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According to national data, 10% of the population of U.S.adult females is left-handed: Suppose that a random sample of size n = 100 U.S. adult females is chosen; Us...

Question

According to national data, 10% of the population of U.S.adult females is left-handed: Suppose that a random sample of size n = 100 U.S. adult females is chosen; Use the Standard Deviation Rule and the properties of the sampling distribution of p-hat. There is a 95% chance that; in any random sample of 100 U.S adult females, the proportion of left-handed females will be between:

According to national data, 10% of the population of U.S.adult females is left-handed: Suppose that a random sample of size n = 100 U.S. adult females is chosen; Use the Standard Deviation Rule and the properties of the sampling distribution of p-hat. There is a 95% chance that; in any random sample of 100 U.S adult females, the proportion of left-handed females will be between:



Answers

In Exercises 69 to $72,$ explain whether the given random variable has a binomial distribution.

Lefties Exactly 10$\%$ of the students in a school are left-handed. Select 15 students at random from the school and define $W=$ the number who are left handed.

Breath problem describes a situation where exactly 10% of the students, the school or left handed and then we select 15 and we define W as the total number of left hand students, and we're supposed to see if it's a binomial distribution or not. So we have to check some properties. The first is that there are in identical trials. This is the case because we're going thio, Um, select these students one by one and check if they're left or right handed and just could be the same process every time to make sure I'm doing these in order of only two possible outcomes. So and this is the case, because every time we do a trial, either the person is left or right handed only two possible outcomes and three you tether. The trials are independent, and we also need to have that the probability of success. You ready? Little difference. But the probability of success is always equal to P, and the probability of failure is always equal to one minus p. And here's where it might be. We gotta be careful. If we If I were to say the school has, like me, think What's your example? Let's say we've said this school has 300 and 50 students, the story of 300 50 students. So then 35 would be lefties? Let's say I pick the first person I pick is a left handed student, so well, the probability the first person being left handed would be, um, 35 out of 3 50 or a 0.1, as it says in the in the direction. But okay, then once you pick this person, then I picked the next person will. So here's one lefty and then I pick another person. Well, now that sample space of students has gone down, told understand, has gone down to choose from. And the, uh, number of lefties has also gone down. And you could imagine this continuing if I want to get, like, probability of getting all left. He's out of my 15 students. Well, these numbers would continue to decrease each time, and that means we're not the same. Like a quick scan on my calculator. 34 divided by 3 49 This guy is equal to 0.97 Where is this guy is point. Once we see that the probability is not the same as we keep going and appoint one that I get one lefty. But then I get Sagan another lefty, so I get to two lefties. Well, then the probability has changed because the first lefty reduced the number of left handed people introduced the number of students. Anyways, this would make it so This is what we would call we're doing with out replacement, because when we do things without replacement and affects the probability, but there's a rule that it is okay to treat, we're allowed to treat without replacement as equivalent to with replacement. If this sample is, um, the sample size is at most 10% or 100.1 of the population. And if we're doing 15 trials out of, ah, 15 trials and we're doing it out of the whole school, in most cases, that would be 15 would definitely be less than 10% of overall school with the, with the couple rare exceptions for really small rural schools. So because it satisfies this 10% rule, which we can't, we're not gonna bother proving here, but you can. There's an argument about why we're allowed to use the 10% rule because they ensures that were within, like some certain decimal of accuracy. It forgets if they know if someone had the exact decimal. Let's see, it means that the variance, it means that this we with that we'll have, like 0.9 accuracy to, ah, hyper geometric. And that's why we're allowed to use this rule. I'm not gonna go further into it right here, but because this rule is satisfied, we're allowed to treat it as if they are independent and as if the probabilities on changing so that we have those two conditions satisfied. And then the last thing is just that we have x equal. A total number of successes. Oh, not ex were using W my bed w equal told number successes. And that is what we're doing. So all the conditions are satisfied because of the special, specifically because of this rule and so we can treat it as a binomial. So w is I know well

So this problem We're comparing lefthanders out of total number of people. We know that there are three left him people out of every 30 and so they don't know how many left handed people would be on a 140. Let's cross multiply. X Times 30 is 30 x 1 40 times three is 4 20 to write by 30 an ex would equal 14.

The following is a solution for # 14. This looks at if there's a difference or if men are more often left handed than women. So they sampled 100 and 68 men and 24 of them were left handed and then they sampled 152 women and they found that nine were left handed. So first let's find the point estimate. The point estimate for P one minus P two is simply P one hat minus P two hats. So we just look at the statistics here. So P one hat, let's just say that this is for males is 20 since that's listed 1st 24/1 68 and you get a long decimal, I'm just going around here but 680.143 and then P two hat is for females. So that's 9/1 52. And whenever you plug that in it should be a point oh 59. So the point estimate would be 590.0.143 minus 0.59. Now on this paper I'm rounding but In my calculator actually did. And so I'm actually gonna go to four decimal places here. It's .08 36. So that's my point estimate for the difference of two population proportions. So now I'm going to find the 95% confidence interval if you go to stat and then air over two tests we're going to go to to prop Z. Ent for interval to proportions the interval. So here you can see my data, 24 out of 1 68 for males nine out of 152 for females The confidence levels 95%. And then we calculate so .01878 and .14851. Okay so let's go in Drop that down. So it's between zero 01878. All the way up. 2.14 851. Okay next up we do a hypothesis test the null hypothesis being. There is no difference between men and women and left handedness. And the alternative it says is there a uh is it that the men are there are more men? The proportion of men is higher than women's. That's why P one minus two P two is greater than zero. We're testing at the five percent level of significance. So really all we need is a test statistic and then I'm gonna go ahead and find the people. I'm going to go to the p value out just because that's what this um part DCs so kind of killing two birds with one stone. Especially because I can use a calculator here. So if I go to Staten tests and I go to the to prop Z. Test here you can see that the data is already pre populated and I just need to change this alternative hypothesis. So P one is greater than P two or P one minus P two is greater than zero. And we calculate and my test statistic is 2.457. And that P values quite small so double oh seven there Not the James Bond kind, so z value is 24 57 and then 0.7 which is less than alpha. Now, whenever the P values less than alpha, we're going to reject H. Knock so will reject the null hypothesis and no hypothesis says that there is no difference. So we're accepting that the alternative is true. So it does appear that men are um left handed more often than women and then part D. Says find the P value and I just found it there so 0.0.

Here in this problem, late is equal to left handed question so I can write. Evaluate. P e is given a 0.13 in the question. Now the probability that both are left handed can be given by P first left handed, first left hand and second left handed, so it can be further return it. We want lab handed multiplication P two left handed. So just putting the value here so I can write 0.13 multiplication 0.13 and on solving it, I get the answer edge 0.169 This is the required probability. Now the probability that at least one is right handed can be calculated by P right is equal to one minus the probability of both are left handed So I can write p one p two. So just putting the value here so going forward and putting it here I can write one minus 0.13 multiplication 0.13 and on solving I get the answer at 0.9831


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