91Ó°ÊÓ

To tackle problems involving normal distribution, calculating the Z-score is an essential step. The Z-score allows us to determine how many standard deviations a data point is away from the mean of the distribution. This is crucial in understanding the position of a value within a dataset.
The formula for the Z-score is given by:

• \( Z = \frac{X - \mu}{\sigma} \)

Here, \( X \) represents the data point we're interested in, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
In this specific problem, we're looking to find out how 8 hours compares to the average of 6.5 hours with a standard deviation of 1 hour.
Plugging in these numbers:

• \( Z = \frac{8 - 6.5}{1} = 1.5 \)

This tells us that 8 hours of sleep is 1.5 standard deviations above the mean sleep duration for the students.

Once we have the Z-score, the next step is to find the cumulative probability. This will tell us the probability of a random data point falling below a particular value.
Using the Z-score we calculated for 8 hours of sleep, 1.5, we refer to standard normal distribution tables or use statistical software for precise values.
For a Z-score of 1.5, the cumulative probability (area under the curve to the left of the Z-score) is approximately 0.9332. This means that 93.32% of the students sleep less than 8 hours.
Cumulative probability gives us a powerful overview of where a particular observation stands in comparison to the rest of the data. It is particularly useful in assessing how typical or unusual a certain observation is within the data distribution.

Understanding complementary probability is key when dealing with probability distributions. After determining the cumulative probability, which shows us how many students sleep less than 8 hours, we can find the complementary probability to answer the question of interest: How many sleep at least 8 hours?
Since cumulative probability informs us of the percentage of occurrences up to a point, complementary probability looks at the opposite side.
To calculate it, we simply subtract the cumulative probability from 1:

• Complementary Probability = \( 1 - 0.9332 = 0.0668 \)

This calculation tells us that approximately 6.68% of students manage to get 8 or more hours of sleep.
Complementary probability highlights the significance of the rare outcomes which are often equally important in understanding the full picture of data distribution.