusive events. The probability off E or me equal probability off e plus probability Off be did remind the probability using this rule and table toe in the appendix. So probability off is bigger than or equal one equal probability of X plus X equals one plus the probability of X equal to and so on toe Probability of X equal 10 equals coin 268 plus 2680.302 plus 0.20 On plus 0.8 eight plus 80.26 plus 0.6 plus 0.1 plus zero plus zero plus zero equal 0.892 Equal 89.2 percent Compliment rule Peak off. Not a equal one minus B off A. Did you mind the probability you think the compliment rule? So probability Off X is bigger than or equal one equal one minus probability off X equal zero equals one minus 10.1 074 equal 0.89 to 6 equal 89.26%. Thes probability should be equal. They aren't equal, but the difference is most likely due to rounding errors in the use deep.


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