Question

construct a data set that has the given statistics.
n = 6
μ = 9
σ = 3

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 9.
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 9 should be 3.
complete the population data set.
6, 6, □, 12, □, 12 (use ascending order.)

Explanation:

Step1: Use the formula for the mean

The formula for the population mean \(\mu=\frac{\sum_{i = 1}^{N}x_{i}}{N}\). Given \(N = 6\) and \(\mu=9\), then \(\sum_{i=1}^{6}x_{i}=N\times\mu=6\times9 = 54\).
We know four of the values: \(6,6,12,12\). Let the two unknown values be \(a\) and \(b\). So \(6 + 6+a + 12+b+12=54\), which simplifies to \(a + b=54-(6 + 6+12+12)=18\).

Step2: Use the formula for the population standard deviation

The formula for the population standard deviation \(\sigma=\sqrt{\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}}\). Given \(\sigma = 3\) and \(N = 6\), then \(\sum_{i=1}^{6}(x_{i}-\mu)^{2}=N\times\sigma^{2}=6\times9 = 54\).
For \(x = 6\), \((6 - 9)^{2}=9\). For \(x = 12\), \((12 - 9)^{2}=9\).
Let \(a\) and \(b\) be the unknowns. We have \(2\times9+2\times9+(a - 9)^{2}+(b - 9)^{2}=54\). Since \(a + b=18\), let \(b = 18 - a\).
Substitute \(b = 18 - a\) into \((a - 9)^{2}+(b - 9)^{2}=54-(2\times9 + 2\times9)=18\).
\((a - 9)^{2}+((18 - a)-9)^{2}=18\), \((a - 9)^{2}+(9 - a)^{2}=18\). Since \((a - 9)^{2}=(9 - a)^{2}\), then \(2(a - 9)^{2}=18\), \((a - 9)^{2}=9\), \(a-9=\pm3\).
If \(a-9 = 3\), then \(a = 12\) (but we already have \(12\)s). If \(a - 9=-3\), then \(a=6\) (but we already have \(6\)s). Wait, let's check another way.
Since \(a + b=18\) and assume \(a=b\) (for simplicity, since the data is symmetric in some sense). Then \(2a=18\), \(a = b = 9\)

Answer:

\(6,6,9,12,9,12\)