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A Z-score, a fundamental concept in statistics, helps to determine how far a specific value is from the mean in terms of standard deviations.
When analyzing normally distributed data, like the weights of hams at Henline Ham Company, the Z-score allows us to find the proportion of hams that weigh less than the label weight.

The formula to calculate the Z-score is as follows:

• \( z = \frac{X - \mu}{\sigma} \)

Where:

• \( X \) is the value of interest (in this case, 9.00 pounds for the label weight).
• \( \mu \) is the mean of the distribution (9.20 pounds).
• \( \sigma \) is the standard deviation (0.25 pounds).

For this scenario, the Z-score calculates to \(-0.80\), implying that a weight of 9.00 pounds is 0.80 standard deviations below the mean.
This Z-score can then be used with a Z-table to find the proportion of the distribution that falls below this point — about 21.19% in this case.

In statistics, the standard deviation measures the dispersion or variability within a set of data.
It tells us how much individual data points differ from the mean.

In the context of the Henline Ham Company, a standard deviation of 0.25 pounds means that the weights of the hams vary by this amount around the average weight of 9.20 pounds.
A smaller standard deviation indicates that the data points (ham weights) are closer to the mean, leading to a more consistent product.

When Glen Henline considers reducing the standard deviation to 0.15 pounds while maintaining the current mean, he would achieve a tighter clustering of ham weights around the mean.

• This means fewer hams would weigh substantially less than the label weight.
• Calculating with the reduced standard deviation, a Z-score of \(-1.33\) is obtained for a 9.00-pound weight.

This adjustment results in only 9.18% of hams weighing less than 9.00 pounds.

The mean, often referred to as the average, is the sum of all values divided by their number.
It provides a central value for a set of data points.

At Henline Ham Company, the mean weight for processed hams is set at 9.20 pounds.
This number signifies the average weight expected across all produced hams, balancing between extremes.
For quality improvement, Glen Henline considers increasing the mean weight to 9.25 pounds.
This change would shift the entire distribution of weights by 0.05 pounds upward.

• Recomputing the Z-score for a 9.00-pound weight gives \(-1.00\).
• Approximately 15.87% of hams would then weigh less than the label weight.

While shifting the mean upward reduces underweight hams, it's less effective than reducing standard deviation at minimizing those falling below label weight.
However, it still provides an easier adjustment to implement for small scale improvements.