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Z-scores are a valuable tool when working with normal distributions. A Z-score represents the number of standard deviations a data point, such as a fuel efficiency measurement, is from the mean of the distribution. It's calculated using the formula: \[Z = \frac{x - \mu}{\sigma}\] where \(x\) is the value we want to convert (in this case, miles per gallon), \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

For our exercise, if you want to find where 29 mpg stands in respect to a car's typical performance, you plug in 29 for \(x\), 30 for \(\mu\), and 1.2 for \(\sigma\). The Z-score gives you an easy way to check how unusual a specific measurement is by showing it on a standard normal distribution curve. Lower or higher Z-scores indicate more unusual data points.

Probability calculations using Z-scores help us find out how likely (or unlikely) it is for a particular event to occur. In the exercise, we sought probabilities for certain fuel efficiencies:
- The probability of fuel efficiency between two values (e.g., 29 and 31 mpg).
- The rarity of exceptionally low values, like less than 25 mpg.

After computing the Z-scores for these values, you consult a Z-table, a tool that tells you how much of the data falls to the left of a given Z-score in a standard normal distribution. To find the probability that a value is between two points, we simply subtract the smaller probability from the larger one. This provides us with insights into the frequency of those readings, which is particularly useful in understanding everyday likelihoods versus rare readings.

Statistical thresholds are boundaries in data, computed to understand the geventy of extreme or typical performances. One common threshold is the 95th percentile. In this exercise, we wanted to find a threshold \(x^{*}\), such that 95% of the car model's fuel efficiencies exceed this value.

To find this, we need to identify the corresponding Z-score first. When a Z-score has 95% of the distribution to its right, it means only 5% is to its left. This particular Z-score is around -1.645. By using the Z-score formula backwards, we can solve for \(x^{*}\). \[x^{*} = -1.645 \times 1.2 + 30 \approx 28.03\]
Such thresholds are valuable in quality control, checking if products meet a standard or how infrequently they fail—a necessity in many industries for regulation compliance.