Stat 301 -- Fall 2015 -- Midterm exam 1
Information and JMP output
Problem 1. Pace of life
Popular belief is that the pace of life differs between U.S. cities. You may be familiar with the
stereotypes: Los Angeles is a “laid back” city, with a slower pace of life; New York City is “hurry up” city
with a faster pace of life. The following data were collected to quantify that pace of life. The
investigators randomly chose 36 US cities from a list of all U.S. cities with populations larger than
100,000 people. Two of the quantities they measured for each city are:
1. walk: the average walking speed of pedestrians in the business district
2. talk: the average number of words per minute spoken by postal clerks
The JMP output includes:
1. walk: The histogram and box plot from Analyze / Distribution.
2. talk: Output from Analyze / Distribution although some numbers have been removed
walk:
-1-
talk:
Quantiles
100.0%
90.0%
75.0%
50.0%
25.0%
10.0%
0.0%
maximum
minimum
27
24.3
23.75
22
18
14.7
10
Summary Statistics
Mean
Std Dev
Std Err Mean
Upper 95% Mean
Lower 95% Mean
N
20.583333
3.8572751
0.6428792
21.888447
19.278219
36
-2-
Test Mean
Hypothesized Value
Actual Estimate
DF
Std Dev
20
20.5833
35
3.85728
t Test
Test Statistic
Prob > |t|
Prob > t
Prob < t
0.3704
0.1852
0.8148
Test Mean
Hypothesized Value
Actual Estimate
DF
Std Dev
22
20.5833
35
3.85728
t Test
Test Statistic
Prob > |t|
Prob > t
Prob < t
0.0342*
0.9829
0.0171*
Test Mean
Hypothesized Value
Actual Estimate
DF
Std Dev
24
20.5833
35
3.85728
t Test
Test Statistic
Prob > |t|
Prob > t
Prob < t
<.0001*
1.0000
<.0001*
-3-
Problem 2: Music and attention span
These data are part of a study of whether music is a potential distraction. Subjects were asked to pay
attention to a sequence of simple tasks. An eye tracking device was used to determine when their
attention wandered. The response is the time in minutes when the subject’s attention wandered. This
measurement was made when a subject was listening to rap music through headphones (the Music
group) or to silence while wearing the headphones (the Control group).
Subjects were randomly assigned to group (Music or Control). There are 22 subjects are in the Music
group and 28 are in the Control group, for a total of 50 subjects.
The JMP output includes, in order:
results from a two-sample t-test assuming equal variances (Fit Y by X / Fit Line / ANOVA Pooled T)
information about the means and standard deviations for each group (Fit Y by X / Fit Line / Means)
results from a two-sample t-test assuming unequal variances (Fit Y by X / Fit Line / T-test)
Certain numbers have been deleted.
Oneway Analysis of AttTime By Group
Oneway Anova
Summary of Fit
Rsquare
Adj Rsquare
Mean of Response
Observations (or Sum Wgts)
0.179816
0.162729
4.1696
50
-4-
t Test
Music-Control
Assuming equal variances
Difference
Std Err Dif
Upper CL Dif
Lower CL Dif
Confidence
-2.2688
0.6994
-0.8626
-3.6750
0.95
t Ratio
-3.24399
Prob > |t|
Prob > t
Prob < t
0.0021*
0.9989
0.0011*
Analysis of Variance
Source
Group
Error
C. Total
DF Sum of Squares
1
63.41474
48
289.24945
49
352.66419
Mean Square
63.4147
6.0260
F Ratio
10.5235
Prob > F
0.0021*
Means for Oneway Anova
Level
Control
Music
Number
28
22
Mean
5.16786
2.89909
Std Error
0.46391
0.52336
Lower 95%
4.2351
1.8468
Upper 95%
6.1006
3.9514
Std Error uses a pooled estimate of error variance
Means and Std Deviations
Level
Control
Music
Number
Mean
Std Dev
28
22
5.16786
2.89909
2.60065
2.25344
Std Err
Mean
0.49148
0.48044
t Test
Music-Control
Assuming unequal variances
Difference
Std Err Dif
Upper CL Dif
Lower CL Dif
Confidence
-2.2688
0.6873
-0.8865
-3.6510
0.95
t Ratio
DF
Prob > |t|
Prob > t
Prob < t
-3.30103
47.49524
0.0018*
0.9991
0.0009*
-5-
Lower 95%
Upper 95%
4.1594
1.9000
6.1763
3.8982
Problem 3: diversity of butterfly species in remnant patches of tropical rain forest
The data for this question come from one of the first experimental studies of the consequences of forest
fragmentation. A logger agreed to leave patches of forest uncut inside a large area that was to be
logged. The experimenters identified 16 locations were a patch would be uncut. The treatment was the
size of the uncut patch, which was randomly assigned to locations. The patches ranged in size from 1
hectare (ha) to 1000 ha. Two years after logging, each patch was surveyed for butterflies. The response
variable is the number of different butterfly species found on the patch. Theory suggests that diversity
is linearly related to the log of the patch area. The variable log Area is the area in log base 10 (so a 1 ha
patch is LogArea = 0, a 10 ha patch is LogArea=1, a 100ha patch is LogArea = 2 and a 1000ha patch is
LogArea = 3).
The JMP output is all computed with Analyze / Fit Y by X / Fit Line. That output includes
a plot of the data with the fitted regression line
information about the fit of the line
parameter estimates
The output is followed by a copy of the data table with:
predicted values,
the columns created by Mean Confidence Interval Formula and
the columns created by Individ Confidence Interval Formula.
Information for “new” observations with areas of 25 ha and 250 ha is included after the observations.
Bivariate Fit of species By log Area
Linear Fit
species = 36.25 + 28.5*log Area
-6-
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.589594
0.560279
23.77799
64.75
16
Analysis of Variance
Source
Model
Error
C. Total
DF Sum of Squares
1
11371.500
14
7915.500
15
19287.000
Mean Square
11371.5
565.4
F Ratio
20.1126
Prob > F
0.0005*
Parameter Estimates
Term
Intercept
log Area
Estimate
36.25
28.5
Std Error
8.701854
6.354935
t Ratio
4.17
4.48
Prob>|t|
0.0010*
0.0005*
Lower 95%
17.58638
14.870019
Data Table with predicted values and intervals
area species log Area Predicted
# species
1
14
0
36.25
1
50
0
36.25
1
55
0
36.25
1
34
0
36.25
1
40
0
36.25
1
57
0
36.25
10
43
1
64.75
10
103
1
64.75
10
33
1
64.75
10
53
1
64.75
10
50
1
64.75
100
110
2
93.25
100
70
2
93.25
100
119
2
93.25
100
60
2
93.25
1000 145
3
121.75
25
.
1.39 76.09
250
.
2.39 104.59
StdErr
Pred. species
8.70
8.70
8.70
8.70
8.70
8.70
5.94
5.94
5.94
5.94
5.94
8.70
8.70
8.70
8.70
14.03
6.46
10.68
-7-
95% for the line
Lower Upper
17.58 54.91
17.58 54.91
17.58 54.91
17.58 54.91
17.58 54.91
17.58 54.91
52.00 77.49
52.00 77.49
52.00 77.49
52.00 77.49
52.00 77.49
74.58 111.91
74.58 111.91
74.58 111.91
74.58 111.91
91.65 151.84
62.23 89.94
81.66 127.51
95% Individ obs
Lower Upper
-18.05 90.55
-18.05 90.55
-18.05 90.55
-18.05 90.55
-18.05 90.55
-18.05 90.55
12.18 117.31
12.18 117.31
12.18 117.31
12.18 117.31
12.18 117.31
38.94 147.55
38.94 147.55
38.94 147.55
38.94 147.55
62.53 180.96
23.24 128.93
48.67 160.50
Upper 95%
54.91362
42.129981