Chapter 11.6, Problem 19PPS

a.

To determine

Graph $90\%$ confidence interval for $n=50,100,250$ on a number line.

Answer to Problem 19PPS

Explanation of Solution

Given information:

Explore how the confidence interval is affected by the sample size and the confidence level. Consider a sample of data where $\overline{x}=25,\text{ s=3 }$ .

Calculation:

Consider the sample data $\overline{x}=25,\text{ s=3 }$

In $90\%$ confidence level $5\%$ of the area lies in each tail, the confidence $z-value$ is $1.645$

Now calculate the value of maximum error estimated, for $n=50,100,250$

  $\begin{array}{l}E\left(50\right)=z\times \frac{s}{\sqrt{n}}\\ =1.645\times \frac{3}{\sqrt{50}}\\ =0.697914\\ \approx 0.70\\ \text{Now},\text{ the confidence interval will be},\\ CI=\overline{x}\pm E\\ =25\pm 0.7\\ =24.30to25.70\end{array}$

  $\begin{array}{l}E\left(100\right)=z\times \frac{s}{\sqrt{n}}\\ =1.645\times \frac{3}{\sqrt{100}}\\ =0.4935\\ \approx 0.49\\ \text{Now},\text{ the confidence interval will be},\\ CI=\overline{x}\pm E\\ =25\pm 0.49\\ =24.51to25.49\end{array}$

  $\begin{array}{l}E\left(250\right)=z\times \frac{s}{\sqrt{n}}\\ =1.645\times \frac{3}{\sqrt{250}}\\ =0.312116\\ \approx 0.31\\ \text{Now},\text{ the confidence interval will be},\\ CI=\overline{x}\pm E\\ =25\pm 0.31\\ =24.69to25.31\end{array}$

Now , the graph of confidence intervals on number line is as ahown,

  

Hence, the resultant graph is plotted.

b.

To determine

Is sample size affecting the confidence interval?

Answer to Problem 19PPS

The sample size increases, the confidence interval decreases.

Explanation of Solution

Given information:

Explore how the confidence interval is affected by the sample size and the confidence level. Consider a sample of data where $\overline{x}=25,\text{ s=3 }$ .

Calculation:

According to the graph plotted in part $\left(a\right)$ , as the of $n$ increases, the size of confidence interval decreases,

Hence, the sample size increases, the confidence interval decreases.

c.

To determine

Graph the $90\%,95\%,99\%$ confidence interval for $n=150$ .

Explanation of Solution

Given information:

Explore how the confidence interval is affected by the sample size and the confidence level. Consider a sample of data where $\overline{x}=25,\text{ s=3 }$ .

Calculation:

Consider the sample data $\overline{x}=25,\text{ s=3 }$ and $n=150$ ,

In $90\%$ confidence level, $5\%$ of the area lies in each tail. Corresponding $z-value$ is $1.645$ , now calculate confidence interval,

  $\begin{array}{l}CI\left(90\%\right)=\overline{x}\pm E\\ =\overline{x}\pm z\times \frac{s}{\sqrt{n}}\\ =25\pm 1.645\times \frac{3}{\sqrt{150}}\\ =25\pm 0.4029\\ =24.60to25.40\end{array}$

Now In $95\%$ confidence level, $2.5\%$ of the area lies in each tail. Corresponding $z-value$ is $1.960$ , now calculate confidence interval,

  $\begin{array}{l}CI\left(95\%\right)=\overline{x}\pm E\\ =\overline{x}\pm z\times \frac{s}{\sqrt{n}}\\ =25\pm 1.960\times \frac{3}{\sqrt{150}}\\ =25\pm 0.48\\ =24.52to25.48\end{array}$

Now In $99\%$ confidence level, $0.5\%$ of the area lies in each tail. Corresponding $z-value$ is $2.576$ , now calculate confidence interval,

  $\begin{array}{l}CI\left(99\%\right)=\overline{x}\pm E\\ =\overline{x}\pm z\times \frac{s}{\sqrt{n}}\\ =25\pm 2.576\times \frac{3}{\sqrt{150}}\\ =25\pm 0.63\\ =24.37to25.63\end{array}$

Now plot all confidence levels on number line,

  

d.

To determine

Is the confidence level affecting the confidence interval?

Answer to Problem 19PPS

Increasing the confidence level will increase the confidence interval.

Explanation of Solution

Given information:

Explore how the confidence interval is affected by the sample size and the confidence level. Consider a sample of data where $\overline{x}=25,\text{ s=3 }$ .

Calculation:

According to the graph of part $\left(c\right)$ as the confidence level increases, the size of confidence interval increases, keeps other constant

Hence, increasing the confidence level will increase the confidence interval.

e.

To determine

Analyse, decreasing the size of the confidence interval affect the accuracy of the confidence interval.

Answer to Problem 19PPS

Decreasing the size of confidence interval, increases the accuracy of the estimate.

Explanation of Solution

Given information:

Explore how the confidence interval is affected by the sample size and the confidence level. Consider a sample of data where $\overline{x}=25,\text{ s=3 }$ .

Calculation:

Expanding the confidence interval reduces the accuracy of the estimate.

so, decreasing the size of confidence interval, increases the accuracy of the estimate.