Math  /  Data & Statistics

QuestionIs it worth pursuing a doctoral degree in education if you already have an undergraduate degree? One way to help make this decision is to look at the mean incomes of these two groups. Suppose that 11 people with bachelor's degrees in education were surveyed. Their mean annual salary was $45,000\$ 45,000 with a standard deviation of $6700\$ 6700. Sixteen people with doctoral degrees in education were found to have a mean annual salary of $40,500\$ 40,500 with a standard deviation of $5200\$ 5200. Assume that the population variances are not the same. Construct a 99%99 \% confidence interval to estimate the true difference between the mean salaries for people with doctoral degrees and undergraduate degrees in education. Let Population 1 be the salaries for people with doctoral degrees and Population 2 be the salaries for people with undergraduate degrees. Round the endpoints of the interval to the nearest whole number, if necessary. Tables Keypad Answer Keyboard Shortcuts

Studdy Solution

STEP 1

What is this asking? We want to find out, with 99% confidence, the difference in average salaries between people with doctoral and undergraduate degrees in education, given some salary data. Watch out! Don't mix up the groups!
We're looking at doctoral minus undergraduate, and the problem tells us which population is which.
Also, the problem says the population variances are *not* the same, so we'll need to use the correct formula for that situation.

STEP 2

1. Find the degrees of freedom.
2. Calculate the standard error.
3. Find the critical t-value.
4. Compute the margin of error.
5. Construct the confidence interval.

STEP 3

Since the population variances are *not* assumed to be equal, we'll use a slightly complicated formula for the degrees of freedom.
It's called Welch's formula, and it looks like this:
[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_

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