Question

The brain volumes ​(cm cubedcm3​) of 50 brains vary from a low of 922922 cm cubedcm3 to a high of 14961496 cm cubedcm3. use the range rule of thumb to estimate the standard deviation s and compare the result to the exact standard deviation of 161.7161.7 cm cubedcm3​, assuming the estimate is accurate if it is within 15 cm cubedcm3.

Answer 1

The estimated standard deviation using the range rule of thumb is 143.5 cm³, compared to the exact value of 161.7 cm³. The absolute difference is 18.2 cm³, exceeding the acceptable range of 15 cm³; therefore, the estimate is not considered accurate.

Step-by-step explanation:

To estimate the standard deviation using the range rule of thumb, we take the range of the data set and divide it by 4. The range is the difference between the highest and lowest values in the set. In this case, the highest brain volume is 1496 cm³ and the lowest is 922 cm³. Therefore, the range is 1496 cm³ - 922 cm³ = 574 cm³.

Now, we estimate the standard deviation (s) as follows:

s = Range/4 = 574 cm³ / 4 = 143.5 cm³.

The exact standard deviation given is 161.7 cm³. To determine if the estimate is accurate, we check if it is within 15 cm³ of the exact standard deviation:

Absolute difference = | Exact standard deviation - Estimated standard deviation | = | 161.7 cm³ - 143.5 cm³ | = 18.2 cm³.

The absolute difference is 18.2 cm³, which is more than the acceptable 15 cm³ difference, so the estimate is not considered accurate in this context.

Answer 2

The range rule of thumb is that the range is four times the standard deviation. The range in this case is 922 cubic cm to 1496 cubic cm, i.e., 574 cubic cm.
Therefore, according to the rule, the standard deviation is 574 cubic cm/4 =
143.5 cubic cm. The actual standard deviation is 161.7 cubic cm. The difference is 161.7 - 143.5 = 18.2. Since this is greater than 15 cubic cm, the rule in this case is not accurate.