Question

construct a data set that has the given statistics.

n = 6
μ = 0
σ = 3

b. the mean of the population data.
c. the number of values in the population data set.
d. the range of the population data.

this means there should be 6 values in the data set.

what does the value μ mean?

a. the range of the population data set.
b. the mean of the population data set.
c. the most common value in the population data set.
d. the difference between all the values in the population data set.

this means the population mean of the 6 values in the data set should be 0.

what does the value σ mean?

a. the number of values that equal the population mean.
b. the range of the population data set.
c. the most common value in the population data.
d. the spread of the data from the population mean.

this means the population standard deviation of the 6 values with a population mean of 0 should be

Explanation:

Step1: Recall the formula for population standard deviation

The formula for population standard deviation is \(\sigma=\sqrt{\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}}\). Given \(N = 6\), \(\mu=0\) and \(\sigma = 3\). Substituting into the formula, we have \(3=\sqrt{\frac{\sum_{i=1}^{6}(x_{i}-0)^{2}}{6}}\).

Step2: Solve for \(\sum_{i = 1}^{6}x_{i}^{2}\)

Squaring both sides of the equation \(3=\sqrt{\frac{\sum_{i=1}^{6}x_{i}^{2}}{6}}\), we get \(9=\frac{\sum_{i=1}^{6}x_{i}^{2}}{6}\). Then \(\sum_{i=1}^{6}x_{i}^{2}=9\times6 = 54\).

Step3: Construct the data - set

Let the data - set be \(x_1,x_2,x_3,x_4,x_5,x_6\). We can try some simple values. For example, consider the data - set \(\{-3,-3,0,0,3,3\}\)

• Calculate the mean: \(\mu=\frac{-3-3 + 0+0 + 3+3}{6}=0\)
• Calculate \(\sum_{i = 1}^{6}(x_{i}-\mu)^{2}=(-3 - 0)^{2}+(-3 - 0)^{2}+(0 - 0)^{2}+(0 - 0)^{2}+(3 - 0)^{2}+(3 - 0)^{2}=9 + 9+0+0 + 9+9=54\)
• Calculate the standard deviation: \(\sigma=\sqrt{\frac{54}{6}}=\sqrt{9}=3\)

Answer:

The data - set \(\{-3,-3,0,0,3,3\}\) satisfies \(N = 6\), \(\mu = 0\) and \(\sigma=3\)