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When working with a normal distribution, the Z-score is a way of standardizing a value, making it possible to compare it to the average of a data set. It represents how many standard deviations an element is from the mean. To calculate it, we use the formula:
\[Z = \frac{X - \mu}{\sigma}\]
Where:

• \(X\) is the value being measured
• \(\mu\) is the mean of the distribution
• \(\sigma\) is the standard deviation

A negative Z-score indicates that the value is below the mean, while a positive score shows it is above. In our cork cutting example, this calculation helps determine how close each produced cork diameter is to its respective machine's average.

Probability gives us a measure of how likely an event is to occur, and in the context of normal distribution, it's crucial to determine how many values fall within a specific range. Using Z-scores, we can find the probability of a value falling between two points by checking Z-tables, which show the percentage of the distribution that lies below a particular Z-score.
In our cork cutting case, once Z-scores are calculated for the boundaries, the probability for the range of acceptable corks can be found. Machine 1, with its boundaries of Z-scores between -1 and 1, shows a probability of 68.27%, indicating how many corks fall within this acceptable range. Machine 2, with scores from -7 to 3, suggests a much higher likelihood at up to 99.87% probability for corks fitting the required size.

Standard deviation is a key indicator in statistics that shows how much variation or dispersion exists from the mean in a data set. The smaller the standard deviation, the closer the data points are to the mean.
Mathematically, it is defined as:
\[\sigma = \sqrt{\frac{1}{N} \sum (X_i - \mu)^2}\]
Where:

• \(N\) is the number of data points
• \(X_i\) is each individual data point
• \(\mu\) is the mean

In our exercise, Machine 1 has a higher standard deviation at 0.1 cm, meaning its cork diameters vary more from the mean compared to Machine 2, which has a brighter precision in its output due to a smaller standard deviation of 0.02 cm.

Cumulative probability refers to the sum of probabilities of a random variable or event occurring up to a certain point. It is used to understand the likelihood of a random variable falling within the range of interest.
In terms of normal distribution, cumulative probability is read from Z-tables and represents the fraction of observations expected to fall below a given Z-score.
In the example: Machine 1’s cumulative probability from Z-scores -1 to 1 leads to a 68.27% chance of corks being the right size. For Machine 2, as the Z-score from -7 to the desired range is beyond standard Z-tables, we consider the cumulative probability up to a Z-score of 3, making it more likely within the acceptable cork diameter range.

Statistical comparison connects two or more data sets, allowing for a decision based on quantitative evidence. In statistical analysis, comparison often relies on both differences in probabilities and variabilities determined through standard deviation and mean.
In the cork example, both machines' products were compared by evaluating probabilities derived from their Z-scores. By comparing these statistical indicators:

• Machine 1 produces acceptable corks with a 68.27% probability.
• Machine 2 shows a higher probability, at approximately 99.87% when compared over a significant Z-score range.

These comparisons allow for a confident conclusion: Machine 2 is more reliable for cutting corks within the specified size constraints based on the evidence from the statistical measures.