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 \begin{center}
  {\Large{\bf {Lab Assignments SBDA131C}}}\\
  Dated: {13-10-17}
 \end{center}

\rightline{Due date: 02 December}
\begin{enumerate}
\item A sample of 25 physically and mentally healthy males participated in a sleep experiment in which the
percentage of each participant’s total sleeping time spent in a certain stage of sleep was recorded. The
variance computed from the sample data was 2.25. Construct 95 percent confidence intervals for $\sigma^2$
and $\sigma$.

Find the following integral numerically
using Trapezoidal rule and compare the value obtained analytically. Obtain the same integral using
Simpson’s $1/3$ rule and compare.

\[ \int_0^1 \frac{3}{1+x^2} dx\]
Write down a routine program for obtaining the integrals. 

\item Compare Euler mid-point and Runge-Kutta method of solving first order differential equations. Solve the following first order differential equation using both methods and compare:

\[\frac{dy}{dx}=tan(y), \qquad y(0)=1.\]

Write computer routines.

\item Use prediction-correction method to solve the following equation:

\[\frac{dy}{dx}=-y^2, \qquad y(0)=1.\]

Write a computer routine.

Krantz et al. investigated dose-related effects of methadone in subjects with torsade
de pointes, a polymorphic ventricular tachycardia. In the study of 17 subjects, nine were being
treated with methadone for opiate dependency and eight for chronic pain. The mean daily
dose of methadone in the opiate dependency group was 541 mg/day with a standard deviation of
156, while the chronic pain group received a mean dose of 269 mg/day with a standard deviation
of 316. Compute a 95\% confidence interval for the difference of the means and clearly state your assumptions. 

\item Explain what is maximum likelihood estimation. 

Use Simpson's $3/8$ rule to obtain the following integral numerically and compare with Simpson's 1/3 rule.. Write down a routine for obtaining the integral. 

\[\int_0^1 \sqrt{x} dx\]


\item The purpose of a study by Moneim et al. was to examine thumb amputations from team roping
at rodeos. The researchers reviewed 16 cases of thumb amputations. Of these, 11 were complete
amputations while five were incomplete. The ischemia time is the length of time that insufficient
oxygen is supplied to the amputated thumb. The ischemia times (hours) for 11 subjects experiencing
complete amputations were\\

4.67; 10.5; 2.0; 3.18; 4.00; 3.5; 3.33; 5.32; 2.0; 4.25; 6.0\\

For five victims of incomplete thumb amputation, the ischemia times were\\

3.0; 10.25; 1.5; 5.22; 5.0

Treat the two reported sets of data as sample data from the two populations as described.
Construct a 95 percent confidence interval for the ratio of the two unknown population
variances.



\item A study is conducted concerning the blood pressure of 60 year old women
with glaucoma. In the study 200 60-year old women with glaucoma are
randomly selected and the sample mean systolic blood pressure is 140 mm
Hg and the sample standard deviation is 25 mm Hg.\\

a. Calculate a 95\% confidence interval for the true mean systolic blood
pressure among the population of 60 year old women with glaucoma.\\

b. Suppose the study above was based on 100 women instead of 200 but
the sample mean (140) and standard deviation (25) are the same.
Recalculate the 95\% confidence interval. Does the interval get wider
or
narrower? Why?

Solve the following equation using Gauss-elimination method. Write down a computer code for the method.


\begin{eqnarray*}
–3x + 2y -6z &=& 6\\
      5x + 7y - 5z &=& 6\\
      x + 4y - 2z &=& 8 \end{eqnarray*}

 
\item The following scores represent a nurse’s assessment (X) and a physician’s assessment (Y) of the
condition of 10 patients at time of admission to a trauma center.\\
X : 18~ 13~ 18~ 15~ 10~ 12~ 08~ 04~ 07~ 03\\

Y : 23~ 20~ 18~ 16~ 14~ 11~ 10~ 07~ 06~ 04\\

(a) Construct a scatter diagram for these data.
(b) Deduce the best curve that fits the data. Try to use Computer for determining the least square fit.  Conduct hypothesis test for reaching a conclusion regarding the relationship between X and Y. 

\item Methadone is often prescribed in the treatment of opioid addiction and chronic pain. Krantz et al.
studied the relationship between dose of methadone and the corrected QT (QTc) interval (shown below) for
17 subjects who developed torsade de pointes (ventricular tachycardia nearly always due to
medications). QTc is calculated from an electrocardiogram and is measured in mm/sec. A higher
QTc value indicates a higher risk of cardiovascular mortality. A question of interest is how well
one can predict and estimate the QTc value from a knowledge of methadone dose. Answer it by means of regression analysis. Draw scatter diagram.

Methadone dose (mg/day): 1000~550~97~90~85~126~300~110~65~650~600~660~270~680~540~600~330\\
QTc (mm/sec): 600~625
560
585
590
500
700
570
540
785
765
611
600
625
650
635
522\\
a) Compute the coefficient of determination.
%(b) Prepare an ANOVA table and use the F statistic to test the null hypothesis that b 1 1⁄4 0. Let
%a% 1⁄4 :05.
(b) Use the t statistic to test the null hypothesis that $\beta_1=0$ at the .05 level of significance.
(c) Determine the p value for each hypothesis test.
(d) State your conclusions in terms of the problem.
(e) Construct the 95 percent confidence interval for $\beta_1$ .

\item Digoxin is a drug often prescribed to treat heart ailments. The purpose of a study by Parker et al.
was to examine the interactions of digoxin with common grapefruit juice. In one experiment, subjects
took digoxin with water for 2 weeks, followed by a 2-week period during which digoxin was
withheld. During the next 2 weeks subjects took digoxin with grapefruit juice. For eight subjects, the
average peak plasma digoxin concentration (Cmax) when taking water is given in the first row of
below. The second row contains the percent change in Cmax concentration when
subjects were taking the digoxin with grapefruit juice [GFJ (\%) change]. Use the Cmax level when
taking digoxin with water to predict the percent change in Cmax concentration when taking digoxin
with grapefruit juice. Draw scatter diagram.

Cmax (ngl/ml) with Water: 2.34
2.46
1.87
3.09
5.59
4.05
6.21
2.34\\

Change in Cmax with GFJ (\%): 29.5
40.7
5.3
23.3
-45:1
-35:3
-44:6
29.5

a) Compute the coefficient of determination.
%(b) Prepare an ANOVA table and use the F statistic to test the null hypothesis that b 1 1⁄4 0. Let
%a% 1⁄4 :05.
(b) Use the t statistic to test the null hypothesis that $\beta_1=0$ at the .05 level of significance.
(c) Determine the p value for each hypothesis test.
(d) State your conclusions in terms of the problem.
(e) Construct the 95 percent confidence interval for $\beta_1$ .

\end{enumerate}

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