Chapter 28, Problem 28.AE

(a)

Interpretation Introduction

Interpretation:

The expected numbers of red and yellow marbles has to be determined.

Explanation of Solution

The given box contains 12000 red marbles and 88000 yellow marbles.

The given random sample contains 1000 marbles from the box.

Now, we have to determine the expected number of red and yellow marbles from the given sample box.

The expected number of red marbles $=n{p}_{red}=(1000)(0.12)=120.$

The expected number of yellow marbles $=n{p}_{yellow}=(1000)(0.88)=880.$

(b)

Interpretation Introduction

Interpretation:

The absolute and relative standard deviations for the numbers given in part (a) has to be determined.

Explanation of Solution

The absolute value of the given red and yellow marbles can be determined by using the following formula.

$\text{Absolute}:{\sigma }_{red}={\sigma }_{yellow}=\sqrt{npq}$

Now, substitute the corresponding values in the above formula and do some simple mathematical calculation to get the final answer.

$\text{Absolute}=\sqrt{(1000)(0.12)(0.88)}=10.28$

Similarly, the relative standard deviations of the given red and yellow marbles can be determined by using the following formula.

$\text{Relative}=\frac{{\sigma }_{red}}{n_{red}}=\frac{10.28}{120}=8.56\%$

$\text{Relative}=\frac{{\sigma }_{yellow}}{n_{yellow}}=\frac{10.28}{880}=1.17\%$

(c)

Interpretation Introduction

Interpretation:

The absolute and relative standard deviations after many drawings of 4000 marbles has to be determined.

Explanation of Solution

For 4000 marbles we have ${\sigma }_{red}=480\text{and}{\sigma }_{yellow}=3520.$

The absolute value of the given red and yellow marbles can be determined by using the following formula.

$\text{Absolute}:{\sigma }_{red}={\sigma }_{yellow}=\sqrt{npq}$

Now, substitute the corresponding values in the above formula and do some simple mathematical calculation to get the final answer.

$\text{Absolute}=\sqrt{(4000)(0.12)(0.88)}=20.55$

Similarly, the relative standard deviations of the given red and yellow marbles can be determined by using the following formula.

$\text{Relative}=\frac{{\sigma }_{red}}{n_{red}}=\frac{20.55}{480}=4.28\%$

$\text{Relative}=\frac{{\sigma }_{yellow}}{{\sigma }_{yellow}}=\frac{20.55}{3520}=0.58\%$

(d)

Interpretation Introduction

Interpretation:

The given set of blank statements has to be completed.

Explanation of Solution

• If we quadruple the size of the sample, we minimize the sampling standard deviation by the factor of $2.$
• If we increase the size of the sample by a factor of n, we minimize the sampling standard deviation by the factor of $\sqrt{n}.$

(e)

Interpretation Introduction

Interpretation:

The required sample size to reduce the sampling standard deviation of red marbles to $\pm 2$ has to be determined.

Explanation of Solution

The required sample size to reduce the sampling standard deviation of red marbles to $\pm 2$ can be achieved as follows.

$\begin{array}{l}\text{Relative}=\frac{{\sigma }_{red}}{n_{red}}=0.02=\frac{\sqrt{n(0.12)(0.88)}}{\left(0.12\right)n}\\ n=1.83\times {10}^4\end{array}$

Therefore, we need the sample size of $n=1.83\times {10}^4$ to reduce the sampling standard deviation of red marbles to $\pm 2.$