Lab Day Exercises
The written portion of the lab day exercises is presented in the first pages and the
accompanying tables from which the conclusions have been drawn are presented in the appendix.
Each problem is referenced by the number appearing in the assignment, and each table’s title
includes the problem number for easy reference.
1a: Table 1 is a frequency table generated from the dataset UCDAVIS2a showing the
frequency of preference for sitting in the front or back of a classroom amongst students. From
Table 1 it can be seen that 53.54 percent of the 99 students surveyed stated a preference for
sitting in the front, while 46.46 percent preferred sitting in back. Table 2 includes both this
proportion and the 99% confidence interval for the proportion of those preferring to sit in the
front. The interval in Table 2 indicates that we can have 99% confidence that the true percentage
of students at UC Davis who prefer to sit in the front of a classroom is between 40.62 percent
and 66.45 percent.
1b: Table 3 is a two-way frequency table also generated from the dataset UCDAVIS2a,
and shows the percent of respondents in the study who favor sitting in the front or back of the
classroom subdivided by gender. In Table 3 it can be seen that 71.70 percent of the students who
preferred to sit in the front were females, while 47.83 percent of students who preferred the back
were females. Table 4 shows the proportions of students who preferred to sit in the front or back
and were female. Also 99% confidence intervals for the proportions and for the difference in the
proportion of people who sit in the front vs. the back that are female are shown. The confidence
interval for the difference ranges from -.91% to 48.65%. Since this confidence interval includes
0% as a possibility in the difference we cannot conclude that there is a statistical difference
between students who chose to sit in the front and are female and those who chose to sit in the
back and are female.
1c: Table 3 indicates that of the 60 females surveyed, 63.33% prefer to sit in the front.
1d: Table 3 also indicates that of the 46 students in the survey who preferred to sit in
back, 52.17% were male.
2a: Table 5 is generated from the Pulsemarch dataset and contains descriptive statistics
for the pulse rates of 40 march participants before the march. The mean pulse rate of participants
before the march was 69.25 and the 90% confidence interval indicates that we can have 90%
confidence that the true mean of pulse rates for those who engage in similar marches is between
66.63 and 71.87.
2b: Table 6 includes descriptive statistics for the pulse rates of march participants both
before and after the march, and the difference in the two. It can be seen that the mean before
march pulse rate for all participants was 69.25. The 90% confidence interval around the mean
before pulse rate is from 66.63 to 71.87. The mean after march pulse rate was 79.68 and the 90
% confidence interval around the mean after march pulse rate is from 76.65 to 82.70. There is no
overlap in the before and after confidence intervals for the mean pulse rate, suggesting there is a
statistically significant difference in the two. Table 6 also indicates that the difference in mean
pulse rates, i.e. before march – after march, for all participants is -10.43; or an increase in pulse
rate of 10.43. The 90% confidence interval for the difference in the mean before and after pulse
rates indicates that we can have 90% confidence that the true mean after march pulse rate for the
population of concern is between 8.50 to 12.35 higher than the before march pulse rate.
2c: Table 7 contains the descriptive statistics for the after march pulse rates, comparing
female and male participants. The mean pulse rate for the female participants was 82.00 and
78.125 for males. The difference in the mean after march pulse rates was 3.875 with females
being higher. Table 8 contains the 90% confidence intervals for the mean after march pulse rate
for females, males, and the difference in females and males. We can have 90% confidence that
the true mean after march pulse rate for females is between 75.83 and 88.17. We can also be
90% confident that the true mean after march pulse rate for males is between 74.95 and 81.30.
There is significant overlap in the confidence intervals for females and males and suggests there
may not be a statistically significant difference in the after march pulse rates between females
and males. Table 9 indicates the p value for the mean female vs. male after march pulse rate is
greater than .05 and therefore the pooled difference in means was used to determine the 90%
confidence interval for the difference in means. The 90% confidence interval around the mean of
the difference in after march pulse rates for females and males ranges from -2.2842 to 10.0342.
As this interval includes the possibility 0 difference, we cannot conclude that there is a
statistically significant difference in the after march pulse rates between females and males.
3: The margin of error contained in the 90% confidence interval for the mean before
march pulse rate is determined by:
MOE = t * s/(√n)
in this case: 2.62 = 1.684 * 9.85/(√40).
If we needed to retain the same margin of error, but have a 99% confidence interval we would
need to change the t in the calculation to 2.704. (This is a conservative value for t. We know that
in order to increase the confidence level we will have to increase the sample size, and as the
sample size increases, the degrees of freedom associated with t increase. Therefore the true value
of t would be somewhat smaller than 2.704 with a larger sample size, but as we cannot know the
actual t value to use until the sample size is known we can use 2.704 as an estimate.) The
calculation to determine the new sample size with the same margin of error is:
N = (t * s/ MOE)2
in this case: N = (2.704 * 9.85/2.62)2 ≈ 104
Therefore in order to achieve a 99% confidence level we would need a sample size of at least
104.
Appendix I: Tables
Table 1: Frequency of Students Who Sit in the Front and Back Seats (1.a)
Seat
Cumulative Cumulative
Frequency Percent Frequency
Percent
Front
53
53.54
53
53.54
Back
46
46.46
99
100.00
Table 2: 99% Confidence Interval of the Proportion of Students Who Sit in the Front (1.a)
Binomial Proportion
Seat = Front
Proportion
0.5354
ASE
0.0501
99% Lower Confidence Limit
0.4062
99% Upper Confidence Limit
0.6645
Exact Confidence Limits
99% Lower Confidence Limit
0.4021
99% Upper Confidence Limit
0.6651
Table 3: Two-Way Frequency Table of Seat Choice by Sex (1.b)
Table of Seat by SEX
Seat
Frequency
Percent
Row Percent
Column Percent
SEX
Female Male Total
Front
38
38.38
71.70
63.33
15
15.15
28.30
38.46
53
53.54
Back
22
22.22
47.83
36.67
24
24.24
52.17
61.54
46
46.46
Total
60
60.61
39
99
39.39 100.00
Table 4: 99% Confidence Intervals of the Proportion of Female Students who sit in the Front vs. the Back and a 99%
Confidence Interval for the Difference in the Proportions (1.b)
Female Risk Estimates
Proportion
(Asymptotic) 99%
ASE Confidence Limits
(Exact) 99%
Confidence Limits
Front
0.7170 0.0619
0.5576
0.8764
0.5339
0.8600
Back
0.4783 0.0737
0.2885
0.6680
0.2886
0.6725
Total
0.6061 0.0491
0.4796
0.7326
0.4719
0.7298
Difference
0.2387 0.0962
-0.0091
0.4865
Difference is (Front - Back)
Table 5: 90% Confidence Interval for the Mean Pulse Rate for All Participants before the March (2.a)
Analysis Variable : BEFORE
Standard Lower 90% Upper 90%
Number Minimum Maximum Mean Median Deviation CL for Mean CL for Mean
40
46.00
96.00
69.25
68.00
9.85
66.63
71.87
Table 6: 90% Confidence Intervals for the Mean Before and After Pulse Rates and the Mean Difference in the
Before and After Pulse Rates for each Participant (2.b)
Variable
Standard
N Minimum Maximum Mean Median Deviation
BEFORE
AFTER
Difference
40
40
40
46.00
60.00
-30.00
96.00 69.25
104.00 79.68
4.00 -10.43
68.00
80.00
-9.50
9.85
11.34
7.22
Lower 90% Upper 90%
CL for Mean CL for Mean
66.63
76.65
-12.35
71.87
82.70
-8.50
Table 7: The Mean of the Difference in Pulse Rates before and after the March for Females and for Males, and the
Difference of the Means of the Females vs. the Males (2.c)
SEX
N
Mean
Standard
Deviation
Standard
Error
Minimum
Maximum
Female
16
-13.3750
8.5703
2.1426
-30.0000
-4.0000
Male
24
-8.4583
5.5245
1.1277
-21.0000
4.0000
-4.9167
6.8896
2.2236
Difference (1-2)
Table 8: 90% Confidence Level for the Difference of the Means of the before vs. after pulse rates of Females vs.
Males (2.c)
SEX
Method
Female
Male
Mean
90% Confidence
Level
Mean
Standard
Deviation
90% CL
Standard
Deviation
-13.3750
-17.1310
-9.6190
8.5703 6.6391 12.3181
-8.4583
-10.3910
-6.5256
5.5245 4.4674
7.3228
6.8896 5.8127
8.5138
Difference (1-2)
Pooled
-4.9167
-8.6655
-1.1678
Difference (1-2)
Satterthwaite
-4.9167
-9.0641
-0.7692
Table 9: P Value for Determining Whether to Use Pooled or Sattethwaite Difference (2.c)
Equality of Variances
Method
Folded F
Number DF
15
Den DF
F Value
23
2.41
Pr > F
0.0564