Question

Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of \(71 \mathrm{mph}\) and a standard deviation of \(8 \mathrm{mph}\). a. The current speed limit is \(65 \mathrm{mph}\). What is the proportion of vehicles less than or equal to the speed limit? b. What proportion of the vehicles would be going less than \(50 \mathrm{mph}\) ? c. A new speed limit will be initiated such that approximately \(10 \%\) of vehicles will be over the speed limit. What is the new speed limit based on this criterion? d. In what way do you think the actual distribution of speeds differs from a normal distribution?

Short answer

a. 22.66% b. 0.43% c. ~81 mph d. Skewed by traffic conditions.

Step-by-step solution

Understand the Problem

We are given a normal distribution for vehicle speeds with a mean of \(71 \text{ mph}\) and a standard deviation of \(8 \text{ mph}\). We need to find proportions related to certain speed limits and determine a new speed limit using the properties of this distribution.

Calculate Proportion for Speed Limit 65 mph

To find the proportion of vehicles driving less than or equal to \(65\text{ mph}\), calculate the z-score using the formula: \[z = \frac{x - \mu}{\sigma}\]where \(x = 65\), \(\mu = 71\), and \(\sigma = 8\). This yields:\[z = \frac{65 - 71}{8} = -0.75\]Using a standard normal distribution table or calculator for \(z = -0.75\), we find the proportion is roughly \(0.2266\) (22.66%).

Calculate Proportion for Speed Less than 50 mph

Again, calculate the z-score for \(50 \text{ mph}\) using:\[z = \frac{50 - 71}{8}\]which gives:\[z = -2.625\]Using the standard normal distribution table, the proportion is about \(0.0043\) (0.43%).

Determine New Speed Limit for 10% Above Limit

For the new speed limit where 10% are above, we need to find the 90th percentile. The z-score corresponding to 90% is approximately \(1.28\). Use the z-score formula to find \(x\):\[x = \mu + z\sigma = 71 + 1.28 \times 8 = 81.24 \]The new speed limit would be approximately \(81 \text{ mph}\).

Discuss Difference in Actual Distribution

Real-world speed distributions may not be perfectly normal due to factors like traffic conditions, road work, weather conditions, and human factors. These can cause the distribution to have more skew (either left or right) compared to a normal distribution.

Key concepts

Z-Score

A z-score is a statistical measurement that describes a value's position in relation to the mean of a group of values. To find the z-score for a number, subtract the mean of the dataset from the number, then divide the result by the standard deviation. Formula: \[ z = \frac{x - \mu}{\sigma}\]Where:

• \(x\) is the value being compared,
• \(\mu\) is the mean of the dataset,
• and \(\sigma\) is the standard deviation.

Z-scores tell us how many standard deviations away from the mean a particular score is located. A negative z-score means the number is below the mean, while a positive z-score means it's above the mean.
For example, in our exercise, the z-score of a speed of 65 mph within our vehicle speed distribution is -0.75. This implies that 65 mph is 0.75 standard deviations below the mean speed of 71 mph.

Percentile

Percentiles show the relative standing of a data point within a data set. The percentile rank of a score is expressed as a percentage of scores within the data set that fall below that score. If you score higher than 85% of the individuals in your group, you are in the 85th percentile.
In the context of the exercise, we calculate the percentile for speeds like 65 mph and 50 mph using their z-scores. For instance, a z-score of -0.75 correlates to the 22.66th percentile. This means that 22.66% of vehicles are traveling at or below 65 mph.
Percentiles are useful for comparing the standings of different scores across various datasets and for determining cut-off points for values within a distribution. For instance, setting speed limits so that only the top 10% exceed the limit involves setting the speed at the 90th percentile.

Mean and Standard Deviation

The mean and standard deviation are fundamental concepts in statistics. The mean, denoted \(\mu\), represents the average of a set of numbers, calculated by dividing the sum of all numbers by the count of numbers.
The standard deviation measures the spread of a dataset, indicating how much each number in a set deviates from the mean. A small standard deviation indicates numbers are close to the mean, whereas a large standard deviation shows a wider spread.
In our example, the average (mean) speed is 71 mph and the standard deviation is 8 mph. This implies that most vehicle speeds will range between 63 mph to 79 mph (one standard deviation from the mean).
These metrics help us understand the central tendency and variability of the data, playing a critical role in determining how normal distributions are formed and interpreted.

Real-World Distribution Differences

While a normal distribution provides an ideal mathematical model, actual distributions often deviate due to external factors:

• **Traffic Conditions**: Congestion could lower average speeds.
• **Road Work**: Construction may create temporary bottlenecks.
• **Weather**: Inclement weather could drastically alter safe driving speeds.
• **Human Behavior**: Drivers' risk-taking and attentiveness vary significantly, impacting the overall speed profile.

These elements can introduce skewness, where one tail of the distribution is longer or fatter than the other, resulting in a non-normal distribution. It's also possible for the data to exhibit kurtosis, where the distribution might be more peaked or flatter compared to a perfect normal distribution. Therefore, when making real-world decisions, it’s essential to consider these deviations from the normal model.