# IGNOU MSTL-002 Solved Assignment 2023 | PGDAST

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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## IGNOU MSTL-002 Assignment Question Paper 2023

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Course Code: MSTL-002
Assignment Code: MSTL-002/TMA/2023
Maximum Marks: 100
Note:
1. All questions are compulsory.
2. Solve the following questions in MS Excel.
3. Take the screenshots of the final output/spreadsheet.
4. Paste all screenshots in the assignment booklets with all necessary interpretation and steps.
Q1 A manager of an amusement park wanted to study the waiting times of visitors for issuing entry tickets during a peak hour. A subgroup of 15 visitors was selected (one at each ten minutes interval during an hour) and the time (in minutes) was measured from the point each visitor entered in the line to when he or she began to be attended. The results of 40 days period are recorded in the following table:
The manager of this amusement park needs to construct suitable control charts for variability as well as average to infer whether the waiting times of visitors for getting entry tickets is under statistical control or not. If it is out-of-control, she also computes the revised control limits, if necessary.
Q 2 A company designs decorative glass wall panels. Each panel is supposed to meet company standards for such things as glass thickness, ability to reflect, size of panel, quality of glass, colour, and so on. To control these features, the company quality people randomly sampled the panels from every shift and determined how many of the panels are out of compliance on at least one feature. The data collected from 25 such samples are shown below:
 Sample No. Sampled Panels Out of Compliance Panels 1 69 2 2 71 3 3 66 3 4 65 9
Sample No. Sampled Panels Out of Compliance Panels 1 69 2 2 71 3 3 66 3 4 65 9| Sample No. | Sampled Panels | Out of Compliance Panels | | :—: | :—: | :—: | | 1 | 69 | 2 | | 2 | 71 | 3 | | 3 | 66 | 3 | | 4 | 65 | 9 |
 5 69 3 6 67 2 7 70 4 8 73 5 9 71 3 10 69 2 11 74 5 12 79 2 13 74 4 14 74 3 15 71 2 16 67 3 17 69 2 18 75 4 19 71 2 20 72 4 21 69 3 22 74 3 23 69 2 24 69 4
5 69 3 6 67 2 7 70 4 8 73 5 9 71 3 10 69 2 11 74 5 12 79 2 13 74 4 14 74 3 15 71 2 16 67 3 17 69 2 18 75 4 19 71 2 20 72 4 21 69 3 22 74 3 23 69 2 24 69 4| 5 | 69 | 3 | | :— | :— | :— | | 6 | 67 | 2 | | 7 | 70 | 4 | | 8 | 73 | 5 | | 9 | 71 | 3 | | 10 | 69 | 2 | | 11 | 74 | 5 | | 12 | 79 | 2 | | 13 | 74 | 4 | | 14 | 74 | 3 | | 15 | 71 | 2 | | 16 | 67 | 3 | | 17 | 69 | 2 | | 18 | 75 | 4 | | 19 | 71 | 2 | | 20 | 72 | 4 | | 21 | 69 | 3 | | 22 | 74 | 3 | | 23 | 69 | 2 | | 24 | 69 | 4 |
Construct a suitable control chart for fraction of out of compliance panels to check whether the process is said to be in a state of control or not using both approaches. Also construct the revised control charts, if necessary.
Q3 A researcher is interested in studying the impact of the weekly working hours and type of machine used ( 0 for Machine $\mathrm{A}$$\mathrm{A}$A\mathrm{A}$\mathrm{A}$ and 1 for Machine $\mathrm{B}$$\mathrm{B}$B\mathrm{B}$\mathrm{B}$ ) on the number of produced items of a particular type. The data were collected for 40 weeks and shown in the following table:
 Week Produced Item Working Hours Machine Type 1 9 48 1 2 15 67 1 3 12 61 1 4 17 86 0 5 19 93 1 6 17 80 1 7 12 55 0 8 9 51 1
Week Produced Item Working Hours Machine Type 1 9 48 1 2 15 67 1 3 12 61 1 4 17 86 0 5 19 93 1 6 17 80 1 7 12 55 0 8 9 51 1| Week | Produced Item | Working Hours | Machine Type | | :—: | :—: | :—: | :—: | | 1 | 9 | 48 | 1 | | 2 | 15 | 67 | 1 | | 3 | 12 | 61 | 1 | | 4 | 17 | 86 | 0 | | 5 | 19 | 93 | 1 | | 6 | 17 | 80 | 1 | | 7 | 12 | 55 | 0 | | 8 | 9 | 51 | 1 |
 9 7 44 0 10 18 89 0 11 13 55 1 12 10 56 0 13 15 67 1 14 13 63 1 15 15 73 0 16 15 73 0 17 14 70 0 18 15 67 1 19 12 57 1 20 14 68 0 21 13 57 1 22 11 57 0 23 11 64 0 24 13 67 0 25 10 56 0 26 7 47 0 27 8 47 0 28 12 64 0 29 7 42 0 30 11 60 0 31 15 67 1 32 13 60 1 33 16 69 1 34 10 44 1 35 18 83 1 36 20 94 1 37 17 82 0 38 19 93 1 39 10 57 0 40 7 35 0
9 7 44 0 10 18 89 0 11 13 55 1 12 10 56 0 13 15 67 1 14 13 63 1 15 15 73 0 16 15 73 0 17 14 70 0 18 15 67 1 19 12 57 1 20 14 68 0 21 13 57 1 22 11 57 0 23 11 64 0 24 13 67 0 25 10 56 0 26 7 47 0 27 8 47 0 28 12 64 0 29 7 42 0 30 11 60 0 31 15 67 1 32 13 60 1 33 16 69 1 34 10 44 1 35 18 83 1 36 20 94 1 37 17 82 0 38 19 93 1 39 10 57 0 40 7 35 0| 9 | 7 | 44 | 0 | | :—: | :—: | :—: | :—: | | 10 | 18 | 89 | 0 | | 11 | 13 | 55 | 1 | | 12 | 10 | 56 | 0 | | 13 | 15 | 67 | 1 | | 14 | 13 | 63 | 1 | | 15 | 15 | 73 | 0 | | 16 | 15 | 73 | 0 | | 17 | 14 | 70 | 0 | | 18 | 15 | 67 | 1 | | 19 | 12 | 57 | 1 | | 20 | 14 | 68 | 0 | | 21 | 13 | 57 | 1 | | 22 | 11 | 57 | 0 | | 23 | 11 | 64 | 0 | | 24 | 13 | 67 | 0 | | 25 | 10 | 56 | 0 | | 26 | 7 | 47 | 0 | | 27 | 8 | 47 | 0 | | 28 | 12 | 64 | 0 | | 29 | 7 | 42 | 0 | | 30 | 11 | 60 | 0 | | 31 | 15 | 67 | 1 | | 32 | 13 | 60 | 1 | | 33 | 16 | 69 | 1 | | 34 | 10 | 44 | 1 | | 35 | 18 | 83 | 1 | | 36 | 20 | 94 | 1 | | 37 | 17 | 82 | 0 | | 38 | 19 | 93 | 1 | | 39 | 10 | 57 | 0 | | 40 | 7 | 35 | 0 |
i) Prepare a scatter plot to get an idea about the relationship among the variables.
ii) Fit a linear regression model and its related analysis at $1\mathrm{%}$$1\mathrm{%}$1%1 \%$1\mathrm{%}$ level of significance.
iii) Does the fitted regression model satisfy the linearity and normality assumptions?
iv) Also, draw both fitted regression lines on the scatter plot. Q 4 A popular café chain wishes to improve customer service and its employee scheduling based on the daily customers’ footfall during past 10 weeks. The numbers of customers served in the restaurants during that period are given as follow:
 Week Monday Tuesday Wednesday Thursday Friday Saturday Sunday $\mathbf{1}$$\mathbf{1}$1\mathbf{1}$\mathbf{1}$ 443 608 371 341 544 460 332 $\mathbf{2}$$\mathbf{2}$2\mathbf{2}$\mathbf{2}$ 279 358 312 377 438 277 402 $\mathbf{3}$$\mathbf{3}$3\mathbf{3}$\mathbf{3}$ 219 288 349 223 375 208 199 $\mathbf{4}$$\mathbf{4}$4\mathbf{4}$\mathbf{4}$ 264 343 190 362 423 202 387 $\mathbf{5}$$\mathbf{5}$5\mathbf{5}$\mathbf{5}$ 204 273 334 208 373 216 392 $\mathbf{6}$$\mathbf{6}$6\mathbf{6}$\mathbf{6}$ 379 292 417 234 303 364 238 $\mathbf{7}$$\mathbf{7}$7\mathbf{7}$\mathbf{7}$ 332 241 348 377 252 432 441 $\mathbf{8}$$\mathbf{8}$8\mathbf{8}$\mathbf{8}$ 321 478 499 478 327 604 429 $\mathbf{9}$$\mathbf{9}$9\mathbf{9}$\mathbf{9}$ 588 649 523 699 499 569 772 $\mathbf{1}\mathbf{0}$$\mathbf{1}\mathbf{0}$10\mathbf{1 0}$\mathbf{1}\mathbf{0}$ 658 848 843 793 751 975 941
Week Monday Tuesday Wednesday Thursday Friday Saturday Sunday 1 443 608 371 341 544 460 332 2 279 358 312 377 438 277 402 3 219 288 349 223 375 208 199 4 264 343 190 362 423 202 387 5 204 273 334 208 373 216 392 6 379 292 417 234 303 364 238 7 332 241 348 377 252 432 441 8 321 478 499 478 327 604 429 9 588 649 523 699 499 569 772 10 658 848 843 793 751 975 941| Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday | | :—: | :—: | :—: | :—: | :—: | :—: | :—: | :—: | | $\mathbf{1}$ | 443 | 608 | 371 | 341 | 544 | 460 | 332 | | $\mathbf{2}$ | 279 | 358 | 312 | 377 | 438 | 277 | 402 | | $\mathbf{3}$ | 219 | 288 | 349 | 223 | 375 | 208 | 199 | | $\mathbf{4}$ | 264 | 343 | 190 | 362 | 423 | 202 | 387 | | $\mathbf{5}$ | 204 | 273 | 334 | 208 | 373 | 216 | 392 | | $\mathbf{6}$ | 379 | 292 | 417 | 234 | 303 | 364 | 238 | | $\mathbf{7}$ | 332 | 241 | 348 | 377 | 252 | 432 | 441 | | $\mathbf{8}$ | 321 | 478 | 499 | 478 | 327 | 604 | 429 | | $\mathbf{9}$ | 588 | 649 | 523 | 699 | 499 | 569 | 772 | | $\mathbf{1 0}$ | 658 | 848 | 843 | 793 | 751 | 975 | 941 |
i) Determine the seasonal indices for these data using a 7-day moving averages.
ii) Obtain the deseasonalised values.
iii) Fit the appropriate trend for the deseasonalised data using the least-squares method by matrix approach that best describes these data.
iv) Project the number of customers on Wednesday of the ${22}^{\text{th}}$22^(“th “)22^{\text {th }} week.
v) Plot the original data, the deseasonalised data, and the trend.

## 1. TUTOR MARKED ASSIGNMENT MSTL-003: Biostatistics Lab

Course Code: MSTL-003
Assignment Code: MSTL-003/TMA/2023
Maximum Marks: 100
Note:
1. All questions are compulsory.
2. Solve the following questions in MS Excel.
3. Take the screenshots of the final output/spreadsheet.
4. Paste all screenshots in the assignment’s booklets with all necessary interpretation and steps.
Q1(a) A random sample of 440 patients of cardiology department of a hospital was taken and their workout timing and severity of heart disease status were recorded. The following table shows the workout timing and severity of heart disease:
 $\begin{array}{c}\text{Workout}\\ \text{(in minutes)}\end{array}${:[” Workout “],[” (in minutes) “]:}\begin{array}{c}\text { Workout } \\ \text { (in minutes) }\end{array} Severity of Heart Disease Low Mild Moderate High Very High No workout 5 13 26 21 23 $\mathbf{0}-\mathbf{1}\mathbf{5}$$\mathbf{0}-\mathbf{1}\mathbf{5}$0-15\mathbf{0}-\mathbf{1 5}$\mathbf{0}-\mathbf{1}\mathbf{5}$ 6 15 19 19 21 $\mathbf{1}\mathbf{5}$$\mathbf{1}\mathbf{5}$15\mathbf{1 5}$\mathbf{1}\mathbf{5}$ to $\mathbf{3}\mathbf{0}$$\mathbf{3}\mathbf{0}$30\mathbf{3 0}$\mathbf{3}\mathbf{0}$ 16 17 14 16 12 $\mathbf{3}\mathbf{0}$$\mathbf{3}\mathbf{0}$30\mathbf{3 0}$\mathbf{3}\mathbf{0}$ to $\mathbf{4}\mathbf{5}$$\mathbf{4}\mathbf{5}$45\mathbf{4 5}$\mathbf{4}\mathbf{5}$ 18 17 13 11 9 $\mathbf{4}\mathbf{5}$$\mathbf{4}\mathbf{5}$45\mathbf{4 5}$\mathbf{4}\mathbf{5}$ to $\mathbf{6}\mathbf{0}$$\mathbf{6}\mathbf{0}$60\mathbf{6 0}$\mathbf{6}\mathbf{0}$ 20 19 15 13 7 $\ge \mathbf{6}\mathbf{0}$$\ge \mathbf{6}\mathbf{0}$>= 60\geq \mathbf{6 0}$\ge \mathbf{6}\mathbf{0}$ 16 22 6 5 6
” Workout (in minutes) ” Severity of Heart Disease Low Mild Moderate High Very High No workout 5 13 26 21 23 0-15 6 15 19 19 21 15 to 30 16 17 14 16 12 30 to 45 18 17 13 11 9 45 to 60 20 19 15 13 7 >= 60 16 22 6 5 6| $\begin{array}{c}\text { Workout } \\ \text { (in minutes) }\end{array}$ | Severity of Heart Disease | | | | | | :—: | :—: | :—: | :—: | :—: | :—: | | | Low | Mild | Moderate | High | Very High | | No workout | 5 | 13 | 26 | 21 | 23 | | $\mathbf{0}-\mathbf{1 5}$ | 6 | 15 | 19 | 19 | 21 | | $\mathbf{1 5}$ to $\mathbf{3 0}$ | 16 | 17 | 14 | 16 | 12 | | $\mathbf{3 0}$ to $\mathbf{4 5}$ | 18 | 17 | 13 | 11 | 9 | | $\mathbf{4 5}$ to $\mathbf{6 0}$ | 20 | 19 | 15 | 13 | 7 | | $\geq \mathbf{6 0}$ | 16 | 22 | 6 | 5 | 6 |
Test at $5\mathrm{%}$$5\mathrm{%}$5%5 \%$5\mathrm{%}$ level of significance whether workout habit and heart disease are associated with to each other or not.
(b) To study the association between the diabetic patients and their family history of diabetes, the following data were obtained on 70 subjects.
 $\begin{array}{c}\text{Diabetes in}\\ \text{Family}\end{array}${:[” Diabetes in “],[” Family “]:}\begin{array}{c}\text { Diabetes in } \\ \text { Family }\end{array} Diabetes in Subject Total Yes No Yes 14 3 17 No 3 50 53 Total 17 53 70
” Diabetes in Family ” Diabetes in Subject Total Yes No Yes 14 3 17 No 3 50 53 Total 17 53 70| $\begin{array}{c}\text { Diabetes in } \\ \text { Family }\end{array}$ | Diabetes in Subject | | Total | | :—: | :—: | :—: | :—: | | | Yes | No | | | Yes | 14 | 3 | 17 | | No | 3 | 50 | 53 | | Total | 17 | 53 | 70 |
Which test is appropriate in this situation? Check whether the diabetes runs with generations in families or not at $5\mathrm{%}$$5\mathrm{%}$5%5 \%$5\mathrm{%}$ level of significance using appropriate test.
$\left(10+15\right)$$\left(10+15\right)$(10+15)(10+15)$\left(10+15\right)$
Q2 A researcher is interested to check the relationship between the serum creatinine (in $\mathrm{m}\mathrm{g}/\mathrm{d}\mathrm{L}$$\mathrm{m}\mathrm{g}/\mathrm{d}\mathrm{L}$mg//dL\mathrm{mg} / \mathrm{dL}$\mathrm{m}\mathrm{g}/\mathrm{d}\mathrm{L}$ ) with the weight (in $\mathrm{k}\mathrm{g}$$\mathrm{k}\mathrm{g}$kg\mathrm{kg}$\mathrm{k}\mathrm{g}$ ) and gender (0 if female and 1 if male). The data were collected from the hospital records to examine the contribution of these variables to serum creatinine. A total of 40 patients were sampled and the data are shown in the following table:
 S. No. $\begin{array}{c}\text{Serum}\\ \text{Creatinine}\end{array}${:[” Serum “],[” Creatinine “]:}\begin{array}{c}\text { Serum } \\ \text { Creatinine }\end{array} Weight Gender S. No. $\begin{array}{c}\text{Serum}\\ \text{Creatinine}\end{array}${:[” Serum “],[” Creatinine “]:}\begin{array}{c}\text { Serum } \\ \text { Creatinine }\end{array} Weight Gender 1 0.7 46 1 21 1.1 55 1 2 1.3 65 1 22 0.9 55 0 3 1 59 1 23 0.9 62 0 4 1.5 84 0 24 1.1 65 0 5 1.7 91 1 25 0.8 54 0 6 1.5 78 1 26 0.5 45 0 7 1 53 0 27 0.6 45 0 8 0.7 49 1 28 1 62 0 9 0.5 42 0 29 0.5 40 0 10 1.6 87 0 30 0.9 58 0 11 1.1 53 1 31 1.3 65 1 12 0.8 54 0 32 1.1 58 1 13 1.3 65 1 33 1.4 67 1 14 1.1 61 1 34 0.8 42 1 15 1.3 71 0 35 1.6 81 1 16 1.3 71 0 36 1.8 92 1 17 1.2 68 0 37 1.5 80 0 18 1.3 65 1 38 1.7 91 1 19 1 55 1 39 0.8 55 0 20 1.2 66 0 40 0.5 33 0
S. No. ” Serum Creatinine ” Weight Gender S. No. ” Serum Creatinine ” Weight Gender 1 0.7 46 1 21 1.1 55 1 2 1.3 65 1 22 0.9 55 0 3 1 59 1 23 0.9 62 0 4 1.5 84 0 24 1.1 65 0 5 1.7 91 1 25 0.8 54 0 6 1.5 78 1 26 0.5 45 0 7 1 53 0 27 0.6 45 0 8 0.7 49 1 28 1 62 0 9 0.5 42 0 29 0.5 40 0 10 1.6 87 0 30 0.9 58 0 11 1.1 53 1 31 1.3 65 1 12 0.8 54 0 32 1.1 58 1 13 1.3 65 1 33 1.4 67 1 14 1.1 61 1 34 0.8 42 1 15 1.3 71 0 35 1.6 81 1 16 1.3 71 0 36 1.8 92 1 17 1.2 68 0 37 1.5 80 0 18 1.3 65 1 38 1.7 91 1 19 1 55 1 39 0.8 55 0 20 1.2 66 0 40 0.5 33 0| S. No. | $\begin{array}{c}\text { Serum } \\ \text { Creatinine }\end{array}$ | Weight | Gender | S. No. | $\begin{array}{c}\text { Serum } \\ \text { Creatinine }\end{array}$ | Weight | Gender | | :—: | :—: | :—: | :—: | :—: | :—: | :—: | :—: | | 1 | 0.7 | 46 | 1 | 21 | 1.1 | 55 | 1 | | 2 | 1.3 | 65 | 1 | 22 | 0.9 | 55 | 0 | | 3 | 1 | 59 | 1 | 23 | 0.9 | 62 | 0 | | 4 | 1.5 | 84 | 0 | 24 | 1.1 | 65 | 0 | | 5 | 1.7 | 91 | 1 | 25 | 0.8 | 54 | 0 | | 6 | 1.5 | 78 | 1 | 26 | 0.5 | 45 | 0 | | 7 | 1 | 53 | 0 | 27 | 0.6 | 45 | 0 | | 8 | 0.7 | 49 | 1 | 28 | 1 | 62 | 0 | | 9 | 0.5 | 42 | 0 | 29 | 0.5 | 40 | 0 | | 10 | 1.6 | 87 | 0 | 30 | 0.9 | 58 | 0 | | 11 | 1.1 | 53 | 1 | 31 | 1.3 | 65 | 1 | | 12 | 0.8 | 54 | 0 | 32 | 1.1 | 58 | 1 | | 13 | 1.3 | 65 | 1 | 33 | 1.4 | 67 | 1 | | 14 | 1.1 | 61 | 1 | 34 | 0.8 | 42 | 1 | | 15 | 1.3 | 71 | 0 | 35 | 1.6 | 81 | 1 | | 16 | 1.3 | 71 | 0 | 36 | 1.8 | 92 | 1 | | 17 | 1.2 | 68 | 0 | 37 | 1.5 | 80 | 0 | | 18 | 1.3 | 65 | 1 | 38 | 1.7 | 91 | 1 | | 19 | 1 | 55 | 1 | 39 | 0.8 | 55 | 0 | | 20 | 1.2 | 66 | 0 | 40 | 0.5 | 33 | 0 |
(i) Prepare a scatter plot to get an idea about the relationship among the variables.
(ii) Fit a linear regression model and its related analysis at $1\mathrm{%}$$1\mathrm{%}$1%1 \%$1\mathrm{%}$ level of significance.
(iii) Does the fitted regression model satisfy the linearity and normality assumptions?
(iv) Also, draw both fitted regression lines on the scatter plot.
Q3 A hypothetical data of 40 patients on age (in years), weight (in kgs) and systolic blood pressure (in $\mathrm{m}\mathrm{m}/\mathrm{h}\mathrm{g}$$\mathrm{m}\mathrm{m}/\mathrm{h}\mathrm{g}$mm//hg\mathrm{mm} / \mathrm{hg}$\mathrm{m}\mathrm{m}/\mathrm{h}\mathrm{g}$ ) denoting 1 for high $\mathrm{S}\mathrm{B}\mathrm{P}$$\mathrm{S}\mathrm{B}\mathrm{P}$SBP\mathrm{SBP}$\mathrm{S}\mathrm{B}\mathrm{P}$ and 0 for normal $\mathrm{S}\mathrm{B}\mathrm{P}$$\mathrm{S}\mathrm{B}\mathrm{P}$SBP\mathrm{SBP}$\mathrm{S}\mathrm{B}\mathrm{P}$ are given in the following table:
 S. No. Age Weight SBP S. No. Age Weight SBP $\mathbf{1}$$\mathbf{1}$1\mathbf{1}$\mathbf{1}$ 52 60 0 $\mathbf{2}\mathbf{1}$$\mathbf{2}\mathbf{1}$21\mathbf{2 1}$\mathbf{2}\mathbf{1}$ 47 48 0 $\mathbf{2}$$\mathbf{2}$2\mathbf{2}$\mathbf{2}$ 56 68 1 $\mathbf{2}\mathbf{2}$$\mathbf{2}\mathbf{2}$22\mathbf{2 2}$\mathbf{2}\mathbf{2}$ 42 45 0 $\mathbf{3}$$\mathbf{3}$3\mathbf{3}$\mathbf{3}$ 51 54 0 $\mathbf{2}\mathbf{3}$$\mathbf{2}\mathbf{3}$23\mathbf{2 3}$\mathbf{2}\mathbf{3}$ 45 57 0 $\mathbf{4}$$\mathbf{4}$4\mathbf{4}$\mathbf{4}$ 63 74 1 $\mathbf{2}\mathbf{4}$$\mathbf{2}\mathbf{4}$24\mathbf{2 4}$\mathbf{2}\mathbf{4}$ 56 83 1 $\mathbf{5}$$\mathbf{5}$5\mathbf{5}$\mathbf{5}$ 54 62 0 $\mathbf{2}\mathbf{5}$$\mathbf{2}\mathbf{5}$25\mathbf{2 5}$\mathbf{2}\mathbf{5}$ 49 63 0 $\mathbf{6}$$\mathbf{6}$6\mathbf{6}$\mathbf{6}$ 51 67 0 $\mathbf{2}\mathbf{6}$$\mathbf{2}\mathbf{6}$26\mathbf{2 6}$\mathbf{2}\mathbf{6}$ 56 94 1 $\mathbf{7}$$\mathbf{7}$7\mathbf{7}$\mathbf{7}$ 51 66 0 $\mathbf{2}\mathbf{7}$$\mathbf{2}\mathbf{7}$27\mathbf{2 7}$\mathbf{2}\mathbf{7}$ 55 87 1 $\mathbf{8}$$\mathbf{8}$8\mathbf{8}$\mathbf{8}$ 54 65 0 $\mathbf{2}\mathbf{8}$$\mathbf{2}\mathbf{8}$28\mathbf{2 8}$\mathbf{2}\mathbf{8}$ 53 67 0 $\mathbf{9}$$\mathbf{9}$9\mathbf{9}$\mathbf{9}$ 59 71 1 $\mathbf{2}\mathbf{9}$$\mathbf{2}\mathbf{9}$29\mathbf{2 9}$\mathbf{2}\mathbf{9}$ 65 70 1 $\mathbf{1}\mathbf{0}$$\mathbf{1}\mathbf{0}$10\mathbf{1 0}$\mathbf{1}\mathbf{0}$ 51 89 1 $\mathbf{3}\mathbf{0}$$\mathbf{3}\mathbf{0}$30\mathbf{3 0}$\mathbf{3}\mathbf{0}$ 44 70 0
S. No. Age Weight SBP S. No. Age Weight SBP 1 52 60 0 21 47 48 0 2 56 68 1 22 42 45 0 3 51 54 0 23 45 57 0 4 63 74 1 24 56 83 1 5 54 62 0 25 49 63 0 6 51 67 0 26 56 94 1 7 51 66 0 27 55 87 1 8 54 65 0 28 53 67 0 9 59 71 1 29 65 70 1 10 51 89 1 30 44 70 0| S. No. | Age | Weight | SBP | S. No. | Age | Weight | SBP | | :—: | :—: | :—: | :—: | :—: | :—: | :—: | :—: | | $\mathbf{1}$ | 52 | 60 | 0 | $\mathbf{2 1}$ | 47 | 48 | 0 | | $\mathbf{2}$ | 56 | 68 | 1 | $\mathbf{2 2}$ | 42 | 45 | 0 | | $\mathbf{3}$ | 51 | 54 | 0 | $\mathbf{2 3}$ | 45 | 57 | 0 | | $\mathbf{4}$ | 63 | 74 | 1 | $\mathbf{2 4}$ | 56 | 83 | 1 | | $\mathbf{5}$ | 54 | 62 | 0 | $\mathbf{2 5}$ | 49 | 63 | 0 | | $\mathbf{6}$ | 51 | 67 | 0 | $\mathbf{2 6}$ | 56 | 94 | 1 | | $\mathbf{7}$ | 51 | 66 | 0 | $\mathbf{2 7}$ | 55 | 87 | 1 | | $\mathbf{8}$ | 54 | 65 | 0 | $\mathbf{2 8}$ | 53 | 67 | 0 | | $\mathbf{9}$ | 59 | 71 | 1 | $\mathbf{2 9}$ | 65 | 70 | 1 | | $\mathbf{1 0}$ | 51 | 89 | 1 | $\mathbf{3 0}$ | 44 | 70 | 0 |
 $\mathbf{1}\mathbf{1}$$\mathbf{1}\mathbf{1}$11\mathbf{1 1}$\mathbf{1}\mathbf{1}$ 56 72 1 $\mathbf{3}\mathbf{1}$$\mathbf{3}\mathbf{1}$31\mathbf{3 1}$\mathbf{3}\mathbf{1}$ 48 54 0 $\mathbf{1}\mathbf{2}$$\mathbf{1}\mathbf{2}$12\mathbf{1 2}$\mathbf{1}\mathbf{2}$ 55 72 1 $\mathbf{3}\mathbf{2}$$\mathbf{3}\mathbf{2}$32\mathbf{3 2}$\mathbf{3}\mathbf{2}$ 61 79 1 $\mathbf{1}\mathbf{3}$$\mathbf{1}\mathbf{3}$13\mathbf{1 3}$\mathbf{1}\mathbf{3}$ 46 57 0 $\mathbf{3}\mathbf{3}$$\mathbf{3}\mathbf{3}$33\mathbf{3 3}$\mathbf{3}\mathbf{3}$ 45 85 0 $\mathbf{1}\mathbf{4}$$\mathbf{1}\mathbf{4}$14\mathbf{1 4}$\mathbf{1}\mathbf{4}$ 42 54 0 $\mathbf{3}\mathbf{4}$$\mathbf{3}\mathbf{4}$34\mathbf{3 4}$\mathbf{3}\mathbf{4}$ 63 98 1 $\mathbf{1}\mathbf{5}$$\mathbf{1}\mathbf{5}$15\mathbf{1 5}$\mathbf{1}\mathbf{5}$ 52 63 0 $\mathbf{3}\mathbf{5}$$\mathbf{3}\mathbf{5}$35\mathbf{3 5}$\mathbf{3}\mathbf{5}$ 49 78 0 $\mathbf{1}\mathbf{6}$$\mathbf{1}\mathbf{6}$16\mathbf{1 6}$\mathbf{1}\mathbf{6}$ 65 67 1 $\mathbf{3}\mathbf{6}$$\mathbf{3}\mathbf{6}$36\mathbf{3 6}$\mathbf{3}\mathbf{6}$ 65 80 1 $\mathbf{1}\mathbf{7}$$\mathbf{1}\mathbf{7}$17\mathbf{1 7}$\mathbf{1}\mathbf{7}$ 50 67 0 $\mathbf{3}\mathbf{7}$$\mathbf{3}\mathbf{7}$37\mathbf{3 7}$\mathbf{3}\mathbf{7}$ 60 70 1 $\mathbf{1}\mathbf{8}$$\mathbf{1}\mathbf{8}$18\mathbf{1 8}$\mathbf{1}\mathbf{8}$ 42 53 0 $\mathbf{3}\mathbf{8}$$\mathbf{3}\mathbf{8}$38\mathbf{3 8}$\mathbf{3}\mathbf{8}$ 53 98 1 $\mathbf{1}\mathbf{9}$$\mathbf{1}\mathbf{9}$19\mathbf{1 9}$\mathbf{1}\mathbf{9}$ 50 68 1 $\mathbf{3}\mathbf{9}$$\mathbf{3}\mathbf{9}$39\mathbf{3 9}$\mathbf{3}\mathbf{9}$ 41 53 0 $\mathbf{2}\mathbf{0}$$\mathbf{2}\mathbf{0}$20\mathbf{2 0}$\mathbf{2}\mathbf{0}$ 39 55 0 $\mathbf{4}\mathbf{0}$$\mathbf{4}\mathbf{0}$40\mathbf{4 0}$\mathbf{4}\mathbf{0}$ 50 70 1
11 56 72 1 31 48 54 0 12 55 72 1 32 61 79 1 13 46 57 0 33 45 85 0 14 42 54 0 34 63 98 1 15 52 63 0 35 49 78 0 16 65 67 1 36 65 80 1 17 50 67 0 37 60 70 1 18 42 53 0 38 53 98 1 19 50 68 1 39 41 53 0 20 39 55 0 40 50 70 1| $\mathbf{1 1}$ | 56 | 72 | 1 | $\mathbf{3 1}$ | 48 | 54 | 0 | | :— | :— | :— | :— | :— | :— | :— | :— | | $\mathbf{1 2}$ | 55 | 72 | 1 | $\mathbf{3 2}$ | 61 | 79 | 1 | | $\mathbf{1 3}$ | 46 | 57 | 0 | $\mathbf{3 3}$ | 45 | 85 | 0 | | $\mathbf{1 4}$ | 42 | 54 | 0 | $\mathbf{3 4}$ | 63 | 98 | 1 | | $\mathbf{1 5}$ | 52 | 63 | 0 | $\mathbf{3 5}$ | 49 | 78 | 0 | | $\mathbf{1 6}$ | 65 | 67 | 1 | $\mathbf{3 6}$ | 65 | 80 | 1 | | $\mathbf{1 7}$ | 50 | 67 | 0 | $\mathbf{3 7}$ | 60 | 70 | 1 | | $\mathbf{1 8}$ | 42 | 53 | 0 | $\mathbf{3 8}$ | 53 | 98 | 1 | | $\mathbf{1 9}$ | 50 | 68 | 1 | $\mathbf{3 9}$ | 41 | 53 | 0 | | $\mathbf{2 0}$ | 39 | 55 | 0 | $\mathbf{4 0}$ | 50 | 70 | 1 |
For this data:
(i) Fit a multiple logistic regression model.
(ii) Test the significance of the individual model coefficients ${\beta }_{1}$${\beta }_{1}$beta_(1)\beta_{1}${\beta }_{1}$ and ${\beta }_{2}$${\beta }_{2}$beta_(2)\beta_{2}${\beta }_{2}$ at $5\mathrm{%}$$5\mathrm{%}$5%5 \%$5\mathrm{%}$ level of significance.
(iii) Obtain the $95\mathrm{%}$$95\mathrm{%}$95%95 \%$95\mathrm{%}$ confidence intervals for ${\beta }_{1}$${\beta }_{1}$beta_(1)\beta_{1}${\beta }_{1}$ and ${\beta }_{2}$${\beta }_{2}$beta_(2)\beta_{2}${\beta }_{2}$.
(iv) Determine the Nagelkerke pseudo R-squared.
Q4 A clinical study was conducted on individuals with advanced stage of Hepatocellular Carcinoma to test three lines of Treatments: T1, T2 and T3. Thirthy-six patients with stage III Hepatocellular Carcinoma who agreed to take part in the experiment were randomly allocated one of three line of Treatments $\mathrm{T}1,\text{}\mathrm{T}2$$\mathrm{T}1,\text{}\mathrm{T}2$T1,T2\mathrm{T} 1, \mathrm{~T} 2 and T3. The primary outcome was mortality, and patients were monitored for up to 60 months (5 years) after recruitment. The data (in months) so obtained are given as follows:
 $\begin{array}{c}\text{Patient}\\ \text{ID}\end{array}${:[” Patient “],[” ID “]:}\begin{array}{c}\text { Patient } \\ \text { ID }\end{array} $\begin{array}{c}\text{Survival}\\ \text{time}\end{array}${:[” Survival “],[” time “]:}\begin{array}{c}\text { Survival } \\ \text { time }\end{array} Outcome Treatment $\begin{array}{c}\text{Patient}\\ \text{ID}\end{array}${:[” Patient “],[” ID “]:}\begin{array}{c}\text { Patient } \\ \text { ID }\end{array} $\begin{array}{c}\text{Survival}\\ \text{time}\end{array}${:[” Survival “],[” time “]:}\begin{array}{c}\text { Survival } \\ \text { time }\end{array} Outcome Treatment ID001 14 Died T3 ID019 50 Died $\mathrm{T}1$$\mathrm{T}1$T1\mathrm{T} 1$\mathrm{T}1$ ID002 27 Unknown T1 ID020 54 Unknown T3 ID003 37 Unknown T3 ID021 57 Died T2 ID004 44 Died T1 ID022 60 Survived T3 ID005 27 Died T2 ID023 20 Died $\mathrm{T}1$$\mathrm{T}1$T1\mathrm{T} 1$\mathrm{T}1$ ID006 29 Died T3 ID024 22 Unknown T2 ID007 50 Died T2 ID025 11 Unknown $\mathrm{T}2$$\mathrm{T}2$T2\mathrm{T} 2$\mathrm{T}2$ ID008 31 Died T1 ID026 12 Unknown T1 ID009 54 Died T2 ID027 57 Unknown $\mathrm{T}3$$\mathrm{T}3$T3\mathrm{T} 3$\mathrm{T}3$ ID010 32 Died T2 ID028 60 Survived $\mathrm{T}3$$\mathrm{T}3$T3\mathrm{T} 3$\mathrm{T}3$ ID011 32 Unknown T2 ID029 44 Died $\mathrm{T}1$$\mathrm{T}1$T1\mathrm{T} 1$\mathrm{T}1$
” Patient ID ” ” Survival time ” Outcome Treatment ” Patient ID ” ” Survival time ” Outcome Treatment ID001 14 Died T3 ID019 50 Died T1 ID002 27 Unknown T1 ID020 54 Unknown T3 ID003 37 Unknown T3 ID021 57 Died T2 ID004 44 Died T1 ID022 60 Survived T3 ID005 27 Died T2 ID023 20 Died T1 ID006 29 Died T3 ID024 22 Unknown T2 ID007 50 Died T2 ID025 11 Unknown T2 ID008 31 Died T1 ID026 12 Unknown T1 ID009 54 Died T2 ID027 57 Unknown T3 ID010 32 Died T2 ID028 60 Survived T3 ID011 32 Unknown T2 ID029 44 Died T1| $\begin{array}{c}\text { Patient } \\ \text { ID }\end{array}$ | $\begin{array}{c}\text { Survival } \\ \text { time }\end{array}$ | Outcome | Treatment | $\begin{array}{c}\text { Patient } \\ \text { ID }\end{array}$ | $\begin{array}{c}\text { Survival } \\ \text { time }\end{array}$ | Outcome | Treatment | | :—: | :—: | :—: | :—: | :—: | :—: | :—: | :—: | | ID001 | 14 | Died | T3 | ID019 | 50 | Died | $\mathrm{T} 1$ | | ID002 | 27 | Unknown | T1 | ID020 | 54 | Unknown | T3 | | ID003 | 37 | Unknown | T3 | ID021 | 57 | Died | T2 | | ID004 | 44 | Died | T1 | ID022 | 60 | Survived | T3 | | ID005 | 27 | Died | T2 | ID023 | 20 | Died | $\mathrm{T} 1$ | | ID006 | 29 | Died | T3 | ID024 | 22 | Unknown | T2 | | ID007 | 50 | Died | T2 | ID025 | 11 | Unknown | $\mathrm{T} 2$ | | ID008 | 31 | Died | T1 | ID026 | 12 | Unknown | T1 | | ID009 | 54 | Died | T2 | ID027 | 57 | Unknown | $\mathrm{T} 3$ | | ID010 | 32 | Died | T2 | ID028 | 60 | Survived | $\mathrm{T} 3$ | | ID011 | 32 | Unknown | T2 | ID029 | 44 | Died | $\mathrm{T} 1$ |
 ID012 60 Unknown $\mathrm{T}1$$\mathrm{T}1$T1\mathrm{T} 1$\mathrm{T}1$ ID030 47 Unknown T2 ID013 2 Unknown $\mathrm{T}2$$\mathrm{T}2$T2\mathrm{T} 2$\mathrm{T}2$ ID031 32 Died $\mathrm{T}2$$\mathrm{T}2$T2\mathrm{T} 2$\mathrm{T}2$ ID014 42 Died $\mathrm{T}3$$\mathrm{T}3$T3\mathrm{T} 3$\mathrm{T}3$ ID032 34 Died $\mathrm{T}1$$\mathrm{T}1$T1\mathrm{T} 1$\mathrm{T}1$ ID015 42 Unknown $\mathrm{T}2$$\mathrm{T}2$T2\mathrm{T} 2$\mathrm{T}2$ ID033 17 Died $\mathrm{T}2$$\mathrm{T}2$T2\mathrm{T} 2$\mathrm{T}2$ ID016 60 Died $\mathrm{T}3$$\mathrm{T}3$T3\mathrm{T} 3$\mathrm{T}3$ ID034 6 Died $\mathrm{T}1$$\mathrm{T}1$T1\mathrm{T} 1$\mathrm{T}1$ ID017 60 Survived $\mathrm{T}3$$\mathrm{T}3$T3\mathrm{T} 3$\mathrm{T}3$ ID035 50 Unknown $\mathrm{T}3$$\mathrm{T}3$T3\mathrm{T} 3$\mathrm{T}3$ ID018 47 Died $\mathrm{T}3$$\mathrm{T}3$T3\mathrm{T} 3$\mathrm{T}3$ ID036 14 Unknown $\mathrm{T}2$$\mathrm{T}2$T2\mathrm{T} 2$\mathrm{T}2$
ID012 60 Unknown T1 ID030 47 Unknown T2 ID013 2 Unknown T2 ID031 32 Died T2 ID014 42 Died T3 ID032 34 Died T1 ID015 42 Unknown T2 ID033 17 Died T2 ID016 60 Died T3 ID034 6 Died T1 ID017 60 Survived T3 ID035 50 Unknown T3 ID018 47 Died T3 ID036 14 Unknown T2| ID012 | 60 | Unknown | $\mathrm{T} 1$ | ID030 | 47 | Unknown | T2 | | :—: | :—: | :—: | :—: | :—: | :—: | :—: | :—: | | ID013 | 2 | Unknown | $\mathrm{T} 2$ | ID031 | 32 | Died | $\mathrm{T} 2$ | | ID014 | 42 | Died | $\mathrm{T} 3$ | ID032 | 34 | Died | $\mathrm{T} 1$ | | ID015 | 42 | Unknown | $\mathrm{T} 2$ | ID033 | 17 | Died | $\mathrm{T} 2$ | | ID016 | 60 | Died | $\mathrm{T} 3$ | ID034 | 6 | Died | $\mathrm{T} 1$ | | ID017 | 60 | Survived | $\mathrm{T} 3$ | ID035 | 50 | Unknown | $\mathrm{T} 3$ | | ID018 | 47 | Died | $\mathrm{T} 3$ | ID036 | 14 | Unknown | $\mathrm{T} 2$ |
For this data,
(i) Construct Kaplan and Meier survival curves.
(ii) Test whether there is a significant difference between the survival distributions of the patients under all treatments at $5\mathrm{%}$$5\mathrm{%}$5%5 \%$5\mathrm{%}$ level of significance.
$$cos\:3\theta =4\:cos^3\:\theta -3\:cos\:\theta$$

## MSTL-002 Sample Solution 2023

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$$cot\:\theta =\frac{cos\:\theta }{sin\:\theta }$$

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