Question

Cathy has obtained a sample mean of $0.7$ and $\sum _{\text{i}=1}^{\text{n}}{x_{\text{i}}}^2=10$ , but she cannot remember if the sample size was $20$ or $25$. She accidentally erased the data file. Find the correct sample size.

Answer 1

To find the correct sample size:

Let $\text{n}$ be a correct sample size.

Using the sample variance formula and calculate the correct sample size in below:

$\begin{array}{rcl}{\text{s}}^2& =& \frac{{\displaystyle \sum _{\text{i}=1}^{\text{n}}{x_{\text{i}}}^2-\frac{{\displaystyle {\left(\sum _{\text{i}=1}^{\text{n}}{x}_{\text{i}}\right)}^2}}{\text{n}}}}{\text{n}-1}\left[\text{Sample variance formula}\right]\\ & \Rightarrow & {\text{s}}^2=\frac{{\displaystyle \sum _{\text{i}=1}^{\text{n}}{x_{\text{i}}}^2-\text{n}{\left(\frac{{\displaystyle \sum _{\text{i}=1}^{\text{n}}{x}_{\text{i}}}}{\text{n}}\right)}^2}}{\text{n}-1}\left[\text{Simplifying}\right]\\ & \Rightarrow & {\text{s}}^2=\frac{{\displaystyle \sum _{\text{i}=1}^{\text{n}}{x_{\text{i}}}^2-\text{n}{\left(\overline{x}\right)}^2}}{\text{n}-1}\\ \therefore {\text{s}}^2& =& \frac{{\displaystyle 10-\text{n}{\left(0.7\right)}^2}}{\text{n}-1}\left[Substitue\sum _{\text{i}=1}^{\text{n}}{x_{\text{i}}}^2=10,\overline{x}=0.7\right]\end{array}$

Since ${\text{s}}^2$ is always positive number, so substitute ${\text{s}}^2\ge 0$ and solve for $\text{n}$:

$\begin{array}{rcl}\frac{{\displaystyle 10-\text{n}{\left(0.7\right)}^2}}{\text{n}-1}& \ge & 0\\ & \Rightarrow & 10-\text{n}{\left(0.7\right)}^2\ge 0\\ & \Rightarrow & \text{n}{\left(0.7\right)}^2\le 10\\ & \Rightarrow & n\le \frac{10}{{\left(0.7\right)}^2}\\ \therefore n& \le & 20\end{array}$

Therefore, the correct possible sample size is $20$.

Hence, the correct sample size is $\mathbf{20}$.