91Ó°ÊÓ

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?

Step by step solution

01

Compute Sample Means (without replacement)

Note that the 12 possible samples are {[1,2],[1,3],[1,4],[2,1],[2,3],[2,4],[3,1],[3,2],[3,4],[4,1],[4,2],[4,3]}. The sample mean for each set can be calculated by adding the two numbers in the set and dividing by 2. For example, for the first sample [1,2], the mean would be \(\frac{1+2}{2}=1.5\). Repeat this procedure for each of the 12 samples.

02

Create the Sampling Distribution (without replacement)

After calculating the mean for each sample, use the results to create a sampling distribution. Note how often each sample mean occurs - this frequency forms the y-axis of the density histogram. Each of the calculated sample mean values represent values on the x-axis.

03

Compute Sample Means (with replacement)

For the case with replacement, there will be 16 possible samples {[1,1],[1,2],[1,3],[1,4],[2,2],[2,3],[2,4],[3,3],[3,4],[4,4],[2,1],[3,1],[4,1],[3,2],[4,2],[4,3]}. Again compute the sample mean for each set as before by summing the two numbers in the set and dividing by 2.

04

Create the Sampling Distribution (with replacement)

As before, create a sampling distribution using the sample means calculated in step 3. Note the frequency of each sample mean for the y-axis of the histogram.

05

Compare the Two Sampling Distributions

The final step involves comparing the two sampling distributions created in steps 2 and 4. Note the similarities and differences between them. Similarities might include a central tendency around the population mean. Differences might stem from the order of sampling and replacement, which can affect the frequency and variability of the means in the distributions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Mean

The concept of population mean is foundational in statistics. It gives you the average value of an entire dataset. Imagine you have a set of numbers, for example, the population \(\{1, 2, 3, 4\}\).
The population mean, denoted as \( \mu \), represents the central value of these numbers. To find \( \mu \), sum up all the numbers and divide by the number of elements in the population:

• Sum = 1 + 2 + 3 + 4 = 10
• Number of elements = 4
• Population mean \( \mu = \frac{10}{4} = 2.5\)

Getting the population mean helps in understanding the overall trend of the data, making it a useful tool in analyzing larger datasets.

Sampling with Replacement

Sampling with replacement means each member of the population can be chosen more than once. After selecting a member, it is "replaced" back into the population pool for potential selection again.
This type of sampling allows for more sample combinations, as seen in this case where a sample size of 2 drawn from the population \(\{1, 2, 3, 4\}\) results in 16 possible samples.

• Each sample sequence like \([1, 1], [1, 2], [1, 3], [1, 4]\)... includes combinations where numbers repeat.
• This repetition leads to a broader range of outcomes.

In real-world applications, sampling with replacement can be useful when the population is small but you need to ensure every sample outcome remains equally likely.

Sampling without Replacement

Sampling without replacement means once a member is selected, it cannot be chosen again within the same sample. This method changes the probability for the remaining selections in your sample.
Using the population \(\{1, 2, 3, 4\}\), sampling without replacement for a sample size of 2 results in 12 unique combinations.

• For instance, each selection like \([1, 2], [1, 3], [1, 4]\)... reflects unique pairings without repetition.
• No element in a sample repeats, reducing the variability compared to sampling with replacement.

This method is often used in situations where the size of the population is manageable, and it is important to ensure that every member has a chance to be part of a sample exactly once.

Sample Mean

The sample mean is the average of observations in a sample taken from a population. Calculating the sample mean helps us estimate the population mean when it's impractical to measure the entire population.
For each possible sample from a population \(\{1, 2, 3, 4\}\), whether sampled with or without replacement, the sample mean is calculated by averaging the numbers in the sample.

• For example, the sample \([1, 2]\) has a mean of \(\frac{1+2}{2} = 1.5\).
• Similarly, for \([2, 3]\), the mean is \(\frac{2+3}{2} = 2.5\).

Such computations of sample means allow us to create a sampling distribution, which can indicate how close our samples are to the population mean. They are essential for inferential statistics, forming the basis for estimations and hypothesis testing.