Question

By doing the following steps, determine a \((1-\alpha) 100 \%\) approximate confidence interval for \(\rho\). (a) For \(0<\alpha<1\), in the usual way, start with \(1-\alpha=P\left(-z_{\alpha / 2}

Short answer

The four steps to calculate the \((1-\alpha) 100%\) confidence interval for \(\rho\) are as follows: expressing \(1-\alpha\) in terms of Z with the bounds \(-z_{\alpha / 2}\) and \(z_{\alpha / 2}\), isolating \(h(\rho)\) and showing that it is strictly increasing in the range \(-1<\rho<1\), showing that its inverse function is \(\tanh(y)\), and finally deriving the confidence interval.

Step-by-step solution

Express 1-α in terms of Z

The first part involves expressing \(1-\alpha\) as \(P\left(-z_{\alpha / 2}<Z<z_{\alpha / 2}\right)\). This is a standard way to represent the probability of a normal random variable Z lying within the range \(-z_{\alpha / 2}\) and \(z_{\alpha / 2}\) where \(z_{\alpha / 2}\) is the z-value corresponding to the tail probability \(\alpha / 2\).

Isolate h(ρ)

Next, we isolate the function \(h(\rho)=(1 / 2) \log [(1+\rho) /(1-\rho)]\) in the middle of the inequality obtained in the step above. Then we find the derivative of \(h(\rho)\). Using quotient and chain rule, \(h^\prime(\rho)= 1/(1-\rho^2)\). This is strictly positive on interval \(-1<\rho<1\), hence, proving the function is strictly increasing in this range.

Identify the Inverse Function

Because \(h(\rho)\) is strictly increasing on \(-1<\rho<1\), it has an inverse function. We are asked to show that the inverse function is the hyperbolic tangent function: \(\tanh (y)=\left(e^{y}-e^{-y}\right) /\left(e^{y}+e^{-y}\right) .\) We show this by applying the inverse function on \(h(\rho)\), isolate \(\rho\) to get the expression for \(\tanh (y)\).

Obtain the Confidence Interval

Finally, we use the function \(h(\rho)\) and its inverse to obtain a \((1-\alpha) 100 \%\) confidence interval for \(\rho\). We express the confidence interval in terms of \(\rho=\tanh (y)\) where y has a normal distribution with mean \(-z_{\alpha / 2}\) and \(z_{\alpha / 2}\).