Question

Brain Volume Using all of the brain volumes listed in Data Set 8 “IQ and Brain Size”, we get this 95% confidence interval estimate: 9027.8< $\sigma$< 33,299.8, and the units of measurement are ${\left(c{m}^3\right)}^2$. Identify the corresponding confidence interval estimate of $\sigma$ and include the appropriate units. Given that the original values are whole numbers, round the limits using the round-off rule given in this section. Write a statement that correctly interprets the confidence interval estimate of $\sigma$.

Short answer

The confidence interval estimate in ${\text{cm}}^3$ is$95.0{\text{cm}}^3<\sigma <182.5{\text{cm}}^3$.

There is 95% confidence that the limits of 95.0 ${\text{cm}}^3$ and 182.5 ${\text{cm}}^3$contain the true value of the standard deviation of brain volumes.

Step-by-step solution

Given information

The level of confidence is 95%.

The confidence interval estimate is $9027.8<\sigma <33299.8$.

Convert the units

The conversion from ${\left({\text{cm}}^3\right)}^2$ to ${\text{cm}}^3$ is done by applying square root to ${\left({\text{cm}}^3\right)}^2$.

Mathematically,

$$\begin{gathered} 9027.8{\left({\text{cm}}^3\right)}^2=\sqrt{9027.8}{\text{cm}}^3 \\ =95{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} 33299.8{\left({\text{cm}}^3\right)}^2=\sqrt{33299.8}{\text{cm}}^3 \\ =182.5{\text{cm}}^3 \end{gathered}$$

Interpret the confidence interval

The confidence interval estimate in ${\text{cm}}^3$ is $95.0{\text{cm}}^3<\sigma <182.5{\text{cm}}^3$

Interpretation:

It can be concluded from the confidence interval thatwe are 95% confident that the limits of 95.0role="math" localid="1648038263318" $c{m}^3$ and 182.5 role="math" localid="1648038249404" $c{m}^3$ contain the true value of the standard deviation of brain volumes.