In this assignment we consider data that examines the effect of two soporific drugs, drugs that induce sleep. These two

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In this assignment we consider data that examines the effect of
two soporific drugs, drugs that induce sleep. These two drugs were
tested on a group of 10 patients. For each of the patients the
increase in hours of sleep was measured both for drug 1 and for
drug 2. The source of the data is the R dataset "sleep", which
quotes the paper by Cushny, A.R. and Peebles, A.R. (1905). (The
action of optical isomers. II. Hyoscines. J. Physiol. 32, 501-510.)
The files "sleep_paired.csv" and
"sleep_unpaired.csv" contain the same data in two
different formats. The first format will be used in the first part
of the assignment and the second format will be used in th second
part. You may download the files sleep_paired.csv and
sleep_unpaired.csv using the links.

A Paired Design
A paired design corresponds to the situation where two different
treatments are given to the same subject and the goal is to assess
the difference between the response to the two treatments. This
design was used in the original experiment. The file
"sleep_paired.csv" reflects this design. It contains two variables
and 10 observations. The variables are:
Save this data set on your computer and read it into R. Compute
the difference between the increase in the first drug and the
increase in the second drug (Hint: If you saved the data in an
object by the name "paired" then you may use the code "d <-
paired$drug1 - paired$drug2" in order to produce the difference.)
In Tasks 1-5 you are asked to examine the distribution of this
difference, apply the t-test and compute a confidence interval for
the mean. For that you may produce a box plot of the difference,
compute the mean and standard deviation, obtain the percentiles of
the t-distribution, and apply the function "t.test".

An Unpaired Design
An unpaired design corresponds to the situation where each
treatments is given to a different subject. The goal is to assess
the difference between the response of the group that obtained the
first treatment and the response of the group that obtained the
second treatment. This design is discussed in the next unit. Part
of that discussion involves the construction of a statistical test
that is based on the t-distribution to examine expected difference
of response between the two treatments. The goal in this part of
the assignment is to develop a different type of test,
called The Permutation Test, to carry out the
same task. As an exercise we will use the same data, but this time
treat it as if each of the drugs were given to a different group of
10 patients. The file "sleep_unpaired.csv" reflects this
(inaccurate) assumption. It contains two variables and 20
observations. The variables are:
Save this data set on your computer, read it into R and apply
the permutation test in order to test the null hypothesis that the
expected increase in hours of sleep in the first group is equal to
the expected increase in the second group. The test is carried out
by the computation of a test statistic and the computation of a
p-value, which corresponds to the probability of obtaining by
random chance outcomes that are more extreme than the computed
statistic.

The test statistic is the absolute value of the difference between
the average of the variable extra for the first 10 observation and
the average for the last 10 observations. The sampling distribution
in the permutation test corresponds to a random assignment of
responses to treatment. Therefore, in order to simulate the
sampling distribution of the statistic the responses are randomly
reordered and the same statistic is computed to the reordered
data.

Specifically, say the data is saved in an object by the name
"unpaired" and the observed response is given in an object by the
name x (using the code "x <- unpaired$extra"). Then the
statistic is computed with th expression
"abs(mean(x[1:10])-mean(x[11:20]))". On the other hand, the
sampling distribution of the statistic is obtained by a random
permutation of the values of x and the application of the same
formula to the permuted values. Repeating this procedure for a
large number of random permutations produces an approximation of
the sampling distribution of the test statistic under the null
hypothesis. (Hint:: An object "X" that contains a random
permutation of the values of x may be obtained using the expression
"X <- sample(x)".)
Submitting the Assignment
For the assignment you should complete the following 8 tasks.
Tasks 1-5 refer to the the paired
design and Tasks 6-8 refer to
the unpaired design. Your answers should be
short and clear. We recommend that you copy and paste the tasks
below into the form titled "Submit your Assignment using this
Form". You can then write you answers to the tasks in the
designated positions that are marked in the text:

Tasks

A Paired Design:
1. The number of outlier observations in the difference between the
response to drug 1 and the response to drug 2 is: _____.
Explain each step in the computation of the number of outlier
observations:

2. The percentile of the t-distribution that should be used in
order to compute an 80%-confidence interval for the expectation of
the difference between the responses to the two drugs is (write the
numerical value): _____.
Attach the R code for conducting the computation:

3. The 80%-confidence interval for the expectation of the
difference between the responses to the two drugs is:
Lower end = _____, Upper end = _____.
Explain each step in the computation of the confidence
interval:

4. The p-value for testing the null hypothesis that the expected
difference is equal to 0 versus the two-sided alternative is equal
to: _____.
Attach the R code for computing the p-value:

5. Do you reject the null hypothesis with a significance level of
5%? __Yes __No.
Explain your choice:

An Unpaired Design:
6. The test statistic for the permutation test is the absolute
value of the difference between the average of the first 10
observations and the average of the last 10 observations. The value
of the test statistic is:_____.
Attach the R code for computing the test statistic:

7. Run a simulation to compute the p-value of the permutation test.
The p-value is:_____.
Attach the R code for computing the p-value:

8. Do you reject the null hypothesis with a significance level of
5%? __Yes __No.
Explain your choice: