Question

How do you find a $95\%$ confidence interval for $\mu $ when $n=32$, x-bar$=137$ft, and $\sigma =7.0$ ft?

Answer 1

Hint: The confidence interval is obtained by adding and subtracting the marginal error from the mean value of x. The marginal error is given by the formula $E={{z}_{\alpha /2}}\dfrac{\sigma }{\sqrt{n}}$. Therefore, the confidence interval is given by $CI=\overline{x}\pm {{z}_{\alpha /2}}\dfrac{\sigma }{\sqrt{n}}$. Here ${{z}_{\alpha /2}}$ is the critical value whose value can be determined by using the normal table, and $\alpha $ is the significance level, given by $\alpha =1-CL$, where $CL$ is the confidence level. On substituting the values given in the above question, and the value of the critical value from the normal table, we will get the required confidence interval.

Complete step by step solution:
We know that the confidence interval is given by
$\Rightarrow CI=\overline{x}\pm E.........\left( i \right)$
The marginal error is given by the formula
$\Rightarrow E={{z}_{\alpha /2}}\dfrac{\sigma }{\sqrt{n}}.......\left( ii \right)$
And the significance level is given by
$\Rightarrow \alpha =1-CL$
According to the above question, we have a confidence level of $95\%$. Therefore, we substitute $CL=0.95$ in the above equation to get
$\begin{align}
  & \Rightarrow \alpha =1-0.95 \\
 & \Rightarrow \alpha =0.05 \\
\end{align}$
So from the equation (ii) the marginal error becomes
$\begin{align}
  & \Rightarrow E={{z}_{\dfrac{0.05}{2}}}\dfrac{\sigma }{\sqrt{n}} \\
 & \Rightarrow E={{z}_{0.025}}\dfrac{\sigma }{\sqrt{n}} \\
\end{align}$
From the normal distribution table, we get ${{z}_{0.025}}=1.96$. Substituting this in the above equation, we get
$\Rightarrow E=1.96\dfrac{\sigma }{\sqrt{n}}$
In the above question, we have $\sigma =7.0$ and $n=32$. Substituting these above we get
$\begin{align}
  & \Rightarrow E=1.96\dfrac{7}{\sqrt{32}} \\
 & \Rightarrow E=1.96\dfrac{7}{4\sqrt{2}} \\
\end{align}$
On solving we get
$\Rightarrow E=2.43$
Substituting this in the equation (i) we get
$\Rightarrow CI=\overline{x}\pm 2.43$
According to the question, we substitute $\overline{x}=137$ in the above equation to get
\[\begin{align}
  & \Rightarrow CI=137\pm 2.43 \\
 & \Rightarrow CI=\left[ 139.43,134.47 \right] \\
\end{align}\]
Hence, the confidence interval is found to be \[\left[ 139.43,134.47 \right]\].

Note:
To solve these types of questions, we need to be familiar with the symbols of the different statistical quantities such as standard deviation, mean etc. In the above question, we were given only the values by denoting the symbols, and were not told what they refer to. Also, we must have access to the normal distribution table and also have the knowledge to read the values from it.