UpStudy Free Solution:

The empirical rule (also known as the 68-95-99.7 rule) states that for a bell-shaped (normal) distribution:

1. Approximately 68% of the data falls within one standard deviation of the mean.

2. Approximately 95% of the data falls within two standard deviations of the mean.

3. Approximately 99.7% of the data falls within three standard deviations of the mean.

Given:

- Mean (\(\mu \)) = 100

- Standard deviation (\(\sigma \)) = 14

Let's use the empirical rule to answer each part of the question:

(a) What percentage of people has an IQ score between 86 and 114?

86 and 114 are one standard deviation below and above the mean, respectively:

- \(86 = 100 - 14\)

- \(114 = 100 + 14\)

According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean. Therefore, approximately 68% of people have an IQ score between 86 and 114.

(b) What percentage of people has an IQ score less than 86 or greater than 114?

Since 68% of the data falls within one standard deviation of the mean, the remaining percentage (100% - 68%) falls outside this range. This remaining percentage is 32%.

Since the distribution is symmetric, half of this 32% will be below 86 and the other half will be above 114:

- Percentage less than 86 or greater than 114 = 32%

(c) What percentage of people has an IQ score greater than 114?

As calculated in part (b), 16% of the data falls above one standard deviation from the mean (since 32% is split into two tails, 16% in each tail).

Therefore, the percentage of people with an IQ score greater than 114 is 16%.

Supplemental Knowledge

The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule that applies to normal distributions (bell-shaped curves). It states that:

- Approximately 68% of the data falls within one standard deviation (\(\sigma \)) of the mean (\(\mu \)).

- Approximately 95% of the data falls within two standard deviations of the mean.

- Approximately 99.7% of the data falls within three standard deviations of the mean.

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