Question

The brain volumes ​(cm cubed​)of 50 brains vary from a low of 904cm cubedto a high of 1490cm cubed.Use the range rule of thumb to estimate the standard deviation s and compare the result to the exact standard deviation of 174.7cm cubed​,assuming the estimate is accurate if it is within 15 cm cubed.The estimated standard deviation is 146.5cm cubed.​(Type an integer or a decimal. Do not​ round.)Compare the result to the exact standard deviation.

Answer 1

x_bar = 1197 cm^3 , s.d_e = 146.5 cm^3

Outside the 15 cm^3 tolerance. Not a good estimation.

Explanation:

Given:

- Lowest value of brain volume L = 904 cm^3

- Highest value of brain volume H = 1490 cm^3

- Exact standard deviation s.d_a = 174.7 cm^3

Find:

Use the range rule of thumb to estimate the standard deviation s and compare the result to the exact standard deviation of 174.7 cm^3 assuming the estimate is accurate if it is within 15 cm^3.

Solution:

- The rule of thumb states that the max and min limits are +/- 2 standard deviations about the mean x_bar. Hence, we will set up two equations.

L = x_bar - 2*s.d_e

H = x_bar + 2*s.d_e

Where, s.d_e is the estimated standard deviation.

- Solve the two equations simultaneously and you get the following:

x_bar = 1197 cm^3 , s.d_e = 146.5 cm^3

- The exact standard deviation is s.d_a = 174.7 cm^3

So, the estimates differs by:

s.d_a - s.d_e = 174.7 - 146.5 = 28.2 cm^3

Hence, its outside the tolerance of 15 cm^3. Not a good approximation.

Answer 2

The Range Rule of Thumb says that the range is about four times the standard deviation. So, if you need to calculate it, you need to divide range (Maximum - Minimum) with 4, S=
`(R)/(4)`.

Explanation:

R=1490 - 904

S = 586 / 4 = 149.5

If you compare the exact standard desviation (149.5 cm) with the estimated (146.5 cm), it is a difference of 3 cm, is not neccesary round the result.

Hope my answer has been useful.