# Statistics-I – BSc.CSIT (TU) Question Paper 2067 | First Semester

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## First Semester | First Year | Tribhuvan University Computer Science and Information Technology (Stat. 108) Subject: Statistics-I , Year: 2067 Old Question Collection | Question Bank

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Full Marks: 60 | Pass Marks: 24 | Time: 3 hours.

Candidates are required to give their answers in their own words as far as practicable.
All notations have the usual meanings.

Group A

Attempt any two:    (2×10=20)

1. Describe in detail stratified random sampling method for drawing a random sample of size n from a population of size N. Write down the expression for an unbiased estimator  $\bar{y_{st}}$ of population mean  $\bar{Y}$ and derive an expression for $Var \bar{(y_{st})}$ when samples were drawn from each stratum by adopting simple random sampling without replacement method. Also find the $Var \bar{(y_{st})}$ under the scheme of proportional allocation of sample sizes to strata.
2. Write down the rationale and method of Wilcoxon matched-pairs signed rank test. Seven prospective graduate students took a test twice with the following scores.
 First Attempt 470 530 610 440 600 590 580 Second Attempt 510 550 600 490 585 620 598

Compute the value of T+ where T+ is the sum of ranks of the positive differences (second attempt – first attempt) using T+ as test statistic carry out the test of the following hypothesis at level 0.05.
H0 : there is no statistical difference between the first and second attempt score
H1 : second attempt score tends to be larger than the first attempt score.

3. To study the effect of age (X1 in years) and weight (X2 in lbs) on systolic blood pressure (Y in mm Hg), the data were recorded for a sample of 15 adult males. The estimated regression model based on data is described below in the box where figures within parenthesis are standard error of the estimate. Further computation shows that
$\sum \left(Y_i - \bar{Y}\right)^2$ = 1835.7 and $\sum \left(Y_i - \hat{Y_l}\right)^2$ = 1101.3

 $\hat{(Y)}$ = 27.4 +0.22X1 +0.56X2 (24.68) (0.248) (0.155)

Explain the meaning of the estimated slope regression coefficients of the model.
What value of Y would you predict if X1 = 55 and X2 = 175?
Compute the value of R2and interpret it.
Carry out the overall goodness-of-fit test of the model at 5% level of significance.
Test the significance of slope regression coefficients at 5% level of significance.

Group B

Answer any eight questions:    (8×5=40)

4. Describe in detail systematic sampling method when N = k x n. Describe problems that will rise in systematic sampling method when N ≠ k x n.
5. If Vsrswr and Vsrswor correspondingly denote that variance of unbiased estimator of the population mean under simple random sampling with and without replacement method, then show that ( Vsrswr – Vsrswor) = $\dfrac{(n-1)}{Nn}$S2 and write your conclusion based on this result.
6. Consider the problem of determining if a die is fair or not. For this a die is rolled for 60 times and observed the following outcomes.
 side 1 2 3 4 5 6 Total Number of times oberved 8 9 13 7 15 8 60

Test the hypothesis H0: the die is fair, that is, all sides have 1/6 chance of appearing against
H1: the die is unfair at level 0.05.

7. Describe the method of Mann Whitney test.
8. Suppose that an IQ test is given to eleven randomly selected pairs consisting of one brother and one sister from the same family. To test the null hypothesis that this sample was drawn from a population in which the median IQ of a brother and sister do not differ against the alternative hypothesis that the sister would score higher than brother. IQ scores are summarized below.
 Sister’s score 129 111 117 120 116 101 107 127 105 123 113 Brother’s score 115 108 123 104 110 98 106 119 95 130 101

Using sign test carry out the above said hypothesis at 5% level of significance.

9. Describe rationale and method of Kruskal – Wallis one-way ANOVA test.
10. Suppose in a multiple regression model problem, the ANOVA table is as follows. How many independent variables are in the model? What is the sample size? What is the value of R2? Carry out the overall goodness-of-fit test of the model at 5% level of significance.
 Source SS Df Regression 36 2 Error 64 32
11. Explain the meaning of multicollinearity. How do you detect the problem of multicollinearity in multiple regressions?
12. Describe the Cobb-Douglas production function model with its application.
13. Define partial correlation coefficient. If r12 = 0.33, r13 = 0.40, and r23 = 0.76, then compute r13.2 and r23.1.
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