In Exercises 51 and 52, a population and sample size are given...

Question

In Exercises 51 and 52, a population and sample size are given. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population.

The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Four students scored 685, and 9 on a quiz determine the mean and standard deviation of the sampling distribution of the sample means for all possible samples of size 2. Awesome. So it appears for this particular prompt. Now we're ultimately trying to determine the mean and standard deviation of the sampling distribution of the sample means for all possible samples of size 2. So it sounds like we're ultimately trying to solve for two separate answers, the mean and standard deviation. So with that in mind, Our first step that we need to take is we need to note that all possible samples of size 2 with replacement and their means is our target that we're trying to solve for. And since the sampling is with replacement and there are 4 data points, that means that the number of samples will be. 4 x 4, which is equal to 16. So now, with that in mind, we now need to list all the possible samples, and we need to note and recall that order matters with replacement. So with that in mind, we can create a table to help us organize our data. So when we create a table to help us organize our data, we will achieve the following. So we have our far left column which has the sample going from 1 to 16, and then our middle column we have our values, and then on our right column, our far right column, we have our mean values. So we need to note and recall that the mean and standard deviation of the sampling distribution of sample means. So, that means that our first major step for solving for this problem is we need to determine the sampling distribution mean. Because that is one of our final answers we're ultimately trying to solve for, so we're trying to solve for mu subscript X bar is equal to some value, and that is our one of our final answers we're ultimately trying to solve for. So, we need to add all of our sample means and divide by 16. So, Once again, it's important to note That we have our mean value for each of our samples, and we're just basically using the table to keep our data organized and to make sure we don't miss anything. So, moving right along here, so as we should recall, in order to determine the mean, we need to add all of our samples together and then divide by the number of samples. So for this particular problem, we need to take 6 + 7 plus 5.5. Plus 7.5 + 7 + 8 + 6.5 plus 8.5. Fantastic. And then we need to add 5.5 plus 6.5. So when you take 5.5. Plus 6.5 + 5, plus 7, plus 7.5 plus 8.5, and then we need 2. Then we need to add. So after 8.5, we need to add 7 plus 9. Awesome. So when we add all of these values together, we will find that it's going to be simply equal to 112. So that would mean that new X bar is equal to 112 divided by 16. These are 16 data points, which will which will give us an answer of 7.0. And that is one of our final answers. Hooray, we did it. So Moving right along here. So that was one of our, that was our one of our first major steps for solving this particular problem. So now, our second step is we need to determine the sampling, distribution, standard deviation or the standard error, and in order to do that, we need to recall and use the following equation that states that sigma subscript X bar is equal to the square root of. Sigma multiplied or multiplied by sigma of parenthesis X bar. I, so X bar subscript I. Minus mu x bar to the power 2 all divided by N. Fantastic. So moving right along here, we need to compute the squared deviations, and when we do that, we could compile our second table. So to make sure we don't get confused since we have to use a separate table, I will draw a line to separate it. But when we compute the squared deviations, we will be able to compile and create our second table, which our second table consists of the following data. So, as you could see, looking at our tables, which I've gone ahead and like plugged all of our data points into an Excel spreadsheet and let the Excel spreadsheet do the math for me. But when you do the total for the squared deviation, you will find that you will get a value of 20. And that will mean that sigma X bar is equal to the square root of 20 divided by 16, which will mean when we plug that into our calculator and we round to two decimal places, we will get an answer of 1.12, and that is our second and final answer. Hooray, we did it. So going back to look at the problem itself to recap what we've learned, that will mean that our final two answers, without a doubt, will have to be. That are mean, Mu X bar is equal to 7.00 and that our standard deviation sigma X bar is equal to 1.12. And that's it. Those are our final two answers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Sampling with Replacement

Mean of a Sample

Sampling Distribution of Sample Means

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