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The normal distribution is a continuous probability distribution that is symmetrical about the mean. This means the data near the mean are more frequent in occurrence than data far from the mean. Many real-world datasets form a bell-shaped curve when plotted. This curve represents the normal distribution.
Key characteristics of a normal distribution include:

• The mean, median, and mode of the data are all equal.
• It is perfectly symmetric around the mean.
• It follows the empirical rule: about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Understanding the normal distribution is crucial for solving many statistical problems, including the one in our exercise regarding the filling of Coke cans.

A Z-score tells us how far a data point is from the mean in terms of standard deviations. The formula for calculating the Z-score is:
\[ Z = \frac{x - \mu}{\sigma} \]
Where:

• \( x \) is the value of the data point
• \( \mu \) is the mean of the dataset
• \( \sigma \) is the standard deviation of the dataset

The Z-score shows how many standard deviations a particular value lies above or below the mean. For instance, in our exercise, we calculated the Z-score to determine the probability of a Coke can containing at least 12.19 oz of soda.
By converting raw data points to Z-scores, we can use the standard normal distribution table (Z-table) to find probabilities associated with those scores.

The sample mean is the average of a set of observations. It provides an estimate of the population mean. The formula to calculate the sample mean \( \bar{x} \) is:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]
Where:

• \( \sum \) signifies the sum of all observations
• \( x_i \) represents each individual observation
• \( n \) is the number of observations in the sample

In our exercise with 36 cans of Coke, the sample mean is used to assess the probability of getting a sample mean of at least 12.19 oz. Ensuring the sample size is sufficiently large is essential for the reliability of the sample mean as an estimator.

Standard deviation measures the dispersion or spread of a dataset relative to its mean. It quantifies how much the values in the dataset differ from the mean. The formula for standard deviation \( \sigma \) is:
\[ \sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n}} \]
Where:

• \( x_i \) is each individual observation
• \( \mu \) is the mean of the dataset
• \( n \) is the number of observations

A smaller standard deviation indicates that data points are close to the mean, whereas a larger standard deviation indicates that data points are spread out over a large range of values.
In our exercise, measuring the standard deviation of the can volumes helps us assess the variability of the fill amount and calculate probabilities based on the normal distribution.

Probability analysis involves calculating the likelihood of various outcomes. Using the normal distribution and Z-scores, we can determine the probability of a single event occurring within a certain range.
In our example, we calculated the probability of a single Coke can containing at least 12.19 oz of soda and found it using the Z-table. For 36 cans, we used the sample mean and adjusted the standard deviation to find the probability of having a sample mean of at least 12.19 oz.
This type of analysis allows us to consider whether it is reasonable to believe the claimed population mean. As demonstrated, the result provided insights into whether the average fill amount was as stated or different, which can be vital for quality control and understanding consumer expectations.