Page 237 - Six Sigma Demystified

Part 3 s i x s i g m a to o l s 217

Interpretation

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We reject the null hypothesis if X > X for (r – 1)(c – 1) degrees of freedom,
α
0
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where X is the chi-square value at the α level of significance.
α
In the preceding example, the calculated p value from Excel and Minitab is
less than 0.05, so the null hypothesis is rejected, and we assert that there is a
difference between the male and female responses. Alternatively, the critical
value of the chi-square statistic at the 0.05 level of significance, with two
degrees of freedom, may be found from Appendix 3 as 5.991. Since the test
statistic exceeds the critical value, the null hypothesis is rejected.

Contour Plot

A contour plot is made up of curves, each having a constant value of a fitted
response. The curves have equally spaced values of the response. Additional
factors are set at selected values (usually their mean).

When to Use

Improve Stage

• Use in response surface analysis to determine where a maximum or min-
imum response is expected within or close to the data range

Methodology

For example, we can generate a contour plot for the model Y = 4.5 + 0.32x –
1

0.63x . For any value of the response (Y), we can determine x given x or x
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1
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given x .
1
When y = 5, then 0.32x – 0.63x – 0.5 = 0. If x = 0, then x = – 0.5/0.63 =
1
2
1
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–0.794. If x = 0, then x = 0.5/0.32 = 1.56.
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Thus the contour line for Y = 5 passes through the points (0, –0.79) and
(1.56, 0). We also can easily determine for Y = 5 the value of x when x = +1
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and x = –1 and the value of x when x = +1 and x = –1. In this way, we then
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1
1
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calculate other contour lines (for different y values).
Overlaid Contour Plots
Overlaid contour techniques provide a method for evaluating joint regions of
optimality for more than one response. In this technique, we lay one contour