Introduction to Chi-Square Procedures March 11, 2010

The Mars Candy Co. claims that the distribution of colors of M&M’s are as follows: Red: 13% Yellow: 14% Green: 16% Blue: 24% Orange: 20% Brown: 13%

Chi-Square Distribution We are going to test whether or not this claim is true. To do so, we will encounter a new probability density: the Chi-Square density with k degrees of freedom (denoted χ 2 (k)) This family of densities will allow us to test whether or not a population has a certain distribution.

Gather Data Each group should add up the number of each color of M&M’s in their bags. Now we will add up the total of each color in the class: RedYellowGreenBlueOrangeBrownTotal

Observed vs. Expected The previous table listed the observed counts for each color. We also need to know the expected count: how many there would be if the claimed distribution is correct: RedYellowGreenBlueOrangeBrownTotal

Summary Table RYGBlOrBrTotal O E (O-E) 2 /E X2=X2=

The Statistic X 2 The statistic X 2 has approximately a χ 2 (read Chi-Square) distribution with k-1 degrees of freedom, where k is the number of outcome categories. The critical numbers are found in Table D at the back of your book.

Significance Test To carry out our inference, we need to formulate the null and alternate hypotheses: H 0 : H a :

P-value We now find our P-value. This is P(χ 2 > X 2 ), which is found in Table D, using 5 degrees of freedom. Our P-value is: We therefore conclude: