Question

1) The brain volumes (cm3) of 50 brains vary from a low of 910 cm3 to a high of 1492 cm3. Use the ange rule of thumb to estimate the standard deviation s and compare the result to the exact standard deviation of 175.3 cm3, assuming the estimate is accurate if it is within 15 cm3.2) The brain volumes (cm3) of 20 brains have a mean of 1169.3 cm3 and a standard deviation of 124.1 cm3. Use the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such​ data, would a brain volume of 1457.5 cm3 be significantly high?

Answer 1

Answer:

1) The estimated standard deviation from the range rule is 145.5 cm3.

As it is 29.8 cm3 smaller than the exact standard deviation, we consider that the estimation is not accurate enough.

2) Brain volumes smaller than 921.1 cm3 can be considered significantly low.

Brain volumes bigger than 1417.5 cm3 can be considered significantly high.

A brain volume of 1457.5 cm3 is bigger than the expected maximum , so it can be considered significantly high.

Step-by-step explanation:

The range rule indicates the approximate relationship between the range (the difference between the maximum and minimum value) and the standard deviation:

$s\approx\dfrac{R}{4}$

1) If we have a sample of 50 brains volumes, and they range from 910 cm3 to 1492 cm3, we can estimate the standard deviation as:

$s\approx\dfrac{\text{Max}-\text{Min}}{4}=\dfrac{1492-910}{4}=\dfrac{582}{4}=145.5$

The exact standard deviation is 175.3 cm3.

The difference is 175.3-145.5=29.8 cm3, so the estimation is not accurate enough.

2) According to the range rule of thumb we expect to have a maximum value around 2 standard deviations above the mean, and a minimum value 2 standard deviations below the mean. We have a mean of 1169.3 cm3 and a standard deviation of 124.1 cm3, so the expected boundaries are:

$\text{Min}=M-2s=1169.3-2*124.1=1169.3-248.2=921.1\\\\\text{Max}=M+2s=1169.3+2*124.1=1169.3+248.2=1417.5$

A brain volume of 1457.5 cm3 is bigger than the expected maximum , so it can be considered significantly high.