Chapter 7: Q1 (page 339)

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< $σ$ < 33,299.8, and the units of measurement are ${(c m^{3})}^{2}$ . Identify the corresponding confidence interval estimate of $σ$ 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 $σ$ .

Short Answer

Expert verified

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

There is 95% confidence that the limits of 95.0 ${cm}^{3}$ and 182.5 ${cm}^{3}$ 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 $9027.8 < σ < 33299.8$ .

Convert the units

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

Mathematically,

$9027.8 {({cm}^{3})}^{2} = \sqrt{9027.8} {cm}^{3} = 95 {cm}^{3}$

$33299.8 {({cm}^{3})}^{2} = \sqrt{33299.8} {cm}^{3} = 182.5 {cm}^{3}$

Interpret the confidence interval

The confidence interval estimate in ${cm}^{3}$ is $95.0 {cm}^{3} < σ < 182.5 {cm}^{3}$

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.