Question

Suppose that 40 and 90 are two elements of a population data set and that their z-scores are -2 and 3, respectively. Using only this information, is it possible to determine the population’s mean and standard deviation? If so, find them. If not, explain why it’s not possible.

Short answer

Yes, it is possible to find the mean and the standard deviation.

The mean is 60.

The standard deviation is 10.

Step-by-step solution

Given information

x = 40, and x = 90.

The z-score for 40 is (-2), and the z-score for 90 is 3

Finding the mean and the standard deviation

As we have two values from the population, it is possible to calculate the mean and the standard deviation.

$$\begin{gathered} \text{z-score=}\frac{\text{x-μ}}{\text{σ}} \\ \text{WhenZ=-2,} \\ \text{-2=}\frac{\text{40-μ}}{\text{σ}} \\ \text{WhenZ=3,} \\ 3\text{=}\frac{\text{90-μ}}{\text{σ}} \\ \text{Sowecanalsowritetwoequationsmentionedaboveasfollows:} \\ \text{-2σ=40-μ} \\ \text{3σ=90-μ} \\ \text{Now,bysubtractingtheequations,weget} \\ \left(\text{-2σ-3σ}\right)=\left(\text{40-μ}\right)-\left(\text{90-μ}\right) \\ \text{-5σ=-50} \\ \text{σ=}\frac{\text{50}}{\text{5}} \\ \text{σ=10} \end{gathered}$$

Therefore, the standard deviation is 10.

Substituting the value of σ in 3σ = 90 - µ, we get

$$\begin{gathered} \left(3\times 10\right)=\left(90-\mu \right) \\ 30=90-\mu \\ \mu =90-30 \\ \mu =60 \end{gathered}$$

Therefore, the mean is 60.