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A Z-score is a way to quantify the position of a specific data point within a normal distribution. It tells us how many standard deviations a particular value is from the mean. This is particularly important when dealing with normally distributed data, such as the service costs in our example.
To calculate the Z-score, we use the formula:
\[ z = \frac{X - \mu}{\sigma} \]
Where:

• \(X\) is the value of interest, here it is \\(200 (the cost threshold).
• \(\mu\) is the mean of the distribution, which is \\)150.
• \(\sigma\) is the standard deviation, which is \\(25.

By inserting these values, we calculate:
\[ z = \frac{200 - 150}{25} = 2 \]
This tells us that \\)200 is 2 standard deviations above the mean, which can help us find the probability of service costs exceeding the contract threshold.

Probability in statistics helps us understand how likely an event is going to occur. In the context of our exercise, we're interested in the probability that a customer's service costs will exceed \\(200, which is the price of the service contract.
Given the Z-score of 2 that we calculated, we can use the standard normal distribution table to find how likely it is to be below this Z-score.
The table tells us that roughly 97.72% of data points fall below a Z-score of 2. Consequently, the probability of service costs being above \\)200 is:
\[ 1 - 0.9772 = 0.0228 \]
This means there is a 2.28% chance that the service costs will exceed the contract price, allowing the company to predictively manage risk and profits.

Expected value is a fundamental concept in probability and statistics. It provides a measure of the center of a probability distribution and in this context, it calculates anticipations for profit or loss.
For Ward Doering Auto Sales, the expected value of the profit from one service contract can be found by subtracting the average cost of service from the contract price:
\[ \text{Expected Profit} = \text{Contract Price} - \text{Mean Service Cost} \]
Inserting the known values:
\[ \text{Expected Profit} = 200 - 150 = 50 \]
So, the expected profit per contract is \$50. This expectation is based on the average, meaning while individual results may vary, this is the typical profit seen over many contracts.