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 □ cm³ or higher. (type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the range rule of thumb

The range rule of thumb states that the limit for significantly high values is \( \text{mean} + 2\times\text{standard deviation} \), and for significantly low values is \( \text{mean} - 2\times\text{standard deviation} \).

Given: mean (\( \mu \)) = 1186.6 \( \text{cm}^3 \), standard deviation (\( \sigma \)) = 120.7 \( \text{cm}^3 \).

Step2: Calculate the limit for significantly high values

We need to find \( \mu + 2\sigma \).

First, calculate \( 2\sigma \): \( 2\times120.7 = 241.4 \)

Then, add this to the mean: \( 1186.6 + 241.4 = 1428.0 \)

Now, we check if 1478.0 is greater than this limit. Since 1478.0 > 1428.0, we also need to confirm the formula for significantly high (though the main calculation for the limit here is \( \mu + 2\sigma \)). Wait, the problem also has a significantly high limit to find? Wait, the first part was significantly low is 945.2 (which is \( 1186.6 - 2\times120.7 = 1186.6 - 241.4 = 945.2 \), which matches). Now for significantly high, it's \( \mu + 2\sigma = 1186.6 + 2\times120.7 = 1186.6 + 241.4 = 1428.0 \). Wait, but the question is about 1478.0, but first, we need to find the limit for significantly high. Wait, the problem's first blank (significantly low) is 945.2 (which we can verify: \( 1186.6 - 2\times120.7 = 945.2 \)), then significantly high is \( 1186.6 + 2\times120.7 = 1428.0 \). Wait, but the user's problem has two blanks: significantly low is 945.2 (given), and significantly high is to be calculated. Wait, the problem says "Significantly low values are 945.2 \( \text{cm}^3 \) or lower. Significantly high values are \( \square \) \( \text{cm}^3 \) or higher."

So we need to calculate \( \mu + 2\sigma \).

Step3: Compute \( \mu + 2\sigma \)

\( \mu = 1186.6 \), \( \sigma = 120.7 \)

\( 2\sigma = 2\times120.7 = 241.4 \)

\( \mu + 2\sigma = 1186.6 + 241.4 = 1428.0 \)

Answer:

1428.0