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Unformatted text preview: STAT 350 – Spring 2009 Lab 7 SOLUTION
Probability & QQ Plots Var 1 FYI the actual distribution of this data is ~Normal(µ = 40, σ = 6)
35 50 30 45 25 40
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uant i l es a. Looking just at the histogram, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) This one is sort of hard to say. It looks normal,
except there is some evidence of being negatively skewed, but this could be an artifact of small
sample sizes. Personally, I probably wouldn’t say this is conclusively NOT normal.
b. Looking just at the QQ plot, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) This one looks pretty normal. The plotted points
follow the normal line with apparently random variation.
c. According to the ShapiroWilk Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p = 0.8529)
d. According to the KolmogorovSmirnov Test for Normality, can we say this sample came from a
normal distribution? Include the pvalue in your response.
Yes, you may assume normality (p > 0.1500)
e. According to the Cramervon Mises Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p > 0.2500)
f. According to the AndersonDarling Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p > 0.2500) Lab #7  Solution Page 1 Var 2 FYI the actual distribution of this data is ~Exponential(λ = 1/20)
40 25 35 20
30 25
15
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uant i l es a. Looking just at the histogram, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) To me, this one appears too skewed to be normal.
b. Looking just at the QQ plot, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) Just looking at the probability plot here, I don’t see
any reason to not believe this sample came from a normal distribution. The points generally seem to
follow the lines with only random variation. There are 4 points in a row below the line, but with such
a small sample size, it’s hard to tell if this is an artifact (“sampling error”) or actually representative of
the true distribution.
c. According to the ShapiroWilk Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p = 0.7336)
d. According to the KolmogorovSmirnov Test for Normality, can we say this sample came from a
normal distribution? Include the pvalue in your response.
Yes, you may assume normality (p > 0.1500)
e. According to the Cramervon Mises Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p > 0.2500)
f. According to the AndersonDarling Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p > 0.2500) Lab #7  Solution Page 2 Var 3 FYI the actual distribution of this data is ~Binomial(n = 20, π = 0.25)
Recall that if nπ > 5 and n(1π) > 5, then the distribution is approximately normal.
Here nπ = 5, so the approximation is borderline appropriate.
35 10 30 8 25 6
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0. 75 2. 25 3. 75 5. 25
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uant i l es a. Looking just at the histogram, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) I would feel comfortable saying this one is normal
just looking at the histogram. There is something kind of funky going on in the middle, but had I used
different cutpoints (or midpoints) that apparent bimodality might go away.
b. Looking just at the QQ plot, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) No – this does not look normal! There is definitely
something other than random deviations from the line going on here. That pattern of horizontal lines
is due to the fact that the data are all integers. It could be that the data are truly inherently discrete, or
it could be that someone rounded to the nearest whole number to obtain the data.
c. According to the ShapiroWilk Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
No, you may NOT assume normality (p = 0.0151)
d. According to the KolmogorovSmirnov Test for Normality, can we say this sample came from a
normal distribution? Include the pvalue in your response.
No, you may NOT assume normality (p < 0.0100)
e. According to the Cramervon Mises Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
No, you may NOT assume normality (p < 0.0050)
f. According to the AndersonDarling Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
No, you may NOT assume normality (p < 0.0050) Lab #7  Solution Page 3 Var 4 FYI the actual distribution of this data is ~Exponential(λ = 1/20)
50 125 40 100 30 75 P
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7. 5 22. 5 37. 5 52. 5 67. 5
x4 82. 5 97. 5 112. 5 3 2 1 0
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uant i l es a. Looking just at the histogram, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) Definitely not, the data is clearly positively skewed.
b. Looking just at the QQ plot, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) No, there is clear evidence of skewness in the QQ
plot. Starting from the left (bottom), all the points are above the line. In the middle all the points are
above the line. Finally all the remaining points are above the line. We clearly have something other
than random deviations from the line here.
c. According to the ShapiroWilk Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
No, you may NOT assume normality (p < 0.0001)
d. According to the KolmogorovSmirnov Test for Normality, can we say this sample came from a
normal distribution? Include the pvalue in your response.
No, you may NOT assume normality (p < 0.0100)
e. According to the Cramervon Mises Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
No, you may NOT assume normality (p < 0.0050)
f. According to the AndersonDarling Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
No, you may NOT assume normality (p < 0.0050) Lab #7  Solution Page 4 Var 5 FYI the actual distribution of this data is ~Uniform[0, 20]
35 30 17. 5 25 P
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uant i l es a. Looking just at the histogram, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) Well, it doesn’t look normal, but there are so few
points, it’s hard to say conclusively.
b. Looking just at the QQ plot, would you feel comfortable saying this sample came from a normal
distribution? Justify/explain your decision. (There really is no right or wrong answer – this is a
judgment call, I just want you to think about it.) Just looking at the QQplot, there is no strong
evidence that the data are not normally distributed.
c. According to the ShapiroWilk Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p = 0.2511)
d. According to the KolmogorovSmirnov Test for Normality, can we say this sample came from a
normal distribution? Include the pvalue in your response.
Yes, you may assume normality (p = 0.0780)
e. According to the Cramervon Mises Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p = 0.1868)
f. According to the AndersonDarling Test for Normality, can we say this sample came from a normal
distribution? Include the pvalue in your response.
Yes, you may assume normality (p = 0.2120) Lab #7  Solution Page 5 ...
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 Spring '08
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 Probability

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