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3 years ago

HELP ASPAP!!!!! WILL MARK BRAINIEST standard deviation

Mathematics

1 answer:

Umnica [9.8K]3 years ago

5 0

Answer:

The standard deviation of the given data is 11.34.

Step-by-step explanation:

We have to find the standard deviation for the given data;

Grade on No. of students

test (X) with that score (f) $X \times f$ $( X - \bar X)^{2}$ $f \times ( X - \bar X)^{2}$

50 1 50 1073.218 1073.2176

57 2 114 663.578 1327.1552

60 4 240 518.018 2072.0704

65 3 195 315.418 946.2528

72 3 216 115.778 347.3328

75 12 900 60.218 722.6112

77 10 770 33.178 331.776

81 6 486 3.098 18.5856

83 6 498 0.058 0.3456

88 9 792 27.458 247.1184

90 12 1080 52.418 629.0112

92 12 1104 85.378 1024.5312

95 2 190 149.818 299.6352

99 4 396 263.738 1054.9504

100 5 500 297.218 1486.088

Total 91 7531 11580.68

Firstly, the mean of the above data is given by;

Mean, $\bar X$ = $\frac{\sum X \times f}{\sum f}$

= $\frac{7531}{91}$ = 82.76

Now, the standard deviation of the data is given by;

Standard deviation, S.D. = $\sqrt{\frac{\sum f(X-\bar X)^{2} }{\sum f-1}}$

= $\sqrt{\frac{11580.68}{91-1}}$

= 11.34

Hence, the standard deviation of the given data is 11.34.

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$\rm \int \dfrac{x^2+7}{x^2+2x+5}~dx$

Derivative of the denominator:
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Hmm our numerator is 2x+7. Ok this let's us know that a simple u-substitution is NOT going to work. But let's apply some clever Algebra to the numerator splitting it up into two separate fractions. Split the +7 into +2 and +5.

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and then split the fraction,

$\rm \int \dfrac{x^2+2}{x^2+2x+5}~dx+\int\dfrac{5}{x^2+2x+5}~dx$

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$\rm \int \dfrac{1}{u}~du+\int\dfrac{5}{x^2+2x+5}~dx$

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$\rm ln|x^2+2x+5|+\int\dfrac{5}{x^2+2x+5}~dx$

Other one is a little tricky. We'll need to complete the square on the denominator. After that it will look very similar to our arctangent integral so perhaps we can just match it up to the identity.

$\rm x^2+2x+5=(x^2+2x+1)+4=(x+1)^2+2^2$

So we have this going on,

$\rm ln|x^2+2x+5|+\int\dfrac{5}{(x+1)^2+2^2}~dx$

Let's factor the 5 out of the intergral,
and the 4 from the denominator,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\frac{(x+1)^2}{2^2}+1}~dx$

Bringing all that stuff together as a single square,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(\dfrac{x+1}{2}\right)^2+1}~dx$

Making the substitution: $\rm \quad u=\dfrac{x+1}{2}\qquad\to\qquad 2du=dx$

giving us,

$\rm ln|x^2+2x+5|+\frac54\int\dfrac{1}{\left(u\right)^2+1}~2du$

simplying a lil bit,

$\rm ln|x^2+2x+5|+\frac52\int\dfrac{1}{u^2+1}~du$

and hopefully from this point you recognize your arctangent integral,

$\rm ln|x^2+2x+5|+\frac52arctan(u)$

undo your substitution as a final step,
and include a constant of integration,

$\rm ln|x^2+2x+5|+\frac52arctan\left(\frac{x+1}{2}\right)+c$

Hope that helps!
Lemme know if any steps were too confusing.

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