Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.


n = 6
x̄ = 7
s ≈ 2

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with constructing a data set of size n = 6 (6 data points) such that the mean (x̄) is 7 and the standard deviation (s) is approximately 2. Recall that the mean is the average of the data points, and the standard deviation measures the spread of the data around the mean.
2. Step 2: Use the formula for the mean to guide the construction of the data set. The formula for the mean is: $\overline{x}=\frac{\sum x}{n}$. Since x̄ = 7 and n = 6, the sum of the data points must be: $\sum x=\overline{x}\times n=7\times 6=42$. This means the data points must add up to 42.
3. Step 3: Use the formula for the standard deviation to ensure the spread of the data is approximately 2. The formula for the sample standard deviation is: $s_n=\sqrt{\frac{\sum (x-\overline{x}{)}^2}{n-1}}$. You will need to choose data points such that the squared deviations from the mean (7) sum to a value that satisfies this formula when s ≈ 2.
4. Step 4: Start constructing the data set. Choose six numbers that add up to 42 and have a spread around the mean of 7. For example, you can start with numbers slightly above and below 7, such as 5, 6, 7, 7, 8, and 9. Verify that the sum of these numbers equals 42 and adjust if necessary.
5. Step 5: Calculate the standard deviation for your chosen data set to ensure it is approximately 2. Use the formula for the standard deviation to check: compute the squared deviations from the mean, sum them, divide by n - 1 (5 in this case), and take the square root. If the standard deviation is not close to 2, adjust the data points slightly and repeat the process until the desired standard deviation is achieved.

Key Concepts

Sample Size (n)

The sample size, denoted as 'n', refers to the number of observations or data points in a data set. In this case, n = 6 indicates that the constructed data set must contain six individual values. The sample size is crucial for statistical analysis as it affects the reliability and validity of the results derived from the data.

Mean (x̄)

The mean, represented as 'x̄', is the average of a data set, calculated by summing all the values and dividing by the number of observations. For this question, x̄ = 7 means that the total of the six data points must equal 42 (7 multiplied by 6). The mean provides a central value around which the data points are distributed.

Standard Deviation (s)

Standard deviation, denoted as 's', measures the amount of variation or dispersion in a set of values. An approximate standard deviation of 2 indicates that most data points will fall within 2 units of the mean. Understanding standard deviation is essential for assessing the spread of the data and determining how closely the data points cluster around the mean.