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Chapter 4 Probability and Probability Distributions
Solution
TABLE 4.12
Sample and normal quantiles
for cholesterol readings
Patient
Cholesterol Reading
(i .5)20
Normal Quantile
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
133
137
148
149
152
167
174
179
189
192
201
209
210
211
218
238
245
248
253
257
.025
.075
.125
.175
.225
.275
.325
.375
.425
.475
.525
.575
.625
.675
.725
.775
.825
.875
.925
.975
1.960
1.440
1.150
.935
.755
.598
.454
.319
.189
.063
.063
.189
.319
.454
.598
.755
.935
1.150
1.440
1.960
A plot of the sample quantiles versus the corresponding normal quantiles is displayed in Figure 4.27. The plotted points generally follow a straight line pattern.
FIGURE 4.27
290
Normal quantile plot
270
Cholesterol readings
250
230
210
190
170
150
130
110
–2
–1
0
Normal quantiles
1
2
Using Minitab, we can obtain a plot with a fitted line that assists us in assessing how close the plotted points fall relative to a straight line. This plot is displayed
in Figure 4.28. The 20 points appear to be relatively close to the fitted line and thus
the normal quantile plot would appear to suggest that the normality of the population distribution is plausible.
Using a graphical procedure, there is a high degree of subjectivity in making
an assessment of how well the plotted points fit a straight line. The scales of the axes
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4.14 Evaluating Whether or Not a Population Distribution Is Normal
FIGURE 4.28
197
Cholesterol = 195.5 + 39.4884 Normal Quantiles
S = 8.30179 R-Sq = 95.9% R-Sq(adj) = 95.7%
Normal quantile plot
280
260
Cholesterol readings
240
220
200
180
160
140
120
100
–2
–1
0
Normal quantiles
1
2
on the plot can be increased or decreased, resulting in a change in our assessment of
fit. Therefore, a quantitative assessment of the degree to which the plotted points
fall near a straight line will be introduced.
In Chapter 3, we introduced the sample correlation coefficient r to measure
the degree to which two variables satisfied a linear relationship. We will now discuss how this coefficient can be used to assess our certainty that the sample data
was selected from a population having a normal distribution. First, we must alter
which normal quantiles are associated with the ordered data values. In the above
discussion, we used the normal quantiles corresponding to (i .5)n. In calculating the correlation between the ordered data values and the normal quantiles, a
more precise measure is obtained if we associate the (i .375)(n .25) normal
quantiles for i 1, . . . , n with the n data values y(1), . . . , y(n). We then calculate the
value of the correlation coefficient, r, from the n pairs of values. To provide a more
definitive assessment of our level of certainty that the data were sampled from a normal distribution, we then obtain a value from Table 16 in the Appendix. This value,
called a p-value, can then be used along with the following criterion (Table 4.13) to
rate the degree of fit of the data to a normal distribution.
TABLE 4.13
Criteria for assessing fit
of normal distribution
p-value
p .01
.01 p .05
.05 p .10
.10 p .50
p .50
Assessment of Normality
Very poor fit
Poor fit
Acceptable fit
Good fit
Excellent fit
It is very important that the normal quantile plot accompany the calculation
of the correlation because large sample sizes may result in an assessment of a poor
fit when the graph would indicate otherwise. The following example will illustrate
the calculations involved in obtaining the correlation.
EXAMPLE 4.28
Consider the cholesterol data in Example 4.27. Calculate the correlation coefficient
and make a determination of the degree of fit of the data to a normal distribution.