Test for Goodness of Fit

The math department at a community college offers 3 classes that satisfy the math requirement for transfer in majors that do not require calculus: College Algebra, Statistics, and Finite Math*. At Saddleback College, we no longer offer Finite Math, as this course proved to be significantly less popular than the other two. College Algebra Statistics Finite Math

The math department chair is trying to determine how many sections of each class to offer. She claims that students show no preference for which class they take; if this proves to be so she will offer equal numbers of each class. She looks at the number of students who enrolled in each class during the previous semester. College Algebra students Statistics students Finite Math students

She finds the following data: College AlgebraStatisticsFinite Math # students enrolled Determine whether it is reasonable to suppose students have no preference between the three classes (and thus to offer the same number of sections of each.) Use the goodness-of-fit test with α =.05.

If you’d like to try this problem on your own and just check your answer when you’re done go ahead. When you’re ready to check your answer click on the genius to the right. If you’d rather work through this problem together click away from the genius or hit the space bar or forward arrow key.

Set-up The table tells us the observed frequency. College AlgebraStatisticsFinite Math # students enrolled (Observed frequency) It’s our job to calculate the expected frequency.

College AlgebraStatisticsFinite Math # students enrolled (Observed frequency) Expected frequency

To do this, we’ll need to calculate the total number of students enrolled in all three classes. College AlgebraStatisticsFinite Math # students enrolled (Observed frequency) Expected frequency

To do this, we’ll need to calculate the total number of students enrolled in all three classes. College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency

To do this, we’ll need to calculate the total number of students enrolled in all three classes. College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency

Now we can calculate the expected frequency. If the students have no preference between the three classes, we would expect the students to be equally distributed between them. College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency

College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency Divide the total number of students by 3, the number of classes they can choose from.

College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency

College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency 360

Step 1: State the hypotheses and identify the claim (if there is one). The claim is that the students have no preference---there’s no math symbol for this, so we’ll just say it in words.

Eeny, meeny, miney, moe Algebra Stat Finite This is the Null since the Null always states there is no difference between things. We care!

Step (*) Draw the chi-square distribution and label the area in the right tail. Can we use this distribution?

Since all the expected frequencies are at least 5, we can use the chi-square distribution!

.05 Remember, the chi-square test is always right- tailed. (In this case, so is the bull.)

Step 2: Mark off the critical value..05 The critical value is the boundary of the right tail. It will go here.

Any time we use the Chi-square distribution, we need to use table G. Remember that the degrees of freedom will be one less than the number of categories (in this case, the number of classes.)

College Algebra Statistics Finite Math Since there were 3 classes, the degrees of freedom is 3-1 = 2

So we look in the row for d.f. = 2. And the column for α = The critical value is

Let’s add this to the picture

Step 3: Calculate the test value.

sum observed frequency expected frequency College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency 360 Refer back to this table to find the observed and expected frequencies

College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency 360

College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency 360

College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency 360

College Algebra StatisticsFinite MathTotal # students enrolled (Observed frequency) Expected frequency 360

Now add the test value to the picture is (much) bigger than 5.991, so it goes to the right.

Step 4: Make the decision The test value is in the critical region. Reject the Null!

RATS! Rejected again! We rats had nothing to do with it.

Step 5: Answer the question in plain English. There is enough evidence to reject the claim that students have no preference among the three math classes.

Here’s a quick summary …

Each click will show you one step. Step (*) is broken up into two clicks. Step 1 Step (*) Step Step 3 Step 4 Reject the Null. Step 5 There is enough evidence to reject the claim that the students have no preference among the three classes.

And there was much rejoicing