Question

Let \(X\) be \(N\left(\mu, \sigma^{2}\right)\) so that \(P(X<89)=0.90\) and \(P(X<94)=0.95\). Find \(\mu\) and \(\sigma^{2}\).

Short answer

The mean (\mu) is approximately 72.42 and the variance (\sigma^{2}) is approximately 193.05.

Step-by-step solution

Find Z-values

Read from the standard normal distribution table for the z-values corresponding to the given probabilities. For \(P(X<89) = 0.90\), \(z_1\) approximately equals 1.28. For \(P(X<94) = 0.95\), \(z_2\) approximately equals 1.64.

Find Mu using the Z-value formulas

Use transformation formulas to solve for \(\mu\). This gives two equations: \(1.28 = (89 - \mu) / \sigma\) and \(1.64 = (94 - \mu) / \sigma\). Solve these two equations by subtraction to find the value of \(\mu\). By subtraction, \(0.36 = (5/\sigma)\). So, \(\mu = 94 - 1.64\sigma\)

Find Sigma using the found Mu

Substitute the value of \(\mu\) from Step 2 into any one of the two original equations. This gives an equation in \(\sigma\): \(1.28 = (89 - (94 - 1.64 \sigma)) / \sigma\), after simplifying becomes \(\sigma = 5 / 0.36\). Hence \(\sigma\) is approximately 13.889

Calculate Sigma Square

Calculate the variance \(\sigma^2\) by squaring the value of standard deviation \(\sigma\). Namely, \(\sigma^2 = (13.889)^2\)

Calculate Mu

Substitute the found value of \(\sigma\) into the equation \(\mu = 94 - 1.64\sigma\) to solve for \(\mu\). This gives \(\mu\) approximately equals 72.42