Question

John Beale of Stanford, \(\mathrm{CA}\), recorded the speeds of cars driving past his house, where the speed limit read \(20 \mathrm{mph}\). The mean of 100 readings was \(23.84 \mathrm{mph}\), with a standard deviation of \(3.56 \mathrm{mph}\). (He actually recorded every car for a two-month period. These are 100 representative readings.) a) How many standard deviations from the mean would a car going under the speed limit be? b) Which would be more unusual, a car traveling \(34 \mathrm{mph}\) or one going \(10 \mathrm{mph}\) ?

Short answer

A car going under the speed limit is -1.08 standard deviations from the mean. A car traveling at 10 mph is more unusual.

Step-by-step solution

Understanding the Question

We are asked to find how many standard deviations a car going under the speed limit (20 mph) is from the mean speed (23.84 mph) and to determine which is more unusual between a car traveling at 34 mph and one going at 10 mph.

Calculating the Z-Score for 20 mph

To find how many standard deviations a car going 20 mph is from the mean, we use the Z-score formula: \( Z = \frac{X - \mu}{\sigma} \) where \( X \) is the speed, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Substitute \( X = 20, \mu = 23.84, \sigma = 3.56 \): \( Z = \frac{20 - 23.84}{3.56} \approx -1.08 \).

Calculating the Z-Score for 34 mph

Now, we calculate the Z-score for a car traveling at 34 mph using the same formula. Substitute \( X = 34 \): \( Z = \frac{34 - 23.84}{3.56} \approx 2.85 \).

Calculating the Z-Score for 10 mph

Next, we calculate the Z-score for a car traveling at 10 mph. Substitute \( X = 10 \): \( Z = \frac{10 - 23.84}{3.56} \approx -3.90 \).

Comparing Unusualness

A measurement is considered more unusual if its Z-score is further from zero. Compare the absolute Z-scores: |2.85| for 34 mph and |3.90| for 10 mph. The Z-score for 10 mph is further from zero, making it more unusual than 34 mph.

Key concepts

Z-Score Calculation

Calculating a Z-score can be a simple but powerful way to understand how unusual or typical a particular data point is within a given data set.
To calculate a Z-score, you apply the formula:

• \( Z = \frac{X - \mu}{\sigma} \)

Here, \( X \) represents the specific data point you're examining, \( \mu \) is the mean or average of your data, and \( \sigma \) is the standard deviation.
This value tells you how many standard deviations away your data point is from the mean.For example, if a car is speeding at 20 mph, and we know the mean speed is 23.84 mph with a standard deviation of 3.56 mph, the Z-score can be calculated as follows:

• Substitute the values into the formula: \( \frac{20 - 23.84}{3.56} \approx -1.08 \).

This means the car is 1.08 standard deviations below the average speed. In general, a negative Z-score indicates a value below the mean, while a positive Z-score indicates a value above the mean.
This understanding helps us to easily compare different data points and their relative positions in a dataset.

Normal Distribution

Normal distribution is a fundamental concept in statistics often referred to as a bell curve due to its bell-shaped appearance. It is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean.Why is this concept important? Here are some reasons:

• Standard deviations define the shape of the normal distribution.
• About 68% of data falls within one standard deviation from the mean, around 95% within two, and 99.7% falls within three standard deviations.

Understanding normal distribution helps in predicting data trends and random variables. In the context of the exercise, by knowing the car speeds are approximately normally distributed, we can more effectively use Z-scores to understand how unusual certain speeds are.For instance, with a speed of 34 mph giving a Z-score of \( 2.85 \), this suggests it's quite an extreme value, falling far from the mean of 23.84 mph. When compared to 10 mph which gives a Z-score of \( -3.90 \), both speeds are unusual, but 10 mph is further from the mean, emphasizing just how rare it is under standard conditions.

Statistical Analysis

Statistical analysis provides methods and tools to analyze and interpret data. It is particularly useful in making decisions based on data. Let's use statistical analysis in this exercise about car speeds to determine unusualness:

• Start by calculating mean and standard deviation, two key statistical metrics.
• Use these metrics to analyze how specific data points deviate from the mean through Z-score calculations.

This systematic approach helps in revealing patterns or anomalies within the data. In the provided solution, the data analysis reveals that the mean car speed is 23.84 mph with a standard deviation of 3.56. By further assessing the Z-scores, we discovered the most unusual car speed recorded in the exercise was 10 mph.
Using statistical methods, we thus conclude this speed is exceptionally low given the spread and center of the dataset.
Learning these concepts can significantly enhance your ability to analyze data effectively and draw meaningful conclusions.