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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 :

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"?

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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

In the examples above, these conditions weresatisfied.

 

 

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