48+49+50+51+...112+113
1 answer:
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To verify the identity we need the following identities:
i)
ii)
iii)
.
Also, we have know that
and 
Thus,
By (ii) and (iii) we have:
by simplifying and
Now,
is clearly -tanx.
i)
ii)
iii)
Also, we have know that
Thus,
By (ii) and (iii) we have:
by simplifying
Now,
Answer:
a) Range = 1.39
b) the sample variance is 0.1597
c) the sample standard deviation is 0.3996
Step-by-step explanation:
Given the data in the question;
a) Find the range.
To determine the range, we simple subtract the smallest value from the largest value. i.e
Range = largest value - smallest value
from data set; our smallest is 0.53 while our largest value is 1.92
so
Range = 1.92 - 0.53 = 1.39
b) Find the sample variance.
To determine our variance, we use the following formula;
∑( -
)² / n - 1 = [ ∑
²/n-1 ] - [
(
)²]
where = ∑
/n
n is sample size = 14 so lets calculate ∑
∑ = 0.99 + 1.92 + 1.23 + 0.85 + 0.65 + 0.53 + 1.41 + 1.12 + 0.63 + 0.67 + 0.69 + 0.60 + 0.60 + 0.66
∑ = 12.55
∑² = 0.99² + 1.92² + 1.23² + 0.85² + 0.65² + 0.53² + 1.41² + 1.12² + 0.63² + 0.67² + 0.69² + 0.60² + 0.60² + 0.66²
∑² = 13.3253
so
our = ∑
/n = 12.55 / 14 = 0.8964
so our Variance will be;
= [ ∑²/n-1 ] - [
(
)²]
= [ 13.3253 / 14-1 ] - [ (0.8964)²]
= 1.025 - 0.8653
= 0.1597
Therefore, the sample variance is 0.1597
c) Find the sample standard deviation.
we know that standard deviation is the square root of variance;
standard deviation = √Variance
standard deviation = √0.1597
standard deviation = 0.3996
Therefore, the sample standard deviation is 0.3996