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Question 2 (a The following table shows the solid reduction and oxygen demand reduction of an exp...

Question 2 (a The following table shows the solid reduction and oxygen demand reduction of an experiment. d Reduction, 1|Oxyg

(b)A random sample of 5 cotton fiber was taken and their absorbency values are shown below. 19.7, 21.5, 17.3, 18.9, 20.6 (i)

Question 2 (a The following table shows the solid reduction and oxygen demand reduction of an experiment. d Reduction, 1|Oxygen Demand Reduction, y(%) 20 16 28 35 27 10 (i) Calculate Σ, Υ ,Σ and (2 marks) (ii) Find the inear regression ne of oxygen demand solid reduction on (6 marks) iii) Estimate the oxygen demand reduction if the solid reduction is 40%. Comment on the reliability of your estimation. (3 marks) iii Find the coefficient of correlation. Hence, comment on the strength of correlation between the two variables. marks) (v) Calculate and comment the coefficient of determination (2 marks)
(b)A random sample of 5 cotton fiber was taken and their absorbency values are shown below. 19.7, 21.5, 17.3, 18.9, 20.6 (i) Find the mean and standard deviation for the sample given (5 marks) (ii) Construct a 90% confidence interval for the true mean absorbency values for cotton fiber. (4 marks) (iii) Test at 2.5% significance level whether the population mean absorbency value for cotton fiber is more than 19 (6 marks)
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Solution : ( 2 )

( i )

x y x * y x * x
16 20 320 256
18 16 288 324
27 28 756 729
30 35 1050 900
33 34 1122 1089
36 37 1332 1296

\sum{x} = 160 ~,~ \sum{y} = 170 ~,~ \sum{x * y} = 4868 ~,~ \sum{x^2} = 4594

------------------

Solution : ( 2 )

( ii )

x y x * y x * x
16 20 320 256
18 16 288 324
27 28 756 729
30 35 1050 900
33 34 1122 1089
36 37 1332 1296

\sum{x} = 160 ~,~ \sum{y} = 170 ~,~ \sum{x * y} = 4868 ~,~ \sum{x^2} = 4594

ΣΥ*ΣΧ2 ㅡ Σ2. * Σ.ry 170 * 4594-160 * 4868 6 * 4594 _ (160)2 Cl

780980 -778880 27564-25600

2100 1964

=1.069

and

=rị * Σ.ry _ ΣΖ. * ΣΥ=6*4868-160 * 170

29208 - 27200 27564-25600

2008 1964

1.022

\mathrm{Substitute\:}a\mathrm{\:and\:}b\mathrm{\:in\:regression\:equation\:formula}

\begin{aligned} y~&=~a ~+~ b * x \\y~&=~1.069 ~+~ 1.022 *x\end{aligned}

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Solution : ( 2 )

( iii )

y=1.069+1.022 *\left ( \frac{40}{100} \right )

=1.069+1.022 *\left ( \frac{2}{5} \right )

=1.069+\left ( \frac{1.022 *2}{5} \right )

=1.069+\left ( \frac{2.044}{5} \right )

=1.069+0.4088

=1.4778

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Solution : ( 2 )

( iv )

x y x * y x * x y * y
16 20 320 256 400
18 16 288 324 256
27 28 756 729 784
30 35 1050 900 1225
33 34 1122 1089 1156
36 37 1332 1296 1369

\mathrm{Here,}

\sum{X}=160 ~,

~ \sum{Y}=170 ~,

~ \sum{X * Y}=4868 ~,

~ \sum{X^2}=4594 ~

and

~ \sum{Y^2}=5190

\mathrm{Now,}

r=\frac{n*\sum{xy} - \sum{x}*\sum{y}}{\sqrt{\left[n \sum{x^2}-\left(\sum{x}\right)^2\right] * \left[n \sum{y^2}-\left(\sum{y}\right)^2\right]}}

=\frac{ 6*4868 - 160 * 170 }{\sqrt{\left[ 6 * 4594 - 160^2 \right] *\left[ 6 * 5190 - (170)^2 \right] }}

=\frac{ 29208-27200 }{\sqrt{\left(27564-25600\right)\left(31140-28900\right)}}

=\frac{ 29208-27200 }{\sqrt{1964*2240}}

=\frac{2008}{\sqrt{4399360}}

=\frac{2008}{2097.46514}

=0.95734

\mathrm{Hence,\quad\:the\:coefficient\:of\:correlation\: \:is\:\:}\bold{0.95734}

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Solution : ( 2 )

( v )

\mathrm{Coefficient\:of\:determination\: \:}r^{2}=(0.95734)^{2}=0.9164998756

==========================================================================

Solution : ( 3 )

( i )

\bold{Mean\: }(\overline{x} )\:\::

\mathrm{Mean\: }(\overline{x} )= \mathrm{\frac{Sum ~ of ~ terms}{Number ~ of ~ terms}=\frac{19.7 + 21.5 + 17.3 + 18.9 + 20.6 }{5}}

\overline{x}=\frac{ 98 }{ 5 }= 19.6

and

\bold{Standard\:deviation\:\:}(\sigma )\::

data data - mean (data - mean)2
19.7 0.1 0.01
21.5 1.9 3.61
17.3 -2.3 5.29
18.9 -0.7 0.49
20.6 1 1

\mathrm{Here,}

\sum{\left(x_i - \overline{x}\right)^2} = 10.4

= 1.6125 3.2249 V10.4 /10.4 2 Σ(zi-T) /10.4 5-1

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Solution : ( 3 )

( ii )

\mathrm{90\%\:confidence\:interval\:for\:the\:mean\:is\:}

=\overline{x}\pm z*\left ( \frac{\sigma }{\sqrt{n}} \right )

=\left (\overline{x}- z*\left ( \frac{\sigma }{\sqrt{n}} \right ),\quad\overline{x}+ z*\left ( \frac{\sigma }{\sqrt{n}} \right ) \right )

=\left (19.6- 1.645 *\left ( \frac{1.6125 }{\sqrt{5}} \right ),\quad\:19.6+ 1.645 *\left ( \frac{1.6125 }{\sqrt{5}} \right ) \right )

=\left (19.6- \left ( \frac{1.645 *1.6125 }{\sqrt{5}} \right ),\quad\:19.6+ \left ( \frac{1.645 *1.6125 }{\sqrt{5}} \right ) \right )

-(19.-2 519620702 2.6525625 2.6525625

=\left (19.6- \frac{2.6525625}{2.23607},\quad\:19.6+ \frac{2.6525625}{2.23607} \right )

=\left (19.6-1.18626,\quad\:19.6+1.18626 \right )

=\left (18.41374,\quad\:20.78626 \right )

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