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In the context of the normal distribution, probability is a measure that tells us how likely an event is to occur. For our exercise, this event is a golf ball weighing less than or equal to 1.620 ounces. The weights of these golf balls, which follow a normal distribution, allow us to calculate the probability using statistical methods.

To find this probability, we need to know several details:

• The mean, which is the average weight of the golf balls.
• The standard deviation, indicating how much the weights deviate from the mean.

By converting the weight limit into a standardized form called the Z-score, we can easily determine the probability using statistical tables or a calculator. In essence, probability in this context quantifies the likelihood that a randomly chosen golf ball will meet the weight regulations.

Standard deviation is a fundamental concept in statistics that represents the dispersion or spread of data points around the mean. In simpler terms, it tells us how much variation or spread exists from the average value. For our golf balls, the standard deviation is 0.09 ounces. This helps us understand how closely the weights are grouped around the mean of 1.361 ounces.

A lower standard deviation means that most golf balls weigh close to the average, whereas a higher standard deviation signifies more weight variation. In our exercise, knowing the standard deviation is crucial because it directly affects the Z-score calculation, which is used to find the probability that a ball weighs less than a given amount.
This statistical measure gives us insight into the consistency of the golf ball manufacturing process, indicating whether most balls are likely to weigh similarly or if there's a wide range in their weights.

The Z-score is an important statistical measurement that tells us how many standard deviations a specific data point is from the mean of a data set. In our example, we calculated the Z-score to transform the question about golf ball weights into a standardized problem.

The formula to find the Z-score is:\[ Z = \frac{X - \mu}{\sigma} \]where:

• \(X\) is the weight we're interested in, which is 1.620 ounces.
• \(\mu\) is the mean weight (1.361 ounces).
• \(\sigma\) is the standard deviation (0.09 ounces).

Plugging these values into the formula gives us a Z-score of approximately 2.8778. The Z-score helps us locate the exact position of 1.620 ounces on the normal distribution curve, converting the exercise into a standard form. We then use this Z-score with a standard normal distribution table to determine the probability we care about. It's a powerful tool that standardizes different scenarios into a common framework for easy comparison and analysis.