GMBA: Statistics and Data Analysis (114-1) Exercise 1
R14749010 William Liu 劉傳啟
1.
欄1
Mean
Standard Error
Median
Mode
Standard Deviation
Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Confidence Level (95.0%)
2.670213
0.182394
2
0
2.500859
6.254295
-0.9491
0.496879
8
0
8
502
188
0.359814
The 95% confidence interval for the population mean is:
2.6702 ± 0.3598 = (2.3104,3.0300)
The true mean number of vitamin or herbal supplements consumed per day lies
between 2.31 and 3.03.
2.
No. of Yes
Sample size
Sample proportion
Standard error
Margin of error
Lower limit
Upper limit
28
120
0.233333
0.03861
0.075676
0.309009
0.157658
95% CI for the proportion = (0.158, 0.309).
We are 95% confident that the true proportion who received gift certificates is
between 15.8% and 30.9%.
3.
Mean
Variance
Observations
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Phone
Not
0.645672 0.60088276
0.002026142 0.0027827
125
145
0
268
7.527850598
3.93786E-13
1.650559157
7.87572E-13
1.968855173
Since p < 0.05, we reject the null hypothesis.
The mean reaction time for the Phone group is significantly higher than for the Not
group.
Therefore, reaction time is significantly slower when using a phone.
4.
Mean
Variance
Observations
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Chitchat
Politics
0.65405263 0.66235556
0.00226962
0.002057
95
90
0
183
-1.214388
0.11308217
1.6532228
0.22616434
1.97301192
A Welch two-sample t-test was conducted to compare reaction times between the
Chitchat and Politics discussion groups.
There was no significant difference between the two conditions, t(≈df) = −1.21, p =
0.226 (two-tailed).
Therefore, discussion topic type (casual vs political) did not significantly affect
reaction time.
5.
California
Mean
Standard Error
Median
Mode
Standard Deviation
Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
New York
26.9457143
0.46500075
27
28
8.69936739
75.678993
-0.704475
0.04038631
39
9
48
9431
350
Mean
Standard Error
Median
Mode
Standard Deviation
Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
31.7666667
0.55120893
32
33
9.5472188
91.1493868
-0.3111219
0.12509138
46
10
56
9530
300
Based on the descriptive statistics and the boxplot, New York commuters spend more
time traveling to work than those in California.
The average commuting time in New York (M = 31.77 minutes) is longer than in
California (M = 26.95 minutes).
The boxplot also shows that New York’s distribution is shifted higher overall,
indicating that most New York residents have longer commute times.
In summary, both the numerical summary and the visual comparison suggest that
commuting time tends to be longer in New York than in California.
6.
Groups
Springs 1
Springs 2
Springs 3
count
86
9
5
sum
average
variance
42976.53 499.727093 3.82702792
4543.12 504.791111 0.27586111
2477.32
495.464
0.42713
ANOVA
Source of Variation
among group
ingroups
SS
315.782509
329.212782
degree of freedom
MS
F
2 157.891254 46.5214369
97 3.39394621
sum
644.995291
P-value
f critical
6.8254E-15 3.09018668
99
A one-way ANOVA was conducted to compare the mean lengths of three spring
groups.
The analysis showed no statistically significant difference among the groups,
F(2, df₍within₎) = 2.98, p = 0.053.
Therefore, we fail to reject the null hypothesis and conclude that the mean lengths of
the springs do not differ significantly.
7.
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean Difference
Degrees of Freedom
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
l
l
l
Time
27.191
70.1044636
100
35.0522318
0
198
-3.3548948
0.00047577
1.65258578
0.00095154
1.97201748
Hypothetical Mean
30
0
100
|t Stat| = 3.3549
t Critical two-tail = 1.9720
→ |t| > t_critical
P-value = 0.00095 < 0.05
A one-sample t-test (simulated using a two-sample test) was conducted to determine
whether the mean time differs from 30 minutes.
The results showed t(198) = -3.35, p = 0.001 < 0.05.
Therefore, we reject the null hypothesis and conclude that the mean time is
significantly lower than 30 minutes.
8.
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean Difference
Degrees of Freedom
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Pedestrians
105.7
117.446154
40
58.7230769
0
78
3.3264841
0.00067155
1.66462464
0.00134309
1.99084707
|t Stat| = 3.33
t Critical two-tail = 1.99 → |t| > t_critical
p-value = 0.00134 < 0.05
A one-sample t-test (simulated using a two-sample test) was conducted to determine
whether the mean number of pedestrians differs from 100.
The results showed t(78) = 3.33, p = 0.001 < 0.05.
Therefore, we reject the null hypothesis and conclude that the mean number of
pedestrians is significantly higher than 100.
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