Question

A statistician, who is not good at math, wants to represent the relative frequencies of the
10
different categories of a categorical variable in a pie chart. He calculated the relative frequency of each category. In order to make a pie chart representing categories, he calculated the angle of slices for the first
9
categories using the wrong formula
π
∗
r
i
radians, where
r
i
represents the relative frequency of the
i
t
h
category. Since he knows that the sum of angles of slices must be
2
π
radians, he calculated the angle of the tenth slice as
2
π
−
π
∗
∑
9
i
=
1
r
i
.

If the total frequency is equal to
500
and the angle of the tenth category using the above formula is
1.02
π
radians in the pie chart, then what is the actual frequency of the tenth category?​

Answer 1

Answer: Answer is 10

Step-by-step explanation:

Given 10th category angle = 1.02π radians.

10th category angle from formula = 2π − π ∗

P9

i=1 ri

.

⇒ 2π − π ∗

P9

i=1 ri = 1.02π

⇒ π ∗

P9

i=1 ri = (2 − 1.02)π

⇒

X

9

i=1

ri = 0.98 (1)

We know that sum of relative frequencies is equal to 1.

⇒

P10

i=1 ri = 1

⇒

X

9

i=1

ri + r10 = 1 (2)

Substituting equation (1) in equation (2), we will get

⇒

P9

i=1 ri + r10 = 1

⇒ 0.98 + r10 = 1

r10 = 0.02

Given total frequency = 500.

Therefore frequency of 10th category is r10 ∗ 500

Substituting r10 value, we will get frequency of 10th category as 0.02×500 = 10

Answer 2

The actual frequency of the tenth category is 10.

Given,

Total frequency, N = 500

no. of categories of a categorical variable = 10

The  tenth angle obtained using the wrong formula = 1.02 radians

To Find,

The actual frequency of the 10th category

Solution,

The statistician who is not good at math has used the formula $\pi r_i$ to find the angle of each category, where ri is the relative frequency of ith category

The relative frequency is given by r = f ÷ N, where f is the actual frequency of each category and N is the total frequency

Also, the 10th angle was found using the formula $2\pi - \pi \sum_{i=1}^{9}(r_i)$, which is evaluated as 1.02π

$\implies 2\pi - \pi \sum_{i=1}^{9}(r_i) = 1.02\pi\\\\\implies \pi (2 - \sum_{i=1}^{9}(r_i)) = 1.02\pi\\\\\implies 2 - \sum_{i=1}^{9}(r_i) = 1.02\\\\\sum_{i=1}^{9}(r_i) = 2 - 1.02\\\\\sum_{i=1}^{9}(r_i) = 0.98$

Now, we know that

$\sum_{i=1}^{10}r_i = 1\\\\\implies \sum_{i=1}^{9}r_i + r_{10} = 1\\\\\implies r_{10} = 1 - \sum_{i=1}^{9}r_i\\\\\implies r_{10} = 1-0.98\\\\\implies r_{10} = 0.02$

Now, the from the formula for relative frequency, we find the actual frequency of the 10th observation

$r = \frac{f}{N}\\\\\implies f = r \times N\\\\\implies f_{10} = r_{10}\times N\\\\\implies f_{10} = 0.02 \times 500\\\\\implies f_{10} = 10$

Therefore, the actual frequency of the tenth category is 10.

For more clarity, the question is rewritten here:

A statistician, who is not good at math, wants to represent the relative frequencies of the 10 different categories of a categorical variable in a pie chart. He calculated the relative frequency of each category. In order to make a pie chart representing categories, he calculated the angle of slices for the first 9 categories using the wrong formula π∗ri radians, where ri represents the relative frequency of the ith category. Since he knows that the sum of angles of slices must be 2π radians, he calculated the angle of the tenth slice as 2π − π∗∑(1to9)ri. If the total frequency is equal to 500 and the angle of the tenth category using the above formula is 1.02π radians in the pie chart, then what is the actual frequency of the tenth category?​

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