91Ó°ÊÓ

The Z-score is a statistical measure that describes a value's relationship to the mean of a group of values. It tells us how many standard deviations a point is from the mean. To calculate a Z-score, you use the formula:

• \( Z = \frac{X - \mu}{\sigma} \)

where \( X \) is the value we're interested in, \( \mu \) is the average value, and \( \sigma \) is the standard deviation.
In the context of our speed limit problem, if we want to find the percentage of vehicles traveling slower than 80 mph, we calculate the Z-score for 80 mph.
The Z-score helps us in identifying where our specific value lies on the normal distribution curve. A positive Z-score means the value is above the mean, while a negative Z-score indicates it's below the mean. Understanding Z-scores is essential for interpreting normal distributions and calculating probabilities.

Probability measures the likelihood of a certain event occurring. It is expressed as a number between 0 and 1, where 0 indicates the event cannot happen, and 1 means the event will certainly happen. In a normal distribution problem, we often use Z-tables to find probabilities linked to Z-scores.
In our exercise, using a Z-table, we found that a Z-score of 1.55 (for 80 mph) roughly corresponds to 93.94%. This percentage indicates the probability that a vehicle travels slower than 80 mph. Similarly, by finding the probability for Z-scores corresponding to 60 mph and for the speed limit of 70 mph, we can determine the percentage of vehicles traveling within specific speed ranges or over the speed limit.
Thus, probability gives us a statistical insight into how values within certain ranges behave relative to their distribution.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation suggests that the values tend to be close to the mean, while a high standard deviation indicates a wide spread around the mean. In the normal distribution of vehicle speeds, the standard deviation is crucial for determining how spread out the speed values are.

• The formula for standard deviation is usually given as \( \sigma \).

In our problem, the standard deviation is 4.78 mph. This number tells us that the vehicle speeds are generally spread around the mean of 72.6 mph by roughly 4.78 mph. When calculating Z-scores, the standard deviation is used to scale the deviation of our specific value from the mean, giving us a clearer understanding of where the value falls on the distribution curve.

In our exercise, the speed limit is a benchmark used to assess whether vehicles are traveling at safe speeds or exceeding recommended limits. The problem's context is significant because normal distribution allows for assessing how many vehicles operate within acceptable limits as defined by the speed limit.
To find how many vehicles are speeding (traveling faster than 70 mph), we calculate a Z-score for 70 mph. This Z-score helps us determine the percentage of vehicles traveling above this speed limit by checking the complementary cumulative probability (the probability of exceeding that Z-score) with the Z-table.
The concept of a speed limit in such statistical problems serves as a practical application of normal distribution in real-life scenarios like traffic management, where determining the frequency and distribution of speeds helps in policy evaluation and safety optimization.