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): \(\begin{array}{cccccc}1,2 & 1,3 & 1,4 & 2,1 & 2,3 & 2,4 \\ 3,1 & 3,2 & 3,4 & 4,1 & 4,2 & 4,3\end{array}\) 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)

First, determine the sample mean for each of the 12 possible samples: (1+2)/2 = 1.5, (1+3)/2 = 2, (1+4)/2 = 2.5, (2+1)/2 = 1.5, (2+3)/2 = 2.5, (2+4)/2 = 3, (3+1)/2 = 2, (3+2)/2 = 2.5, (3+4)/2 = 3.5, (4+1)/2 = 2.5, (4+2)/2 = 3, (4+3)/2 = 3.5.

02

Construct Sampling Distribution (Without Replacement)

Next, use the sample means to construct a frequency table. Each unique mean represents a value on the x-axis of our histogram, with its frequency representing the height of that bar. Here we need to create a histogram, and to do that, we plot the values of sample means on the x-axis and their frequencies on the y-axis. For this dataset, the frequencies of 1.5, 2, 2.5, 3, 3.5 are 2, 2, 4, 2, 2, respectively. This histogram represents the sampling distribution of \(\bar{x}\) in this case.

03

Compute Sample Means (With Replacement)

Now, we are to assume sampling with replacement, which means a number is placed back into the population after being selected, thus it can be selected again. The possible samples and their means are as follows: (1+1)/2 = 1, (1+2)/2 = 1.5, (1+3)/2 = 2, (1+4)/2 = 2.5, (2+1)/2 = 1.5, (2+2)/2 = 2, (2+3)/2 = 2.5, (2+4)/2 = 3, (3+1)/2 = 2, (3+2)/2 = 2.5, (3+3)/2 = 3, (3+4)/2 = 3.5, (4+1)/2 = 2.5, (4+2)/2 = 3,(4+3)/2 = 3.5, and (4+4)/2 = 4.

04

Construct Sampling Distribution (With Replacement)

Similarly create another histogram with the new sample means. The frequencies of 1, 1.5, 2, 2.5, 3, 3.5, 4 are 1, 2, 3, 4, 3, 2, 1, respectively. This histogram represents the sampling distribution of \(\bar{x}\) in this case.

05

Compare Both Sampling Distributions

Compare the two histograms. They are both symmetrical and have the same mean, but they differ in their shape and spread. The distribution for sampling with replacement has a higher peak and narrower spread, while the distribution for sampling without replacement is flatter and wider.

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Key Concepts

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

Population Mean

The population mean is a central concept in statistics that helps us understand the average value of a dataset. It is denoted by the Greek symbol \( \mu \) and is calculated by summing up all the values in the population and then dividing by the number of values. In our example, the population \( \{1, 2, 3, 4\} \) has a population mean of \( \mu = \frac{1+2+3+4}{4} = 2.5 \). This value represents the average of all individual values in the population.

Understanding the population mean is essential because it serves as a benchmark that sample means are often compared against. When taking random samples from a population, the sample means can vary, but collectively they provide insight into how close our sample is to representing the whole population. The population mean provides a measure of the central tendency and helps guide decisions in predicting or explaining phenomena based on sample data.

Random Sampling

Random sampling is a process used in statistics to select a subset of individuals from a population. It is crucial for ensuring that the sample is representative of the population, reducing bias and thereby increasing the reliability of the results.

• This method involves selecting samples randomly, meaning each member of the population has an equal chance of being chosen.
• In our exercise, we select two numbers from the population \( \{1, 2, 3, 4\} \).
• This can be done either with or without replacement, influencing the possible samples and their characteristics.

Random sampling's major benefit is its ability to generate unbiased estimators, such as estimating the population mean. It is crucial for building accurate sampling distributions, which help infer population parameters from sample statistics. This random selection is at the heart of inferential statistics and plays a vital role in many research designs.

Histogram

A histogram is a graphical representation of data distribution, particularly useful for visualizing the spread and frequency of data points within a dataset. In statistics, it's commonly used to illustrate sampling distributions.

For our exercise, we construct a histogram to display the sampling distribution of \( \bar{x} \), the sample mean. Here's how it helps:

• Each bar represents the frequency of a particular sample mean on the x-axis.
• The y-axis indicates how often each mean appears in our samples.
• This visual serves as a quick summary for understanding the distribution’s shape, center, and spread.

By examining the histogram, one can easily observe patterns such as symmetry or skewness, and compare different sampling methods. In parts a and b of the exercise, histograms help illustrate differences in spread and peak height resulting from sampling with versus without replacement, which is key in statistical inference.

Sampling with Replacement

Sampling with replacement is a method of random sampling where each selected individual is returned to the population, allowing them to be chosen again in subsequent selections. This approach can significantly affect the characteristics of the sample and, consequently, the resulting sampling distribution.

In the given exercise, when selecting two numbers from the population \( \{1, 2, 3, 4\} \) with replacement:

• The total number of possible samples increases because any selected number can appear again in the same sample.
• This leads to 16 different possible combinations, compared to 12 without replacement.
• The sampling distribution becomes more sharply peaked and narrower due to repeated values contributing to sample means.

Sampling with replacement aligns with scenarios where it's feasible to select the same individual more than once, which is often used in bootstrapping techniques in statistics. Understanding this concept is vital for analyzing how sample data might reflect or differ from population data and plays an integral role in simulations and in simplifying complex calculations.