Chapter 7.1, Problem 2CYU

a.

To determine

Find the levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=1.96$.

Answer to Problem 2CYU

The levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=1.96$ is 95%.

Explanation of Solution

Calculation:

Critical value:

If $\alpha$ is a number between 0 and 1 or $0<\alpha <1$, then

• The notation $z_{\alpha }$ represents the z-score with an area $\alpha$ to the right.
• The notation $z_{\frac{\alpha }{2}}$ represents the z-score with an area $\frac{\alpha }{2}$ to the right

Here the critical value is provided as 1.96.

Software procedure:

Step-by-step software procedure to obtain the probability value using MINITAB software is as follows,

• Choose Stat > Graphs > Probability Distribution Plot.
• Choose View Probability.
• From Distribution, choose ‘Normal’ distribution.
• Enter the Mean as 0 and Standard deviation as 1.
• Click Shaded Area tab.
• Choose X-value and both tail for the region of the curve to shade.
• Enter the X-value as 1.96.
• Click OK in all the dialogue boxes

Output using MINITAB software is as follows,

The significance level is 0.05 $\left(=0.025+0.025\right)$.

The confidence level for the critical value $z_{\frac{\alpha }{2}}=1.96$ is,

$\begin{array}{c}\text{Confidence level}=1-\alpha \\ =1-0.05\\ =0.95\\ =95\%\end{array}$

Thus, the levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=1.96$ is 95%.

b.

To determine

Find the levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=2.17$.

Answer to Problem 2CYU

The levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=2.17$ is 97%.

Explanation of Solution

Calculation:

Software procedure:

Step-by-step software procedure to obtain the probability value using MINITAB software is as follows,

• Choose Stat > Graphs > Probability Distribution Plot.
• Choose View Probability.
• From Distribution, choose ‘Normal’ distribution.
• Click Shaded Area tab.
• Choose X-value and both tail for the region of the curve to shade.
• Enter the X-value as 2.17.
• Click OK in all the dialogue boxes

Output using MINITAB software is as follows,

The significance level is 0.03 $\left(=0.015+0.015\right)$.

The confidence level for the critical value $z_{\frac{\alpha }{2}}=2.17$ is,

$\begin{array}{c}\text{Confidence level}=1-\alpha \\ =1-0.03\\ =0.97\\ =97\%\end{array}$

Thus, the levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=1.96$ is 97%.

c.

To determine

Find the levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=1.28$.

Answer to Problem 2CYU

The levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=1.28$ is 80%.

Explanation of Solution

Calculation:

Software procedure:

Step-by-step software procedure to obtain the probability value using MINITAB software is as follows,

• Choose Stat > Graphs > Probability Distribution Plot.
• Choose View Probability.
• From Distribution, choose ‘Normal’ distribution.
• Click Shaded Area tab.
• Choose X-value and both tail for the region of the curve to shade.
• Enter the X-value as 1.28.
• Click OK in all the dialogue boxes

Output using MINITAB software is as follows,

The significance level is 0.20 $\left(=0.1003+0.1003\right)$.

The confidence level for the critical value $z_{\frac{\alpha }{2}}=1.28$ is,

$\begin{array}{c}\text{Confidence level}=1-\alpha \\ =1-0.20\\ =0.80\\ =80\%\end{array}$

Thus, the levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=1.28$ is 80%.

d.

To determine

Find the levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=3.28$.

Answer to Problem 2CYU

The levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=3.28$ is 99.9%.

Explanation of Solution

Calculation:

Software procedure:

Step-by-step software procedure to obtain the probability value using MINITAB software is as follows,

• Choose Stat > Graphs > Probability Distribution Plot.
• Choose View Probability.
• From Distribution, choose ‘Normal’ distribution.
• Click Shaded Area tab.
• Choose X-value and both tail for the region of the curve to shade.
• Enter the X-value as 3.28.
• Click OK in all the dialogue boxes

Output using MINITAB software is as follows,

The significance level is 0.001 $\left(=0.000519+0.000519\right)$.

The confidence level for the critical value $z_{\frac{\alpha }{2}}=3.28$ is,

$\begin{array}{c}\text{Confidence level}=1-\alpha \\ =1-0.001\\ =0.999\\ =99.9\%\end{array}$

Thus, the levels of confidence intervals that have the critical value $z_{\frac{\alpha }{2}}=3.28$ is 99.9%.