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Hypothesis Testing Within One Set of Numbers

Statistics & Risk Management

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Comparison WithinWe need a way to measure the difference between

obtained results and what would occur randomly.For example: At the Mall of America we asked

everyone their age as they entered the store for one hour on Saturday.

We discarded any ages under 16 or over 65 (trimmed). We collected a sample of 100 useable responses. We grouped them in 10 year blocks.

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Our null-hypothesis is that we will get an equal number of people in each of the five age groups.

H0: P1 = P2 = P3 = P4 = P5

Comparison Within

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Age Expect Actual Diff Diff2 Diff2/Expect

16-25 20 13 -7 49 2.45

26-35 20 28 8 64 3.20

36-45 20 17 -3 9 .45

46-55 20 27 7 49 2.45

56-65 20 15 -5 25 1.25

Totals 100 100 0 196 * 9.80

The 9.80 is the chi-square test value.Degrees of Freedom ( df ) = Groups – 1 = 5 –

1 = 4We will use a Type I (alpha) Error Rate of .05

Comparison Within

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df .10 .05 .01 df .10 .05 .01

1 2.71 3.84 6.64 16 23.54 26.30 32.00

2 4.00 5.99 9.21 17 24.77 27.60 33.41

3 6.25 7.82 11.34 18 25.99 28.87 34.80

4 7.78 9.49 13.28 19 27.20 30.14 36.19

5 9.24 11.07 15.09 20 28.41 31.41 37.57

6 10.64 12.59 16.81 21 29.62 32.67 38.93

7 12.02 14.07 18.48 22 30.81 33.92 40.29

8 13.36 15.51 20.09 23 32.01 35.17 41.64

9 14.68 16.92 21.67 24 33.20 36.42 42.98

10 16.99 18.31 23.21 25 34.38 37.65 44.81

11 17.28 19.68 24.72 26 35.56 38.88 45.64

12 18.65 21.03 26.22 27 36.74 40.11 49.96

13 19.81 22.36 27.67 28 37.92 41.34 48.28

14 21.06 23.68 29.14 29 39.09 52.56 49.59

15 22.31 25.00 30.58 30 40.26 43.77 50.89

Critical Values for Chi-Squared Test

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What Does This Mean?We found that the Chi-Squared value of 9.80 is

larger than the Critical Value of 9.49 So we will reject the Null-Hypothesis– Because there is a significant difference

between the obtained results and what would be expected as a result of just random responses.

Notice how close that was!

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What Does That Mean?

That there is a significant difference of ages in the people we interviewed beyond what we would expect (Probability).

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Chi-Square Formula?

Chi-square value

Obtained Score(s)

Expected Score(s)

Expected Score(s)

9

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Excel Formula

=CHITEST(B1:B5,C1:C5)

This function uses two lists of Expected (column B) and Observed (Column C)