91Ó°ÊÓ

The concept of normal distribution is foundational in statistics. It is a probability distribution that is symmetric about its mean, meaning that data near the mean are more frequent in occurrence than data far from the mean. Most of the naturally-occurring phenomena fit this distribution model.

Key characteristics of normal distribution include:

• Bell Shape: The graph of the distribution curve is bell-shaped, with the highest point at the mean.
• Mean, Median, Mode: In a perfectly normal distribution, these three measures of central tendency are all the same.
• Symmetry: It is symmetric around the mean.
• Standard Deviation: Determines the distribution's spread or width. Most data falls within three standard deviations of the mean.

In our exercise, the time for both wheel throwing and firing follows a normal distribution with specified means and standard deviations. By understanding these properties, we can compute the probabilities of a piece of pottery being finished in certain time frames.

A random variable is a variable whose values depend on outcomes of a random phenomenon. It's a concept that allows us to quantify uncertainty with numbers.

There are two types of random variables:

• Discrete Random Variables: These can take on a finite number of values, such as the roll of a die.
• Continuous Random Variables: These can take on an infinite number of values between any two specific values, like the time it takes to polish a surface or exactly how much liquid fills up a bottle.

In the given exercise, both wheel throwing and firing time are represented as continuous random variables because they can take any value within a range. We denote this as \( X \) for wheel throwing time and \( Y \) for firing time, both of which follow normal distributions.

Z-scores are a statistical measurement that describe a value's relation to the mean of a group of values. They are particularly useful in comparing results from different data sets or different scores.

The formula for the Z-score is:

\[ Z = \frac{X - \mu}{\sigma} \]

Where:

• \( X \) is the value in question.
• \( \mu \) is the mean of the distribution.
• \( \sigma \) is the standard deviation.

In the problem, the Z-scores help convert the times (like 95 and 110 minutes) into standard form so we can find the probabilities using the standard normal distribution table.

For example, by calculating the Z-score for 95 minutes as \( Z = \frac{95 - 100}{3.6} \approx -1.39 \), it's established how far and in what direction the 95-minute mark is from the mean of 100 minutes. This process enables us to determine the likelihood or percentage of pottery being completed within that time frame.