Questions: A set of exam scores is normally distributed with a mean = 72 and standard deviation = 5. Use the Empirical Rule to complete the following sentences. 68% of the scores are between and . 95% of the scores are between and . 99.7% of the scores are between and .

Solution Steps

To solve this problem, we will use the Empirical Rule, which states that for a 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 the mean and standard deviation, we can calculate the ranges for each percentage by adding and subtracting the appropriate multiples of the standard deviation from the mean.

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule that applies to normal distributions. It states that:

• \(68\%\) of the data falls within one standard deviation (\(\sigma\)) of the mean (\(\mu\)).
• \(95\%\) of the data falls within two standard deviations (\(2\sigma\)) of the mean.
• \(99.7\%\) of the data falls within three standard deviations (\(3\sigma\)) of the mean.

For \(68\%\) of the scores, we calculate the range as: \[ \mu \pm \sigma = 72 \pm 5 \] This gives us the range: \[ (72 - 5, 72 + 5) = (67, 77) \]

For \(95\%\) of the scores, we calculate the range as: \[ \mu \pm 2\sigma = 72 \pm 2 \times 5 \] This gives us the range: \[ (72 - 10, 72 + 10) = (62, 82) \]

For \(99.7\%\) of the scores, we calculate the range as: \[ \mu \pm 3\sigma = 72 \pm 3 \times 5 \] This gives us the range: \[ (72 - 15, 72 + 15) = (57, 87) \]

Final Answer