Problem 8 - Basic Statistics for Business and Economics with Student

Question

A population consists of the following five values: 0,0 , 1,3,6 a. List all samples of size \(3,\) and compute the mean of each sample. b. Compute the mean of the distribution of sample means and the population mean. Compare the two values. c. Compare the dispersion in the population with that of the sample means.

Step-by-Step Solution

Answer

Samples have less dispersion; sample means and population mean are different.

Step 1: Identify All Samples of Size 3

A sample of size 3 means that we must choose any 3 numbers from the given population of \(0, 0, 1, 3, 6\). The total number of samples here can be determined by combinations, \( \binom{n}{k} = \binom{5}{3} = 10\). We list out each possible sample: (0, 0, 1), (0, 0, 3), (0, 0, 6), (0, 1, 3), (0, 1, 6), (0, 3, 6), (1, 3, 6), (0, 1, 0), (0, 3, 0), and (0, 6, 0).

Step 2: Calculate the Mean of Each Sample

For each sample, add the values and divide the sum by 3 to obtain the sample mean. For example, the mean of (0, 0, 1) is \( \frac{0+0+1}{3} = \frac{1}{3} \). Calculate similarly for other samples: - (0, 0, 1): mean = \( \frac{1}{3} \) - (0, 0, 3): mean = 1 - (0, 0, 6): mean = 2 - (0, 1, 3): mean = \( \frac{4}{3} \) - (0, 1, 6): mean = \( \frac{7}{3} \) - (0, 3, 6): mean = 3 - (1, 3, 6): mean = 3.33 - (0, 1, 0): mean = \( \frac{1}{3} \) - (0, 3, 0): mean = 1 - (0, 6, 0): mean = 2.

Step 3: Compute the Mean of the Distribution of Sample Means

Add all the sample means together and divide by 10 to get the overall mean of the sample means: \( \frac{\frac{1}{3} + 1 + 2 + \frac{4}{3} + \frac{7}{3} + 3 + 3.33 + \frac{1}{3} + 1 + 2}{10} \) which equals \( 1.5 \).

Step 4: Compute the Population Mean

To find the population mean, sum all values of the population and divide by the number of values: \( \frac{0 + 0 + 1 + 3 + 6}{5} = 2 \).

Step 5: Compare the Means

The mean of the sample means \(1.5\) and the population mean \(2\) are different, highlighting a difference between the samples dispersed over smaller and larger values.

Step 6: Compare Dispersion of Population and Sample Means

The variance or the spread in the dataset can be compared using variance formulas, but qualitatively, the smaller range of values in sample means compared to the full range of population suggests sample means have less dispersion than the population values.

Key Concepts

Population Mean

The population mean is a central concept in statistics, representing the average of all data points in a given population. To calculate the population mean, sum up all the individual values within the population and divide that sum by the total number of values. For example, using the population values provided: 0, 0, 1, 3, and 6, we compute the mean by adding these numbers \[\frac{0 + 0 + 1 + 3 + 6}{5} = 2\]This figure, 2, is the population mean. It serves as a benchmark for assessing how typical or atypical specific data points are, and it helps in understanding the overall distribution of data by providing a clear, simplified view of what is considered *average* in the population.

Sample Size

Sample size is an important factor when studying statistics as it determines the number of data points taken from the population. A larger sample size can better represent the population because it captures more of its variability. In our exercise, the sample size is 3, meaning we take sets of three from our population values of 0, 0, 1, 3, and 6. From these five values, many possible samples can be formed. The number of samples can be calculated using combinations, as demonstrated: \[ \binom{5}{3} = 10 \]This calculation tells us there are 10 unique samples of size 3. Each sample is analyzed to draw conclusions that approximate characteristics of the larger population.

Dispersion

Dispersion refers to how spread out the values in a data set are. It's a measure of variability and helps determine how much variation exists from the average (mean). Two key indicators of dispersion are range and variance.
In comparing the dispersion of the population and the sample means: - The range of the population (from 0 to 6) indicates broader dispersion. - The range of sample means implies less dispersion, notably because the averages of combinations of numbers tend to cluster near the population mean.
This reduced spread of sample means suggests they tend to minimize extremes and underscores one reason why averaging is a useful statistical tool.

Combinations in Statistics

Combinations in statistics involve selecting items from a group, where the order does not matter, which is key in calculating the possible samples. Given the population values in our exercise, the task was to choose samples of a specific size.
To determine how many ways we can choose 3 items from 5, we use the formula for combinations:\[\binom{5}{3} = \frac{5!}{3!(5-3)!}\]This equation simplifies to 10, showing that there are ten different possible groups or samples of size 3. Understanding these combinations is crucial for constructing valid samples, making statistical assessments, and ensuring that selected samples represent the population comprehensively.