Stat 401G
Lab 2
Solution
Last week's lab looked at the data on nonstructural carbohydrates (NSC) concentrations
(mg/g) in sapling stems in moist semi-evergreen and dry deciduous tropical forests in
Bolivia. The data are available on the course Web site:
http://www.public.iastate.edu/~wrstephe/stat401.html
Use JMP to look at the comparison of moist to dry forests. It is not sufficient to simply
hand in JMP output. You must answer the questions completely. Turn in the JMP output
attached at the end of your written (word processed) solutions.
1. Two-sample problem.
a) Summarize the NSC concentrations graphically and numerically for each type of
forest. How do dry forests compare to moist forests in terms of NSC
concentrations?
Distributions Forest=Moist
Distributions Forest=Dry
14
12
12
6
0
25
50
75
100
125
8
Count
8
6
4
4
2
2
150
0
25
NSC
maximum
quartile
median
quartile
minimum
Mean
Std Dev
N
75
100
125
150
NSC
Dry Forests
100.0%
75.0%
50.0%
25.0%
0.0%
50
Count
10
10
Moist Forests
92.000
62.000
52.000
42.000
19.000
52.965517
16.242543
29
100.0%
75.0%
50.0%
25.0%
0.0%
maximum
quartile
median
quartile
minimum
Mean
Std Dev
N
137.00
94.75
69.00
46.00
22.00
71.375
31.235817
40
1
Dry forests tend to have lower NSC concentrations than moist forests. This
is shown in the lower means (dry: 53.0 mg/g, moist: 71.4 mg/g) and lower
medians (dry: 52 mg/g, moist: 69 mg/g). The distribution of dry forest NSC
concentrations is fairly symmetric and mounded in the middle while the
distribution of moist forest NSC concentrations is skewed toward higher
values (skewed right).
b) Do the dry forests differ significantly from the moist forests in terms of average
NSC concentration? Perform the appropriate test of hypothesis. Use α=0.05.
H 0 : dry   moist
H A :  dry   moist
t  2.898
P  value  0.0051 *
Because the P-value is so small, we reject the null hypothesis. The difference
between the sample mean NSC concentration for the dry forest trees/shrubs
and that for the moist forest trees/shrubs is statistically significant.
c) Report the value of a 98% confidence interval for the difference between the
mean NSC concentration for dry and moist forests. Give an interpretation of this
interval use the interval to test to see if the difference in mean NSC concentrations
between dry and moist forests is statistically significant.
The 98% confidence interval for the difference between the mean NSC
concentration for dry and moist forests goes from 3.27 mg/g to 33.55 mg/g.
On average, trees/shrubs in moist forests have from 3.27 mg/g to 33.55 mg/g
higher concentrations of NSC than those in dry forests.
Because the confidence interval contains only positive values, it does not
contain zero, this indicates that there is a statistically significant difference
between the mean NSC concentrations for the two types of forest.
2
3
.99
2
.95
.90
1
.75
.50
0
Normal Quantile Plot
d) Use JMP to create the residuals for the two types of forest, dry and moist. Use
Analyze + Distribution to create JMP output that looks at the distribution
(histogram, box plot and Normal Quantile Plot) of the resulting 69 residuals.
.25
.10
.05
-1
-2
.01
-3
20
10
Count
15
5
-60
-40
-20
0
20
40
60
80
e) Refer to the histogram of residuals and indicate what you see and what this tells
you about the condition that the random errors are normally distributed.
The histogram of residuals is mounded between –20 and 0. There is a slight
skew toward larger values (skewed right). The skew is not as pronounced as
what was seen in the one-sample problem (Lab 1). With such a slight skew,
the random errors might still be normally distributed.
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f) Refer to the box plot of residuals. Are there any possible outliers? If so, what are
the corresponding NSC concentrations, species and type of forest? What does the
box plot tell you about the random errors being identically distributed?
There are no apparent outliers displayed in the box plot. This indicates that
the condition that random errors should be identically distributed is most
likely met.
g) Refer to the Normal Quantile Plot of residuals and indicate what you see and what
this tells you about the condition that the random errors are normally distributed?
Although there is a slight curve, the residuals basically follow the diagonal
line representing a normal model. This indicates that random errors could
be normally distributed.
h) Comment on whether or not the data are consistent with the condition that both
populations (dry and moist) have the same standard deviation. This will require
further analysis with JMP
80
60
Residual
(two-sample)
40
20
0
-20
-40
-60
Dry
Moist
Forest
The plot of residuals for the two types of forest shows that the dry forest
residuals are less spread out than those for the moist forest. The values for
sample standard deviations support this but the difference is probably not
great enough for us to worry.
Dry Forest
Std Dev = 16.243
Moist Forest
Std Dev = 31.236
4
i) Summarize what you have learned about the conditions necessary for statistical
analysis of data from your analysis of residuals.
The residuals for the two-sample problem are a little bit better behaved than
for the one-sample problem (Lab 1). The histogram and normal quantile
plot indicate that random errors could be normally distributed.
Additionally, because there are no outliers the condition of identically
distributed errors is most likely met. Although the sample standard
deviations are different (16.2 mg/g for dry forests and 31.2 mg/g for moist
forests) this is probably not a big enough difference to make us change our
conclusion that moist forests have a population mean NSC different from
that of dry forests.
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2. On a separate piece of paper write a brief summary of what you have learned about
NSC concentrations in tree/shrubs in moist and dry tropical forests.
Dry forests tend to have lower NSC concentrations than moist forests. This is
shown in the lower sample mean for dry forests, 53.0 mg/g, compared to that of
moist forests, 71.4 mg/g. There is also more variation in moist forests compared
to dry forests. This is supported by the larger interquartile range of 48.75 mg/g
for moist forests compared to 20 mg/g for dry forests. Also the sample standard
deviation for moist forests is 31.2 mg/g compared to 16.2 mg/g for dry forests.
The difference in sample mean NSC concentrations, 18.4 mg/g, is statistically
significant (P-value = 0.0051). With 98% confidence the population mean NSC
concentration for moist forests is from 3.27 mg/g to 33.55 mg/g higher than the
population mean NSC concentration for dry forests.
Although the conditions for statistical inference may not be met perfectly, slight
right skew in the shape of the distribution of residuals and the difference in
sample standard deviations, the statement that the population mean NSC for
moist forests is different from that of dry forests is still supported by the data.
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