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Generating a Sampling Distribution. Let's illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the scores of 10 students on an exam: The parameter of interest is the mean score \(\mu\) in this population. The sample is an SRS of size \(n=4\) drawn from the population. The Simple Random Sample applet can be used to select simple random samples of four numbers between 1 and 10 , comesponding to the students. (a) Make a histogram of these 10 scores. (b) Find the mean of the 10 scores in the population. This is the population mean \(\mu-\) (c) Use the Simple Random Sample applet to draw an SRS of size 4 from this population. What are the four scores in your sample? What is their mean \(x^{-} \bar{x}\) ? This statistic is an estimate of \(\mu\). (If you prefer not to use applets, use Table \(B\) beginning at line 121 to chose an SRS of size 4 from this population.) (d) Repeat this process nine more times, using the applet (or Table B, continuing on line 121 , if you are not using applets). Make a histogram of the 10 values of \(x^{-} \bar{x}\). You are constructing the sampling distribution of \(x^{-} \bar{x}\). Is the center of your histogram close to \(\mu\) ? How does the shape of this histogram compare with the histogram you made in part (a)?

Step by step solution

01

List and Analyze the Population Scores

Assume the 10 students' scores are as follows: 65, 70, 75, 80, 85, 90, 95, 100, 105, 110. First, we need to look at these scores and understand their distribution. Creating a histogram will help visualize this distribution. Manually, we plot these scores against their frequencies.

02

Calculate the Population Mean

To find the population mean \( \mu \), sum all of the scores and divide by the number of scores. \( \mu = \frac{65 + 70 + 75 + 80 + 85 + 90 + 95 + 100 + 105 + 110}{10} = 87.5 \). This mean is the true average score of the population.

03

Draw a Simple Random Sample (SRS) of Size 4

Using a random number generator or a table of random numbers, select 4 scores from the list of 10 scores to form an SRS. For instance, suppose the selected scores are 70, 85, 90, and 100. Calculate the sample mean \( \bar{x} \) for this sample: \( \bar{x} = \frac{70 + 85 + 90 + 100}{4} = 86.25 \). This sample mean estimates the population mean \( \mu \).

04

Repeat Sampling

Repeat Step 3 nine more times to create a total of 10 sample means. This might involve selecting different samples like (75, 80, 85, 110) or (65, 70, 95, 100) etc., ensuring variety in selection. Calculate \( \bar{x} \) each time.

05

Construct the Sampling Distribution Histogram

Create a histogram using the 10 sample means obtained from Step 4. This histogram represents the sampling distribution of the sample mean \( \bar{x} \). Compare its center to the population mean \( \mu \), which should be close if the samples are properly random.

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Key Concepts

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

Simple Random Sample (SRS)

Understanding the Simple Random Sample (SRS) is crucial for sampling distribution, as it ensures that every individual in a population has an equal chance of being selected in a sample. This removes biases and makes statistical analysis more reliable.
Imagine a class of 10 students, each with distinct scores on an exam. To draw a random sample of 4 students, you could use the Simple Random Sample applet or a random number table. Each student's score has an equal chance of being chosen, ensuring fair representation.
Some important points about SRS:

• No repetition - once a score is chosen, it cannot be picked again in the same sample.
• Every combination of scores has the same probability of selection.
• It's foundational in generating valid sampling distributions.

Using an SRS is like drawing names from a hat, where each name has the same odds of being picked. It's a fair, unbiased method that aligns closely with true randomness.

Population Mean

The population mean, denoted as \( \mu \), is the average of all data points in a population. Calculating it gives us a central value that represents the entire group.
In the context of our example, with 10 students' exam scores: 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, the population mean involves summing all scores and dividing by the total number of scores.
This can be calculated as follows:
\[ \mu = \frac{65 + 70 + 75 + 80 + 85 + 90 + 95 + 100 + 105 + 110}{10} = 87.5 \]
This mean (87.5) is and remains constant because it encompasses the entire population. It's crucial because other statistics, like sample mean and variance, are often measures of how they compare to the population mean.

Sample Mean

The sample mean, represented by \( \bar{x} \), is an estimate of the population mean derived from sample data. Its calculation is similar to that of the population mean, but only uses scores from a subset of the population.
To calculate the sample mean, sum the scores from the chosen sample and divide by the number of scores in that sample. For example, if your sample scores are 70, 85, 90, and 100, the sample mean is calculated as:
\[ \bar{x} = \frac{70 + 85 + 90 + 100}{4} = 86.25 \]
Remember, the sample mean can vary because each sample is different. This variability is what makes multiple samples and their average (like in a sampling distribution) important because collectively they tend to hover around the population mean \( \mu \).

Histogram

A histogram is a graphical representation that organizes a group of data points into specified ranges. It's one of the key tools in identifying the underlying frequency distribution of a dataset.
In both the original problem and solution, histograms are used twice: first, to display the distribution of the population scores, and second, to illustrate the sampling distribution of the sample means.
Creating a histogram involves:

• Grouping data into bins or intervals.
• Counting how many data points fall into each bin.
• Drawing bars for each bin, where the height represents the frequency.

When you make a histogram of the 10 student scores, you'll see how scores are spread across low to high ranges. Later, when you create a histogram of the sample means (estimates of the population mean), you'll observe how these means distribute around the true population mean, \( \mu \). A well-constructed histogram is a powerful way to visualize and interpret data.