Question

Consider the following population: $\{1,2,3,4\} .$ Note that the population mean is $$ \mu=\frac{1+2+3+4}{4}=2.5 $$ a. Suppose that a random sample of size 2 is to be selected without replacement from this population. There are 12 possible samples (provided that the order in which observations are selected is taken into account): $$ \begin{array}{llllll} 1,2 & 1,3 & 1,4 & 2,1 & 2,3 & 2,4 \\ 3,1 & 3,2 & 3,4 & 4,1 & 4,2 & 4,3 \end{array} $$ Compute the sample mean for each of the 12 possible samples. Use this information to construct the sampling distribution of $\bar{x}$. (Display the sampling distribution as a density histogram.) b. Suppose that a random sample of size 2 is to be selected, but this time sampling will be done with replacement. Using a method similar to that of Part (a), construct the sampling distribution of $\bar{x}$. (Hint: There are 16 different possible samples in this case.) c. In what ways are the two sampling distributions of Parts (a) and (b) similar? In what ways are they different?

          Consider the following population: $\{1,2,3,4\} .$ Note that the population mean is 
$$
\mu=\frac{1+2+3+4}{4}=2.5
$$
a. Suppose that a random sample of size 2 is to be selected without replacement from this population. There are 12 possible samples (provided that the order in which observations are selected is taken into account):
$$
\begin{array}{llllll}
1,2 & 1,3 & 1,4 & 2,1 & 2,3 & 2,4 \\
3,1 & 3,2 & 3,4 & 4,1 & 4,2 & 4,3
\end{array}
$$
Compute the sample mean for each of the 12 possible samples. Use this information to construct the sampling distribution of $\bar{x}$. (Display the sampling distribution as a density histogram.)
b. Suppose that a random sample of size 2 is to be selected, but this time sampling will be done with replacement. Using a method similar to that of Part
(a), construct the sampling distribution of $\bar{x}$. (Hint:
There are 16 different possible samples in this case.)
c. In what ways are the two sampling distributions of Parts (a) and (b) similar? In what ways are they different?

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Added by Tyler B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Christopher Stanley

01/07/2022

Step 1

The sample means for the 12 possible samples are as follows: 1,2: (1+2)/2 = 1.5 1,3: (1+3)/2 = 2 1,4: (1+4)/2 = 2.5 2,1: (2+1)/2 = 1.5 2,3: (2+3)/2 = 2.5 2,4: (2+4)/2 = 3 3,1: (3+1)/2 = 2 Show more…

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Consider the following population: $\{1,2,3,4\} .$ Note that the population mean is $$ \mu=\frac{1+2+3+4}{4}=2.5 $$ a. Suppose that a random sample of size 2 is to be selected without replacement from this population. There are 12 possible samples (provided that the order in which observations are selected is taken into account): $$ \begin{array}{llllll} 1,2 & 1,3 & 1,4 & 2,1 & 2,3 & 2,4 \\ 3,1 & 3,2 & 3,4 & 4,1 & 4,2 & 4,3 \end{array} $$ Compute the sample mean for each of the 12 possible samples. Use this information to construct the sampling distribution of $\bar{x}$. (Display the sampling distribution as a density histogram.) b. Suppose that a random sample of size 2 is to be selected, but this time sampling will be done with replacement. Using a method similar to that of Part (a), construct the sampling distribution of $\bar{x}$. (Hint: There are 16 different possible samples in this case.) c. In what ways are the two sampling distributions of Parts (a) and (b) similar? In what ways are they different?

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00:01 Now here on part a, you'll notice that we have been given the 12 possible samples.

00:07 Now, we want to use this information to construct the sampling distribution of x bar.

00:13 Okay, so here on a, we have 1 -2, 1 -3, 1 -14, 2 -1, 2 -3, 2 -3, 2 -4, 3 -1, 3 -2, 3 -4, 4 -1, 4 -2, 4 -3.

00:42 Now for x bar here, if it's 1 -2, the average of 1 and 2 is 1 .5.

00:52 1 -3, we find their average, that's 2.

00:54 We're just adding them together and dividing by 2.

00:59 For 1 -4, add them together and divide by 2, that's 2 .5.

01:02 We're just going to keep repeating this process.

01:04 2 -1 is 1 .5, 2 -3 is 2 .5.

01:11 2 -4 is 3 .3 -1, their average is 2 .3 .3, their average is 2 .3 .3 .3, their average is 2 .5.

01:21 3 -4, their average is 3 .5.

01:26 4 -1, their average is 2 .5.

01:29 4 -2, their average is 3.

01:33 Then 4 -3, their average is 3 .5.

01:38 And so what we have done here is found the sampling distribution of x -bar.

01:42 We want to display this as a histogram.

01:49 The lowest value is 1 .5, and we get that 1 -2 times.

01:56 So 1 .5 occurred twice.

02:00 And as we have a value of 2, that occurs 1, 2 times as well.

02:05 And this will be the frequency and this would be the xxx.

02:12 We have 2 .5 and then occurs four times.

02:25 We have 3, which occurs twice.

02:32 And then we have 3 .5 which occurs twice.

02:37 And so here's our histogram that represents the sampling distribution of x bar...

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