Question

find for the 95% confidence interval (draw the graph and show all your work)
c = 0.95

Explanation:

Step1: Determine the significance level

The confidence level \(C = 0.95\). The significance level \(\alpha=1 - C\). So, \(\alpha=1 - 0.95=0.05\).

Step2: Find \(\alpha/2\)

\(\alpha/2=\frac{0.05}{2}=0.025\).

Step3: Find the \(z\) - value

We want to find \(z_{\alpha/2}\) such that \(P(Z>z_{\alpha/2})=\alpha/2 = 0.025\). Looking up in the standard normal table (or using a calculator with a normal - distribution function), \(z_{0.025}\) is the \(z\) - value for which the area to the right of \(z\) is \(0.025\). The area to the left of \(z\) is \(1 - 0.025=0.975\). From the standard normal table (or using a calculator: for example, in Excel \(=NORM.S.INV(0.975)\) or in R qnorm(0.975)), \(z_{\alpha/2}=1.96\).

The \(95\%\) confidence interval for the population mean \(\mu\) (when the population standard deviation \(\sigma\) is known) is given by \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). If we are just asked for the critical \(z\) - value for the \(95\%\) confidence interval (which is often the first step in constructing the confidence interval), the critical values are \(z=- 1.96\) and \(z = 1.96\).

For the graph:

• Draw the standard normal curve \(y = f(z)=\frac{1}{\sqrt{2\pi}}e^{-\frac{z^{2}}{2}}\).
• The total area under the curve is \(1\).
• Shade the two tails: each tail has an area of \(\alpha/2 = 0.025\). The non - shaded (middle) area is \(C = 0.95\). Mark the points \(z=-1.96\) and \(z = 1.96\) on the \(z\) - axis.

Answer:

The critical \(z\) - values for a \(95\%\) confidence interval are \(z=-1.96\) and \(z = 1.96\).