A survery of 80 randomly selected companies asked them to report the annual income of their presidents. Assuming that incomes are normally distributed with a standrad deviation of #30,000, determine the 90% confidence interval estimate of the mean annual income of all company presidents.

To determine the 90% confidence interval estimate of the mean annual income of all company presidents, we will use the following formula:

Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / Square Root of Sample Size)

Step 1: Find the Sample Mean
The survey asked 80 companies, so the sample size is 80. Let's assume that the sample mean is denoted as x̄.

Step 2: Find the Critical Value
Since we want a 90% confidence interval, we need to find the critical value that corresponds to a 90% confidence level. We can look up this value in the Z-table or use a calculator. For a 90% confidence level, the critical value is approximately 1.645.

Step 3: Calculate the Standard Deviation and Square Root of Sample Size
Given that the standard deviation is $30,000, let's denote it as σ. The square root of the sample size is the square root of 80, which is 8.94 (rounded to two decimal places).

Step 4: Plug in the values into the formula
Confidence Interval = x̄ ± (1.645) * (σ / √n)

Confidence Interval = x̄ ± (1.645) * ($30,000 / 8.94)

Now we can calculate the upper and lower bounds of the confidence interval.
Lower Bound = x̄ - (1.645) * ($30,000 / 8.94)
Upper Bound = x̄ + (1.645) * ($30,000 / 8.94)

By substituting the values, we can find the 90% confidence interval estimate of the mean annual income of all company presidents.

90% confidence interval = mean ± 1.645 SEm

SEm (Standard Error of the mean) = SD/√(n-1)