"We don't know a millionth of onepercent about anything."

Thomas Alva Edison

13.1 TEST FOR GOODNESS OF FIT (Pages 702 - 716)

OVERVIEW: A chi-square test goodnessof fit test is used to see if an observed sample distribution isdifferent from a hypothesized population distribution. In otherwords, it is used to see if what you got is statistically differentfrom what you expected to get.

The chi-square (x2 ) statistic is calculated asfollows:

x2 = sum[(observed - expected)2/expected]

Properties of x2 :

• x2 is nonnegative in value.
• The x2 distribution is not symmetrical. It is skewed to the right.
• All x2 tests are 1-tail tests.
• In a goodness of fit test, the degrees of freedom is number of categories - 1.

Example #1:

A die is tossed 120 times with theresults displayed in the following table. Is there statisticalevidence to suggest that the die is "loaded"?

 Up Face--> 1 2 3 4 5 6 Observed frequency 25 17 15 23 24 16 Expected frequency 20 20 20 20 20 20

In this situation, the degrees of freedom is 6 - 1= 5.
[Note that 5 of the categories are free to vary, but the sixthis not, since all categories have to add up to 120.]

The calculated x2 is

x2 =(25-20)2/20+ (17-20)2/20 + (15-20)2/20 +(23-20)2/20+ (24-20)2/20 + (16-20)2/20 = 5.

At the 5% level of significance, the criticalregion is x2 > 11.1. Since ourcalculated x2 is not in this region, we would not reject a nullhypothesis that says "The die is fair."

Alternate approach:Using the TI-83, the P-value x2 cdf(5,1E99,5) = .4158801852,which is approximately 41.6%. There is no evidence to suggest thatthe die is loaded.

Example #2:

Suppose I flip a coin 100 times andget 80 heads and 20 tails.

 Number of HEADS Number of TAILS Observed 80 20 Expected 50 50

The x2 statistic for this experiment is

x2 =(80-50)2/50+ (20-50)2/50 = 18 + 18 = 36.

The degrees of freedom is 2-1 = 1.

At the 1% level of significance, the criticalregion for x2 is x2 > 6.63. Our calculated value of 36 is well into thisregion. There is strong evidence to suggest that the coin is notfair.

Alternate approach: Using the TI-83, the P-value is x2 cdf(36, 1E99,1) = .00000000197. This is the probabilitythat one would get 80 or more heads when flipping a fair coin 100times. This supports the previous statement suggesting that the coinis not fair.

The x2 goodness of fit test can be used when

• All individual expected counts are at least 1.
• No more than 20% of the expected counts are less than 5.

In the examples above, these conditions weresatisfied.