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Calculating the z-score is a key step when dealing with normal distributions in statistics. The z-score allows us to determine how far away a specific score is from the mean, measured in terms of standard deviations. The formula to calculate the z-score is given by: \[Z = \frac{(X - \mu)}{\sigma}\]Here’s what the formula means:

• X: The value of the score (such as the number of bilingual officers we're considering).
• \(\mu\): The mean or average number of bilingual officers, which is 196 in this case.
• \(\sigma\): The standard deviation, which is given as 22.

To calculate the z-score for any value, simply substitute these numbers into the formula. For example, if we want to find the z-score for 150 officers, it would look like:\[Z = \frac{(150 - 196)}{22}\]This calculation tells us how many standard deviations the value 150 is away from the average 196.

Standard deviation is a measure that indicates the amount of variation or dispersion in a set of values. When a dataset has a high standard deviation, it means the values are spread out widely around the mean. Conversely, a low standard deviation means the data points are close to the mean. In our case, we have a standard deviation of 22 for the number of bilingual officers in army battalions. This tells us that the typical battalion's count of bilingual officers tends to deviate by about 22 from the average, which is 196. Understanding the standard deviation helps in understanding the z-score calculation, because the z-score essentially measures how many standard deviations away from the mean a particular value lies. When calculating percentages or probabilities related to a normal distribution, knowing the standard deviation is crucial as it scales the z-scores appropriately.

The z-table, also known as the standard normal table, is a mathematical table that allows us to find the probability or percentile of a z-score in a standard normal distribution. Once the z-scores for the values of interest are calculated, the next step involves using them to find the corresponding area under the normal curve. For example, if you calculate a z-score and get a value such as -2.09, the z-table will help you find the probability that a value will be less than X. The numbers in the table essentially represent the cumulative probability up to a point. To find the area between two z-scores, find each score's cumulative probability using the z-table and subtract the smaller probability from the larger one. This gives us the percentage of the population within that range.

After calculating the z-scores and finding their corresponding probabilities using the z-table, the final step is to determine the percentage or probability that falls between certain values. Here's how you do it:

• Calculate the z-scores for the lower and upper bounds of your interest range (like 150 and 200).
• Use the z-table to find the cumulative probabilities for each z-score.
• Subtract the cumulative probability at the lower bound from the cumulative probability at the upper bound to get the percentage in between.

This subtraction tells you how likely it is for a random selection to fall within this range of values under the normal distribution. This is particularly useful in scenarios where interest lies in knowing how often certain conditions occur.