Question

All eight companies in the aerospace industry were surveyed as to their return on investment last year. The results are (in percent) 10.6, 12.6, 14.8, 18.2, 12.0, 14.8, 12.2, and 15.6.

A. Calculate the variance.
B. Calculate the range.
C. Calculate the arithmetic mean.

Answer 1

`Variance = 5.2575`

`Range = 5`

`Mean = 13.85`

Explanation:

Given:

10.6, 12.6, 14.8, 18.2, 12.0, 14.8, 12.2, and 15.6

Number of companies (n) = 8

Required

A. Calculate the variance.

B. Calculate the range.

C. Calculate the arithmetic mean.

Calculating the variance. ...

We start by calculating the mean of the given data

`Mean = (\sum x)/(n)`

`Mean = (10.6 + 12.6+ 14.8+ 18.2+ 12.0+ 14.8+ 12.2+ 15.6)/(8)`

`Mean = (110.8)/(8)`

`Mean = 13.85`

Subtract the mean from each data

`10.6 - 13.85 = -3.25\\12.6 - 13.85 = -1.25\\14.8 - 13.85 = 0.95\\18.2 - 13.85 = 4.35\\12.0 - 13.85 = -1.85\\14.8 - 13.85 = 0.95\\12.2 - 13.85 = -1.65\\15.6 - 13.85 = 1.75`

Square these results

`(-3.25)^2 = 10.5625 \\(-1.25)^2 = 1.5625 \\0.95^2 = 0.9025\\4.35^2 = 18.9225\\(-1.85)^2 =3.4225 \\0.95^2 =0.9025 \\(-1.65)^2 =2.7225 \\1.75^2 =3.0625`

Add these results

`10.5625 + 1.5625 + 0.9025 + 18.9225 + 3.4225 + 0.9025 + 2.7225 + 3.0625 =42.06`

Divide result by n

`Variance = (42.06)/(8)`

`Variance = 5.2575`

Calculating the range. ..

Range is calculated as thus

`Range = Highest - Lowest`

From the given data;

`Highest = 15.6; Lowest = 10.6`

So,

`Range = 15.6 - 10.6`

`Range = 5`

Calculating the arithmetic mean....

`Mean = (\sum x)/(n)`

`Mean = (10.6 + 12.6+ 14.8+ 18.2+ 12.0+ 14.8+ 12.2+ 15.6)/(8)`

`Mean = (110.8)/(8)`

`Mean = 13.85`

Answer 2

Variance is 5.2575

Range is 7.6

Arithmetic Mean is 13.85

Explanation:

Given the sample:

10.6, 12.6, 14.8, 18.2, 12.0, 14.8, 12.2, and 15.6.

(A) To calculate Variance

- Find the mean of the numbers, let the mean be M = (Sum of samples)/(number of samples)

M = (10.6 + 12.6 + 14.8 + 18.2 + 12.0 + 14.8 + 12.2 + 15.6)/8

= 110.8/8

= 13.85

- Subtract M from each sample, and square the result.

(10.6 - 13.85)² = 10.5625

(12.6 - 13.85)² = 1.5625

(14.8 - 13.85)² = 0.9025

(18.2 - 13.85)² = 18.9225

(12.0 - 13.85)² = 3.4225

(14.8 - 13.85)² = 0.9025

(12.2 - 13.85)² = 2.7225

(15.6 - 13.85)² = 3.0625

- Finally, variance is

V = (10.5625 + 1.5625 + 0.9025 + 18.9225 + 3.4225 + 0.9025 + 2.7225 + 3.0625)/8

= 42.06/8

= 5.2575

(B) Range = (Highest number in the sample) - (lowest number in the sample)

R = 18.2 - 10.6

= 7.6

(C) Arithmetic mean is M, which we have obtained earlier in (A)

M = 13.85