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Understanding probability calculations can be crucial when dealing with normally distributed data. In statistics, finding how likely an event will happen involves certain steps, especially with a normal distribution.
First, when you're asked about the probability of a shoe weighing more than a certain amount, you're essentially looking for the area under the tail of a bell curve.

• You take the known mean (average measure) and standard deviation (how spread out the weights are) to figure this out.
• Using these, you calculate something called a Z-score. This tells you how far and in what direction your specific measure (like 13 ounces) deviates from the mean.

Once you have the Z-score, you can use a standard normal distribution table or a calculator to find the probability related to that score. For instance, in part a of this exercise, the probability a shoe weighs more than 13 ounces was found using these steps, leading to a result of 2.28%.

Standard deviation is a measure of variation or dispersion in a set of data values. When you need most weights to fall below a certain threshold, adjusting the standard deviation becomes necessary.
Say, you want 99.9% of shoes to weigh less than a certain amount; you first need to know the z-score relating to 99.9%. This score is approximately 3.09, found in a standard normal distribution table.

• With this z-score, you rearrange the formula for a standard normal distribution.
• You solve for what the standard deviation should be, if your mean remains the same, which in this case would be 12 ounces.

In the exercise given, working through the formula revealed that adjusting the standard deviation to approximately 0.324 ounces ensured that 99.9% of the shoes didn't exceed 13 ounces.

The Z-score helps you understand how a specific data point relates to the mean of a set of data. Specifically, it tells you how many standard deviations away a data point is from the mean. This is valuable when interpreting the probability that certain data appears in your normal distribution.
For instance, if a shoe weighing 13 ounces corresponds to a Z-score of 2, it indicates that it trails 2 standard deviations above the mean of 12 ounces.

• To use Z-scores effectively, start by calculating it using the formula \( z = \frac{X - \mu}{\sigma} \).
• This formula takes your specific observation \( X \), subtracts the mean \( \mu \), and divides by the standard deviation \( \sigma \).

In part c of the exercise, if we want 99.9% of weights to be under 13 ounces with a given standard deviation, the Z-score changes how you'd adjust your mean to meet this condition. Calculating this yielded a new mean weight of around 11.455 ounces, ensuring the desired distribution.