4 Marks Questions

Question 14 Marks

Compute Cov(x, y) for the following pair of observations:
(15, 44), (20, 43), (25, 45), (30, 37), (40, 34), (50, 37)

Answer

Here,
$\bar{x}=\frac{15+20+25+30+40+50}{6}=\frac{180}{6}=30$
and
$\bar{y}=\frac{44+43+45+37+34+37}{6}=\frac{240}{6}=40$
Now, construct the following table:

$x$	$x-\bar{x}$	$y$	$y-\bar{y}$	$(x-\bar{x})(y-\bar{y})$
15	-15	44	4	-60
20	-10	43	3	-30
25	-5	45	5	-25
30	0	37	-3	0
40	10	34	-6	-60
50	20	37	-3	-60
Total				-235

So, $\quad \begin{aligned} \operatorname{Cov}(x, y) & =\frac{1}{N} \Sigma(x-\bar{x})(y-\bar{y}) \\ & =\frac{1}{6}(-235)=-39.17\end{aligned}$

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Question 24 Marks

Compute rank correlation from the following data relating to ranks given by two judges in a contest:

Answer

The rank correlation coefficient is given by
$\begin{aligned} r_{ s } & =1-\frac{6 \Sigma d^2}{n\left(n^2-1\right)} \\ & =1-\frac{6 \times 166}{10\left(10^2-1\right)} \\ & =1-\frac{996}{990}\end{aligned}$
$
\begin{array}{l}
=\frac{990-996}{990} \\
=\frac{-6}{990}=-0.006
\end{array}
$
The very low value (almost 0 ) indicates that there is hardly any agreement between the ranks given by two Judges in the contest.

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Question 34 Marks

For the given set of data, calculate the percentile rank and quartile rank for observation 6.
5, 6, 6, 9, 11, 13, 15, 18, 20

Answer

The given data is already arranged in ascending order, so there is no need to arrange the data in ascending order.
Here, we can see that observation 6 comes under the repeated rank
So, here Y = 9 R = 2 and M = 2
(Here we choose the 6 which is at the farthest side. so, here we take 6 which is at 3^sd rank, since, 6 is at 3 ^sd rank, therefore the ranks below to 6 are 2 i.e., the value of M = 2 )
We apply formula,
$P R=\left[\frac{M+(0.5 \times R)}{Y}\right] \times 100$
$\begin{array}{l}=\frac{2+(0.5 \times 2)}{9} \times 100 \\ =\frac{3}{9} \times 100=0.33 \times 10=33 \% 1 \end{array}$
Now, locating this percentile rank on quartile scale, we observe that, it lies between Q₁ and Q₂ Therefore, quartile ranking for score 6 is Q₂

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Applied Maths 4 Marks Questions

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