Chapter 15.1, Problem 45E

To determine

To Find: The range about the sample mean that gives a 99% chance that the true mean lies within it.

Answer to Problem 45E

The interval about the sample mean is $\left[15.69,16.71\right]$ and it is 99% true that mean is within it.

Explanation of Solution

Given:

The mean diameter of the tree is 16.2 millimetres and the standard deviation of 1.4 millilitres.

The range about the sample mean is 99%.

Calculation:

Consider the standard deviation and the mean is,

  $\begin{array}{l}s=1.4\\ \overline{X}=16.2\end{array}$

To obtain the interval about the sample mean that has 99% chance that the true mean lies with the interval that is,

  $\begin{array}{l}P=99\%\\ t=2.58\end{array}$

The range of the interval about the sample shows that the mean is 99% chance that the true mean lies is,

  $\begin{array}{c}\left[\overline{X}-t{\sigma }_{\overline{X}},\overline{X}+t{\sigma }_{\overline{X}}\right]=\left[16.2-2.58\left(0.198\right),\left(16.2\right)+2.58\left(0.198\right)\right]\\ =\left[15.69,16.71\right]\end{array}$

Thus, the interval about the sample mean is $\left[15.69,16.71\right]$ and it is 99% true that mean is within it.