91Ó°ÊÓ

In statistics, the mean is often referred to as the average. It represents the central point or typical value of a set of data. In the context of normal distribution, it is the point around which the data is symmetrically distributed. For this exercise, the mean (\(\mu\)) is \(6.5\) hours.
To understand the data's variability, we use standard deviation. Standard deviation measures how spread out the numbers in a data set are from the mean. A smaller standard deviation indicates that the data points tend to be closer to the mean, while a larger one signifies a more spread-out distribution. Here, the standard deviation (\(\sigma\)) is \(1\) hour.
These two metrics, the mean, and standard deviation, fully describe a normally distributed curve. They help us understand not only the center of the data but also its spread, which is crucial for statistical analysis.

The Empirical Rule is a handy shortcut for understanding the spread of data in a normal distribution. It is sometimes known as the 68-95-99.7 rule, providing a simple way to know where most of your data points lie.
This rule states:

• 68% of data falls within one standard deviation (\(\mu \pm 1\sigma\)) of the mean.
• 95% of data falls within two standard deviations (\(\mu \pm 2\sigma\)) of the mean.
• 99.7% of data falls within three standard deviations (\(\mu \pm 3\sigma\)) of the mean.

In the given problem, we are interested in the range that contains 95% of hours slept. Using the Empirical Rule, we quickly calculate that 95% of the data is within two standard deviations from the mean (\(6.5 \pm 2 \times 1\)), resulting in a range of \(4.5\) to \(8.5\) hours. This helps make quick, informed decisions without complex calculations.

Solving statistical problems often involves a structured approach that helps break down complex questions into manageable steps. In the context of the exercise, it's about leveraging known statistical principles to find solutions.
First, understand the distribution. Recognize that the problem describes a normal distribution; this knowledge tells you that mean and standard deviation are key.
Then, apply relevant rules. The Empirical Rule was crucial in our problem because it provided a direct method to identify the range incorporating 95% of the data.
Finally, verify and interpret the results. Once you calculate the range, check against given options to confirm the correct answer. This validation ensures that you've correctly applied statistical concepts to arrive at the right solution.

• Identify the distribution type.
• Use appropriate statistical rules.
• Calculate solutions and verify them.

A follow-through on these steps helps in solving many statistical problems with confidence.