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Ch. 2 – Modeling Distributions of Data Sec. 2.2 – Assessing Normality
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Ch. 2 – Modeling Distributions of Data
Sec. 2.2 – Assessing Normality
Assessing Normality
Later on in this course, we will be using various statistical inference procedures to answer questions. These tests involve sampling individuals and recording data to gain insights about the populations from which they come (INFERENCE!!!!!).
When using the procedures, we usually have to assume the population has an approximately Normal distribution.
Today, we look at a strategy for assessing Normality.
Example, p. 122
By plotting the data, we can see that the
distribution is roughly symmetric and unimodal.
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
Example, p. 122Using computer software, the following numerical summaries were found:
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
Example, p. 122
Counting the number observations within 1, 2, & 3 standard deviations
of the mean:
36/50 = 72%
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
Example, p. 122
Counting the number observations within 1, 2, & 3 standard deviations
of the mean:
48/50 = 96%
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
Example, p. 122
Counting the number observations within 1, 2, & 3 standard deviations
of the mean:
15.357 50/50 = 100%
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
Example, p. 122
36/50 = 72%
48/50 = 96%
15.357 50/50 = 100%
The percents are quite close to 68, 95, and 99.7.
4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7
6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0
8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9
9.1 9.2 9.5 9.6 9.6 9.7 10.2
10.3
10.5
10.6
10.6
10.8
10.9
11.1
11.5
12.3
12.3
12.3
12.7
14.7
Assessing Normality Just because a distribution looks Normal,
doesn’t mean we can say it is. 68-95-99.7 Rule can help to provide additional
evidence for or against Normality Normal Probability Plots can also be a useful tool
when assessing Normality.
Example, p. 123 – Normal Probability Plot How it’s made:
1. Arrange the observed data values form smallest to largest and record the percentile for each one.
2. Use the standard Normal distribution to find the z-scores at these same percentiles.
3. Plot each observation x against its expected z-score from Step 2.
Example, p. 123 – Normal Probability Plot
Interpreting a Normal Probability Plot
If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal.
Systematic deviations from a straight line indicate a non-Normal distribution.
Outliers appear as points that are far away from the overall pattern of the plot.
Example p. 124 – Guinea Pig Survival
Example p. 124 – Guinea Pig Survival
Homework: Due Tuesday P. 132 #65, 66, & 74
