Question

the brain volumes (cm³) of 20 brains have a mean of 1186.6 cm³ and a standard deviation of 120.7 cm³. use the given standard deviation and 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 1478.0 cm³ be significantly high? significantly low values are 945.2 cm³ or lower. (type an integer or a decimal. do not round.) significantly high values are 1428 cm³ or higher. (type an integer or a decimal. do not round.) is 1478.0 cm³ significantly high? a. no, because it is above the upper limit separating value. b. yes, because it is above the upper limit separating value. c. yes, because it is below the lower limit separating value. d. no, because it is between the limits separating values. e. no, because it is below the lower limit separating value. f. yes, because it is between the limits separating values.

Explanation:

Step1: Find lower limit

The range rule of thumb for lower limit: \( \text{Mean} - 2\times\text{Standard Deviation} \)
Given mean \( \mu = 1186.6 \), standard deviation \( \sigma = 120.7 \)
\( \text{Lower Limit} = 1186.6 - 2\times120.7 = 1186.6 - 241.4 = 945.2 \)

Step2: Find upper limit

The range rule of thumb for upper limit: \( \text{Mean} + 2\times\text{Standard Deviation} \)
\( \text{Upper Limit} = 1186.6 + 2\times120.7 = 1186.6 + 241.4 = 1428 \)

Step3: Compare 1478.0 with limits

We have upper limit \( 1428 \). Now, \( 1478.0>1428 \), so it is above the upper limit. But wait, the question is about significantly high. Wait, no, wait: Wait, the significantly high values are above upper limit (1428) or higher? Wait, the upper limit is 1428. Now, 1478.0 is above 1428. But let's check the options. Wait, the options for "Is 1478.0 cm³ significantly high?"
Wait, the upper limit is 1428. So 1478.0 is above upper limit. But let's check the options:
Option B: Yes, because it is above the upper limit separating value. Wait, but wait, the upper limit is 1428. 1478>1428, so it is above upper limit. But let's recheck the steps.

Wait, first, calculate lower and upper limits:

Lower limit: \( \mu - 2\sigma = 1186.6 - 2\times120.7 = 1186.6 - 241.4 = 945.2 \) (matches the given significantly low value)

Upper limit: \( \mu + 2\sigma = 1186.6 + 2\times120.7 = 1186.6 + 241.4 = 1428 \) (matches the given significantly high value)

Now, the brain volume is 1478.0. Compare with upper limit (1428). 1478.0 > 1428, so it is above the upper limit separating value. So the answer should be B? Wait, no, wait the option B says "Yes, because it is above the upper limit separating value." Wait, but let's check the options again.

Wait, the options:

A. No, because it is above the upper limit separating value. (No, if above, it should be significantly high, so A is wrong)

B. Yes, because it is above the upper limit separating value. (Yes, because upper limit is 1428, 1478>1428, so above upper limit, so significantly high. So B is correct? Wait, but wait, the upper limit is 1428, which is the limit for significantly high. So values above 1428 are significantly high. 1478 is above 1428, so yes, because it is above upper limit. So option B.

Wait, but let's confirm:

Significantly low: below lower limit (945.2)

Significantly high: above upper limit (1428)

Values between 945.2 and 1428 are not significant.

1478.0 is above 1428, so it is significantly high. So the reason is it is above upper limit. So option B.

Answer:

B. Yes, because it is above the upper limit separating value.