Births: Sampling Distribution of Sample Proportion When two bi...

Question

Births: Sampling Distribution of Sample Proportion When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg (where and Assume that those four outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from two births. Does the mean of the sample proportions equal the proportion of girls in two births? Does the result suggest that a sample proportion is an unbiased estimator of a population proportion?

Accepted Answer

Hello. In this video, we are going to suppose that there are 3 coins that are being flipped, and each coin is equally likely to land on heads or tails. We are going to let H represent a success. We want to determine what the mean of the sam, what is the mean of the sample proportions. We want to determine if it is equal to the population proportion of heads, and we want to determine if the sample proportion is unbiased and estimator of the population proportion. OK, so, let's just go ahead and quickly analyze the scenario. We have 3 coins that we are flipping. Now, there are two sides to the coins, heads or tails, and each of them are likely to land on heads. Now, the important thing about this problem is that we are letting H represent a successful coin flip. So, in order to approach the first part of this problem, let's just go ahead and list out all the probabilities or all the likely outcomes of these three coin flips. So The first outcome, best case scenario, is that all 3 coins land on heads. Or we can have the first two coins be heads, and the last one be tails. First is heads. Second is tails. Third is heads, or 1 is tails, and the last two are heads. We can also have the first two coins be tails with the last one being heads. We can have the first one be tails, second one be heads, 3rd 1 be tails, or we can have the first one be heads while the last two are tails. Or worst case scenario, we have all coins land on tails. So there are 8 likely outcomes to the 3 coin flips. Now, for the first part of the problem, we want to determine the probability of a success. So we are going to label that as Phat, and this is going to be defined as the number of heads divided by the total amount of flips. So Since there are 8 likely outcomes, the total amount of flips for each of these scenarios is going to be 8. Now, we need to determine the amount of scenarios in which we get all three heads. Now, for the scenario in which we have all heads, That is only 1 likely outcome from the total of 8 outcomes. So that is going to be 1 out of 8. Then, what is the probability that we get all tails? Well just like the heads, there's only one possible outcome for all this being tails from the eight outcomes. So we're going to label that as 1/8. Now, what is the possibility that we get two tails? And one heads. Well, in this case here, there are 3 likely likely outcomes for getting two tails with 1 head, so that is going to be 3 out of 8. And finally, what is the probability of getting two heads and 1 tails? Well, it's going to be the same as getting two tails with one head, there are only 3 outcomes from the 8 total outcomes. OK. So now what we want to do is we want to go ahead and take the sum of all the possibilities. Now, the sum of all these possibilities are going to be defined as the following. It is the sum which is equal to the probability of that outcome happening multiplied by the possibility of the total outcome. And so this is going to be the the summation that defines the sample proportion. So, that is going to be the possibility of getting all heads, and again, what's the probability of success for all heads? Well, the probability for success is 3 out of 3, so we're going to take 3 out of 3. And multiply this by 1 out of 8. Then we are going to add the probability of getting all tails. So what's the probability of success in all tails while it's 0, because we don't have any heads, and multiply this by the outcome of 1/8. Now we need to ask ourselves, what is the likelihood of success with two tails and one heads? Well, from the 3 coin flips, we only get 1 success out of the 3, and we multiply this by the likeliness of that outcome, which is 3 out of 8. And for the final sum, we take the likeliness of success from two heads, one tails, which is 2/3, and multiply this by the outcomes, which is 3/8. Now, simplifying this is going to give us 1/8 + 2/8 + 1/8, which is equal to 4/8, which is equal to 1/2. And this is going to be our sample proportion. And the answer to the first part of the problem. Now, the second part of the problem asks us to determine if the to determine the population proportion of heads. Well, what this is asking us is, what is the probability of landing on heads when flipping one coin? Well, since there are only two outcomes, the likelihood of success is 1/2, since you can either have heads or you can have tails. So the so the population proportion from heads to tails is going to be 1/2, which is the solution to the second part of the problem. And for the third part of the problem, the 3rd part of the problem is asking us if the sample proportion is unbiased estimator to the population proportion. Well, since these sample proportion and the population proportion are both the same. That means that they are unbiased. And with that being said, the solution to this problem is going to be C. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

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Key Concepts

Sampling Distribution

Sample Proportion

Unbiased Estimator

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