In Exercises 11–14, use the population of {2, 3, 5, 9} of the ...

Question

In Exercises 11–14, use the population of {2, 3, 5, 9} of the lengths of hospital stay (days) of mothers who gave birth, found from Data Set 6 “Births” in Appendix B. Assume that random samples of size n = 2 are selected with replacement.

Sampling Distribution of the Sample Mean

a. After identifying the 16 different possible samples, find the mean of each sample, and then construct a table representing the sampling distribution of the sample mean. In the table, combine values of the sample mean that are the same. (Hint: See Table 6-3 in Example 2.)

Accepted Answer

Welcome back, everyone. Consider the population 1468, representing the number of days patients stayed in a rehabilitation center. Suppose random samples of size N equal to are selected with replacement. After listing all possible samples, calculate the mean of each sample and construct a table showing the sampling distribution of the sample mean. Combined sample means that are the same. Let us tackle the first part of this problem. We want to list all possible samples and then calculate the mean of each sample to help us create the sampling distribution. Now if we think about it here, we're given the population and we're told that we're drawing random samples of size N equal to with replacement. That means then that each of the two positions in the sample can be any of the four values, so there are 4 multiplied by 4 or 16 possible samples. So if we were to write out what all of those samples were, for example, OK. Then we know that we could have 11, OK. One could be combined with all of the numbers in the population. So that's 1114, 1618, OK. And then we're basically just repeating that for the rest of values. So then we could have 4144, 46 and 48, OK. And then let me continue this table down here, OK. Then we could have starting with 6, so that's 61, OK, 64. 66 and then 68 and you'll see why I put the table like this in a minute. And then with 8, we could have 81. 84. 86 And ate it. So these are all of our possible samples, OK? Next, we're supposed to calculate the mean of each sample so we could find the sample mean. OK. Let me make sure I say sample mean. Uh, that would be X bar, OK? So, no, that just means we're gonna calculate the sample mean for each of these samples. So, for example, the mean of 1 and 1 would be 1 because 1 + 1 divided by 2 equals 1. The meaning of 1 and 4 is 2.5, of 1 and 6 is 3.5, of 1 and 8 is 4.5, 4 and 1 would be 2.5, like we had here before. 4 and 4 is 44 and 654 and 8 would be 6. if we continue with our sample means down here. OK. Then 6 and 1 would be 3.5. 6 and 4 would be 56 and 6 would be 6, and 6 and 8 would be 7. And then 8 and 1 would be 4.5, 8 and 4 it would be 68 and 6, it would be 7, and 8 and 8, it would be 8. So these are all of the samples and their sample means, OK? No, remember that the next part of our problem tells us that we are going to construct a table showing the sampling distribution of the sample mean. Now if you look at our table you will notice here that some sample means are repeated, for example. The sample mean 2.5 is the mean for the sample 1 and 4 and 4 and 1. So we can use this to create a table here. So let's do a table, right? Let's have another table where we're going to have the sample mean. Expire, OK. We're gonna have the frequencies of the sample means. And then for that. We're going to find the probabilities. Yeah So let's write down a list in our table of all the unique sample means that we had found. So we know that for a sample means we had 1, 2.5, 3.5, or 567, and 8. OK? And if we go back to our list of values that we had, we know only one sample had a sample mean 12 had a sample mean of 2.5, 2 also had a sample mean of uh 3.5. 4 is 1, 4.5, oops, I forgot, 4.5 actually, so let's just make some space for that one in our table. OK. So 4.5 had 25 had 26 had 3. 7 had 2 and 8 had 1. So these are all of the frequencies for our sample means. Now we need to find their probabilities. Note that the total frequency is equal to 16. There are 16 samples. So to calculate each probability, we can divide the frequency by the total frequency. So for example, the probability the sample mean is 1 would be 1 divided by 16. The probability is 2.5 would be 2 divided by 16 or 1/8. The probability it's 3.5 would also be 1/8. The probability is 4 would be 1/16. 4.5 is 1/8, 5 is 1/8, 6 would be 3/16th, OK? 7 would also be 1/8 and then 8 would be 1. Thus this would be the table of our sample mean and its probabilities. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Sampling Distribution

Sample Mean

Combining Values in a Distribution Table

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