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Three randomly selected households are surveyed_ The numbers of people in the households are 2 7, and 12. Assume that samples of size n = 2 are randomly selected wi...

Question

Three randomly selected households are surveyed_ The numbers of people in the households are 2 7, and 12. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 7 , and 12. Construct probability distribution table that describes the sampling distribution of the proportion of even numbers when samples of sizes n = 2 are randomly selected_ Does the mean of the sample proportions equal the proportion of even numbers in the population? Do the sampl proport

Three randomly selected households are surveyed_ The numbers of people in the households are 2 7, and 12. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 7 , and 12. Construct probability distribution table that describes the sampling distribution of the proportion of even numbers when samples of sizes n = 2 are randomly selected_ Does the mean of the sample proportions equal the proportion of even numbers in the population? Do the sampl proportions target the value of the population proportion? Does the sample proportion make good estimator of the population proportion? Listed below are the nine possible samples_ 2,2 2,7 2,12 7,2 7,7 7,12 12,2 12,7 12,12 Construct the probability distribution table _ Sample Probability Proportlon (Type an integer or fraction )



Answers

Sampling Distribution of Sample Proportions For a random sample of size $n,$ the sample proportion is the number of individuals in the sample with a specified characteristic divided by the sample size. The sampling distribution of sample proportions is the distribution formed when sample proportions of size $n$ are repeatedly taken from a population where the probability of an individual with a specified characteristic is $p .$ The sampling distribution of sample proportions has a mean equal to the population proportion $p$ and a standard deviation equal to $\sqrt{p q / n}$. Assume the sampling distribution of sample proportions is a normal distribution. Construction About $63 \%$ of the residents in a town are in favor of building a new high school. One hundred five residents are randomly selected. What is the probability that the sample proportion in favor of building a new school is less than $55 \% ?$ Interpret your results.

And question number 42. The central proportion is normally distributed with the mean P and start The division is square root off peak you over in, does it? The score is the value decreased by the mean and divided by the standard divisions. Or we have that status equal to pick up minus peak over square Toto pick you boy and which is equal to 4.8 E minus 4.74 over square root, hopefully in 74 by one minus whole 0.74 bye 110 which is equal to 1.43 40 street. So the two, determined, of course, wanted probability using the normal probability table in the appendix that that the p equal toe X hurry to than 80 person, which is equal to be off that greater than 1.43 equal toe one minus B off that greater less than one point for history, which was 3 April two one minus 2.9 thousands, and 236 which is equal to 7.64

So this question and the population is the population is 459 and then the sample size is too because to pick two room samples of pigs without replacements decided to and then it from the proportion off. The, um the question say, is that to a page randomly and the first question wants us to find the proportion of all numbers. So the progression of all numbers, yeah, is going to be from the origin of odd numbers from 45 And I is going to be, um, so which is five and nine, because for is an even number. Yeah. And the next question the next question wants us to describe, um to construct a similar table. The table that describes the sampling distribution of the sample mean and, um represents the sampling, the distribution of the sample profession of all numbers. So, um, to start, we were gonna have to put the samples right here. So the table, we're going to start with the samples and we know that the samples for 52 a paper and on me we fall color for on the four call of five for common nine five comma four five Come on. Five five Kalmunai nine. Common four. Nine Come of five, Michael, my nine and then the other side is going to be the progression of all numbers. Proportion off all numbers. So informed for the zero old number Your own numbers in this is one in this. This one in this, this one in five or five days. So five a night. This too. Nine or four. There's 19 or five. There's two. Um, I'm sorry. In nine or five, there's two numbers in 19 nineties, two numbers and then the probability off each one, I will tell each one is going to be one about night because they're night possible. Some samples and that's what's gonna be all over the table. And then the question also wants us to combine the values off sample proportions that the same and to find that we just do proportions. Um, sorry about that. Just do proportions off all numbers, compressions of all numbers and then the probability. And then we have it all. You have serial old number one outnumber or through our numbers. The probability of zero numbers, if you can see, is one of a night they probably just one or number is 4/9 in poverty. Have to. Our numbers are for one night, so that's the answer. The next question. The next question wants us to find the mean of the sampling distribution off the sample proportions off good number the mean the mean of the sample proportions. The sample distribution of the sample question of our number. Already total number off proportions of old Number, which is 0111 suits who wanted to divided by the number of samples, which is nine. So adding all these numbers off gives 12 and the number of samples nine night, and this is a call to 1.333 So I just not pull 1.3 1.333 That's the answer. Been the last question this question asked, Um, is the sample profession and unbiased estimator off the pop population this progression? Why, or why not? P. S O is no. The sample proportion is not unbiased estimator off the population population because the mean off the proportion, which is 1.3 the mean of the proportion community published in 1.3 is not equal. So Yes. So the meat of the progression is not equal to the population progression which is equal to the population progression is gonna see it too. And that's the answer.

So this question stays that two birds chosen a random on and choosing randomly on between boys and girls. So it has to be either a boy or girl. And we're supposed Thio constructed savior that describes the sampling distribution off the sample proportions off girls from two birds. So our goal is to have two birds and two girls, basically. So we're gonna start by drawing the table well, and, oh, before I draw the table just for just for clarification, I'm gonna be writing on I'll go because we the probability of having to birds and ah, boy or girl is either two boys on We're gonna let BB boys and G B girls. So it's either be two boys, one girl, one boy, one girl, one girl, one boy or two girls, and that way, no are supposed to do so we draw the table and the first side is gonna be proportion off girls. In example, the other side is gonna be for a meaty and then the proportion of zero girls is on one of four because the hotel, having zero girls because we have four options, writes a number of probabilities off the possible probabilities off four on board. They'll probably still having on zero girl is one because we have BB. So we have one over fall the the put the proportion off one girl, As you can see your these two, it occurs twice. So that's gonna be tour for or one of us to, um, the proportion of two girls is one Yorkers once, and that's one of our four. So this is basically table that shows the sampling distribution of the sample proportions off girls from two birds. And then the second question, um, wants us to find out if the mean second question was, find out if the mean of the sample proportions are equal to the proportions off girls into birds. So to find out, we have to know because we just found out, um, we just found out the table. So we have to find out the meaning of the sample proportions. And we also have to find out the proportions of girls from two girls forced to know it is going to be the same. So we can go to the question so on e. So we're going to start with propositions or girls into birds proportions. And then this is gonna be a proportion of girls in two birds is gonna be the number of girls into birds divided by the total number divided by the total number. And then this is equal to one about two or 0.5, and then we're gonna find the meat off the sample proportion. And this is equal to the some off some of fourth sample proportions invited by the number of samples. And this is gonna be, um, equal to remember we had from the table. We had one before we had we had 11 probability before we had before we had one before. So to find out, to find out their own the main off the sample proportions are basically going tohave to figure out how many girls way each of the equation. So in BB, we had zero girls in BG, we have have 10.5 girls in G B. We have 0.5 girls and g, we have two girls like one girl. So we're gonna have zero those zero time to buy Does your plane for small what in my school And if you find if you calculate guy that's gonna b e 0.5. So we can tell that the meat of the sample professions and the professions off girls into birds. Ah, bull equal to 0.5. So, um, the question says, does the mean equal the proportion? So yes, yes, your equal. And then the next question say is, does the results. So just sample proportion is an unbiased estimator off the population profession. And the answer is gonna be yes. This is because the sample proportion off girls and sue birds is an estimate on the unbiased estimator off population proportions because the proportion of girls and sue birds equal to me. So we're going to say yes because the proportions Jim, he called soup the mean the sample, and that's the answer.

Deals with um estimators figuring out whether they're unbiased, unbiased and um normal distributions. So the problem gives us that given a population of numbers of population data set of 4, 5 and nine. So let me just highlight this four or 5 and nine highlight actually a neutral colour. Um What proportion of odd numbers are there out of this population? Well you know there are two Odd numbers you see five and 9 And there are three numbers. So the answer would be two divided by three and it's 66%. So 66% of the population is odd. Now it's asking us to find like the the estimator for all the defined this sort of proportion of odd numbers for all the um N equals two samples and find the mean of those. So if I were to take a N equals two sample from this sort of uh three element data set, I can either pick 444549 and so on. And so all these this this this box right here is all the combinations that I can pick of N equals two from a set of three. Now let me find the proportion of odd numbers within this sample. So 4, 4. How many odd numbers are there? Well there's zero odd numbers. Therefore the proportion of odd numbers in this first sample is zero 45. How many odd numbers are here? Five right here is an odd number. Therefore the proportion of odd numbers for this particular sample is .5. And I keep repeating this. Right. So for example for the 55 there's two odd numbers and therefore the proportion is one because there's two odd numbers and there's two numbers in total inside the inside the sample. I keep doing this for the D9 samples and this is what I get right now. I want to create a probability distribution for this. This um proportion dataset, that's what I do here. So there are zero is zero value, there's a 00.5 value and there's a one value. How many zeros are in this nine dataset? There's only one, so one divided by nine is equal to that 1.111 .5. There are 1,234, 4 values of .5. And these four values translate to four divided by nine because there's nine elements here and repeat that for the number one, there are four number ones. And so I get 40.444. Now, given this probability distribution, I could even sort of like graphic if I would like but I would have to find a means of this probability distribution. So to do that I'm just multiplying the value, the category times the probability and you get this value and I sum this entire thing and I get .666 Notice how this .666 is the exact same as the first number. So this number is a reminder is the proportion of odd numbers within the population. And this number is here is the mean of the proportion of odd numbers of all the N equals two samples possible. I know this is a lot of lot of steps to go through, but it comes to a point where because these two are, these two numbers are the same. The proportions are proportions, estimators of population are a unbiased estimator of the population, because these two values are the same.


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