Question

Use Pencil Only Be Neat & Organized 1. The score of 15 randomly selected exams are given below: 82 98 70 65 90 76 67 85 70 89 100 65 72 90 80 (a) (2 points) Find Round your answer to one decimal place (b) (2 points) Find s Round your answer to one decimal place (c) (2 points) Find the exact value in reduced fraction for s2.

Answer 1

Solution :

$82,\:\:98,\:\:70,\:\:65,\:\:90,\:\:76,\:\:67,\:\:85,\:\:70,\:\:89,\:\:100,\:\65,\:\:72,\:\:90\:\:\mathrm{and}\:\:80&$

( a )

$\mathrm{Mean\:}(\bar{x}) = \mathrm{\:\frac{Sum ~ of ~ terms}{Number ~ of ~ terms}}$

$=\frac{82+98+70+65+90+76+67+85+70+89+100+65+72+90+80}{15}$

$=\frac{1199}{15}$

$=79.9333\approx 79.9$

$=79.9$

$\mathrm{Hence,\quad\:}\bar{x}=79.9$

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( b )

$\mathrm{Standard\: deviation\:\:(s )\::}$

$s = \sqrt{ \frac{ \sum{\left(x_i - \overline{x}\right)^2 }}{n-1} }$

$\mathrm{Now,}$

data	data - mean	(data - mean)2
82	2.0667	4.27124889
98	18.0667	326.40564889
70	-9.9333	98.67044889
65	-14.9333	223.00344889
90	10.0667	101.33844889
76	-3.9333	15.47084889
67	-12.9333	167.27024889
85	5.0667	25.67144889
70	-9.9333	98.67044889
89	9.0667	82.20504889
100	20.0667	402.67244889
65	-14.9333	223.00344889
72	-7.9333	62.93724889
90	10.0667	101.33844889
80	0.0667	0.00444889

$\sum{\left(x_i - \overline{x}\right)^2} = 1932.9333$

$s = \sqrt{ \frac{ \sum{\left(x_i - \overline{x}\right)^2 }}{n-1} }$

$= \sqrt{ \frac{ 1932.9333 }{ 15 - 1} }$

$=\sqrt{\frac{1932.9333}{14}}$

$=\sqrt{138.06666 }$

$=11.75018\approx 11.8$

$=11.8$

$\mathrm{Hence,\quad\:}s= 11.8$

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( c )

$\mathrm{Variance\:\:}(s^{2})\:\::$

$s^{2}= \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1}$

$\mathrm{Now,}$

data	data - mean	(data - mean)2
82	2.0667	4.27124889
98	18.0667	326.40564889
70	-9.9333	98.67044889
65	-14.9333	223.00344889
90	10.0667	101.33844889
76	-3.9333	15.47084889
67	-12.9333	167.27024889
85	5.0667	25.67144889
70	-9.9333	98.67044889
89	9.0667	82.20504889
100	20.0667	402.67244889
65	-14.9333	223.00344889
72	-7.9333	62.93724889
90	10.0667	101.33844889
80	0.0667	0.00444889

$\sum{\left(x_i - \overline{x}\right)^2} = 1932.9333$

$s^{2}= \frac{ \sum{\left(x_i - \overline{x}\right)^2 }}{n-1}$

$=\frac{ 1932.9333 }{ 15 - 1}$

$=\frac{1932.9333}{14}$

$=138.0667$

$\mathrm{Hence,\quad\:}s^{2}= 138.0667$