Download HW #2 Keys

1
Short Course: Biostatistics in Practice Fall 2015
HW #2 (total 40 pts, Due by Oct 13)
Last Name: ____________________
First Name: _____________________
1. The following statements are based on the side-by-side box plot. Circle T (true) or F (false) in the following
set of statements. Each question is worth 2 points
(1) T
F
Parameters usually are used to describe the characteristics of the sample.
(2) T
F
Confidence intervals are p-values are inferential statistics.
(3) T
F
μ is often referred as a population mean.
(4) T
F
The sample mean is more sensitive to the extreme values than the sample median.
(5) T
F
When the data is skewed to the right, the median is greater than the mean when the
distribution is bell shaped.
2. Which of the following statements in NOT correct in regard to the normal distribution? (5pts)
(a) The normal distribution is bell shaped
(b) The normal distribution is symmetric around the mean
(c) If X are normally distributed with a mean of 135 and a stand deviation of 15, then Prob(X ≤ 105) = 0.05,
approximately.
(d) If X are normally distributed with a mean of 135 and a stand deviation of 15, then Prob(X ≤ 120) = 0.16,
approximately.
(e) None in the above
Answer: _____c_______
Mean ± 1 SD = [120, 150] thus 68% lies within this range that means 32% lies outside of this range. But the
normal distribution is symmetric around the mean thus, 16% is less than 120, 16% is greater than 150
Mean ± 2 SD = [105, 165] thus 95% lies within this range that means 5 % lies outside of this range. But the
normal distribution is symmetric around the mean thus, 2.5% is less than 105, 16% is greater than 165
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3. An experiment was conducted at the University of California-Berkeley to study the effect of psychological
environment on the anatomy of the brain. A group of 19 rats was randomly divided into two groups. Animals in
the treatment group lived together in a large cage furnished with playthings that were changed daily; animal in
the control group live in isolation with no toys. After a month, the experimental animals were sacrificed and
dissected to obtain the cortex weights (the thinking part of the brain) in milligrams. Using the posted data
(CortexWeight.syz), answer the following questions. Refer the posted MYSTAT guidance
(MYSTATguide.docx) for step by step procedures. (6 pts for each question)
(1) Compute basic descriptive statistics of the cortex weight by Group. Basic descriptive statistics should
include means, standard deviations, medians, and ranges. Run MYSTAT (Follow the first part of
instructions on MYSTATguide.docx). Cut and paste output from MYSTAT in the space below.
Results for GROUP$ = Treat
Cortex_Weight
N of Cases
12
Minimum
649.000
Maximum
749.000
Range
100.000
Median
703.000
Arithmetic Mean
701.917
95.0% Lower Confidence Limit 681.074
95.0% Upper Confidence Limit 722.759
Standard Deviation
32.804
Results for GROUP$ = Contr
Cortex_Weight
N of Cases
7
Minimum
627.000
Maximum
698.000
Range
71.000
Median
651.000
Arithmetic Mean
656.143
95.0% Lower Confidence Limit 635.357
95.0% Upper Confidence Limit 676.929
Standard Deviation
22.475
(2) Compute the 95% confidence intervals for the cortex weight by Group. Do you think these two
confidence intervals overlap each other? why or why not?
95% CI for Treatment = [681.074, 722.759]
95% CI for control = [635.357, 676.929]
Since the upper limit of the CI for control is less than the lower limit of CI for treatment, two confidence
intervals do not overlap each other.
(3) Based on the computed confidence intervals from (2), can you claim the true mean cortex weight of rats
that lived together in a large cage furnished with playthings is higher than the true mean cortex weight of
rats that live in isolation with no toys? Why or why not?
3
Based on 95% CI for the population mean cortex weight, the true mean cortex weight for control rats
can be as high as 678 but for rats in treatment, it is at least 681. Thus, we can conclude that the true
mean cortex weight of rats in treatment is higher than that of rats in control.
(4) Compute the sample difference in the mean cortex weight between two groups and the 95% confidence
interval for that mean difference. Provide one under the assumptions of difference variance and one
under the assumptions of the equal variance. (Follow the second part of instructions on
MYSTATguide.docx).
Two-sample t-test on Cortex_Weight Grouped by Group$ vs Alternative = 'not equal'
GROUP N Mean
Contr
Treat
Standard
Deviation
7 656.143 22.475
12 701.917 32.804
Separate Variance
Difference in Means
95.00% Confidence Interval
t
df
p-value
: -45.774
: -72.691 to -18.856
: -3.598
: 16.380
: 0.002
Pooled Variance
Difference in Means
95.00% Confidence Interval
t
df
p-value
: -45.774
: -75.448 to -16.100
: -3.254
: 17.000
: 0.005
Sample Mean difference = -45.774( control – treatment)
95% CI for the true mean difference = [-75,448, -16,10] when equal variance assumed, [-72.691, 18.856 ] when unequal variance assumed.
(5) Can you claim that rats’ brains are developed better when they live together in a large cage furnished
with playthings than rats without those things and live in isolation? Why or why not?
This study shows that the average cortex weight for rats with toys is higher by 16 ~18 mg at least and by
73 ~ 75 mg at most, compared with the one for rats without such environments. Thus, rats’ brains are
developed better in such a enriching environments and this can be applied to humans also.
(6) When you compare two groups, small p-values that are less than 0.05 mean that differences between
groups are statistically significant. Did you find statistically significant p-values for the comparison of
the mean cortex weight between two groups? Then report those p-values.
Since both p-values are less than 0.05 (0.002, 0.005), we find that the observed difference is statistically
significant.