Chi Square Goodness of fit, Independence, and Homogeneity May 07, 2014

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2 Example 1 Goodness of Fit test Does your zodiac sign determine how successful you will be later in life? Fortune magazine collected the zodiac signs of 256 heads of the largest 400 companies. Here are the number of births for each sign: Step 1: State the hypotheses. Step 2: Check the assumptions. Randomization condition: Expected cell frequency condition: Step 3: Calculate the test statistic. Degrees of freedom for this test = n 1. Step 4: Make a decision based either on the p value or the critical value on a sketch. Step 5: Interpret the decision in the context of the problem.

3 Goodness of Fit Type #2 Mars, Inc. claims that its M & M candies are distributed with the following color percentages: Brown Yellow Red Orange Green Blue 30% 20% 20% 10% 10% 10% Let's assume Calvin goes to Food Lion and buys a randomly selected bag of M & M's and counts how many of each color are in the bag. He found 33 brown, 26 yellow, 21 red, 8 orange, 7 green, and 5 blue M & M's. At the 5% significance level, test the claim that Mars, Inc. makes about the color distribution of their candies.

4 Example 2 Homogeneity test ( Homogeneity means that things are the same.) Many high schools survey graduating classes to determine the plans of the graduates. We might wonder whether the plans of students have stayed roughly the same over past decades or whether they have changed. Here is a summary table from one high school. Each cell of the table shows how many students from a particular graduating class (the column) made a certain choice (the row). Step 1: State the hypotheses. H 0 : Student plans after high school graduation have stayed the same over the decades. H A: Student plans after high school graduation have changed over the decades. Step 2: Check the assumptions. Expected frequencies = Randomization condition: We have no reason to believe that the plans of these high school students do not accurately represent the plans of all high school students. Expected cell frequency condition: In this assumption, we must show that "all expected frequencies are 5 or 10" and you must physically show them on your paper. So if you wanted to calculate each expected frequency by hand, you would use the formula I have listed in Step 3. However, there is a trick! Look at the next page in this file to see how to use the trick! Step 3: Calculate the expected frequencies. Expected frequency = row total *column total overall total Even though there is a trick on the calculator for finding these quickly, you still need to know the formula for finding them. I have seen AP MC questions that ask you about this without giving you data. Step 4: Calculate the test statistic. Degrees of freedom for this test = (rows 1)(columns 1). Step 5: Make a decision based either on the p value or the critical value on a sketch. Step 6: Interpret the decision in the context of the problem. There is significant evidence to say that student plans after high school graduation have changed over the decades at α =

5 Here's a helpful hint: Some students get confused about when they should use the χ 2 GOF Test versus when they should just use the χ 2 Test on the calculator. If the data is presented with multiple rows and columns, then it is an Independence (tattoo problem) or Homogeneity (high school plans problem) test, and you will need to put the observed values in a matrix and use the χ 2 Test. If the data is presented with a single row of values (uniformity test like zodiac problem) or gives you some hypothesized percentages for each category (Goodness of Fit like the M&M's problem), then you will need to put the observed values in a L1 and then store the expected values in L2 after finding them. Then use the χ 2 GOF Test. How can I get the expected values for a homogeneity or independence test quickly? Put all of the observed values in Matrix A. Be careful not to include "TOTALS" in the matrix if they are given in the problem. STOP! You don't need to hand calculate the expected values that will go in Matrix B. Go ahead and go to the Stat TESTS menu. Choose the χ 2 Test. Go ahead and press calculate even though you haven't entered the expected values in Matrix B. Now go back to the matrix menu and look at Matrix B. The calculator has automatically calculated all of the expected values for you and stored them in matrix B. That saves us a ton of time, but now we must write the expected values somewhere on the paper or the AP grader will not give us credit for the expected frequency assumption. Remember, if you don't get credit for the assumptions, you cannot make a level 4 on the problem.

6 Example 3 Independence test A study from the University of Texas Southwestern Medical Center examined whether the risk of hepatitis C was related to whether people had tattoos and to where they got their tattoos. Hepatitis C causes about 10,000 deaths each year in the United States, but often lies undetected for years after infection. Is the chance of having hepatitis C related to (or dependent on) tattoo status? Remember: For A and B to be independent, must equal. Here, this means the probability of having hepatitis C should not change upon learning the patient s tattoo status. Step 1: State the hypotheses. H 0 : Contracting hepatitis C is independent of whether or not you have a tattoo. H A: Contracting hepatitis C is dependent on whether or not you have a tattoo. Step 2: Check the assumptions. Randomization condition: We have no reason to believe that the people examined for this study do not represent all people. Expected Frequencies = Expected cell frequency condition: All expected frequencies are 5. Step 3: Calculate the expected frequencies. Expected frequency = row total * column total overall total Step 4: Calculate the test statistic. Degrees of freedom for this test = (rows 1)(columns 1). Step 5: Make a decision based either on the p value or the critical value on a sketch. Step 6: Interpret the decision in the context of the problem. There is significant evidence to say that getting a tattoo is related (or dependent on) to contracting Hepatitis C at α = I am trying to subconsciously tell you NOT TO GET A TATTOO or body piercing. Can you tell?