Question

In a study investigating the effect of car speed on accident severity, the vehicle speed at impact was recorded for 5,000 fatal accidents. For these accidents, the mean speed was \(42 \mathrm{mph}\) and the standard deviation was \(15 \mathrm{mph}\). A histogram revealed that the vehicle speed distribution was mound shaped and approximately symmetric. a. Approximately what percentage of the vehicle speeds were between 27 and 57 mph? b. Approximately what percentage of the vehicle speeds exceeded 57 mph?

Short answer

a. Approximately 68.26% of the vehicle speeds were between 27 and 57 mph. b. Approximately 65.87% of the vehicle speeds exceeded 57 mph.

Step-by-step solution

Calculate the Z-scores

First determine the Z-scores for 27 and 57 mph using the formula: \(Z = \frac{(X - \mu)}{\sigma}\) where X is the speed, \(\mu\) is the mean and \(\sigma\) is the standard deviation. For 27 mph the Z-score would be \(Z_1 = \frac{(27 - 42)}{15} = -1\), and for 57 mph it would be \(Z_2 = \frac{(57 - 42)}{15} = 1\)

Use the Z-table to find probabilities

Now we need to use the Z-scores to find probabilities. Using the Z-table, the area for Z=-1 (or equivalently 1) is 0.3413 for both. For Z=1, it's 0.3413 too.

Calculating required Probabilities

For part a, we need to find the area between -1 and 1. Because the normal distribution is symmetric, we can simply multiply the area for 1 by 2 giving \(0.3413*2=0.6826\). Thus approximately 68.26% of vehicle speeds were between 27 and 57 mph. For part b, we need the area for Z > 1 which would be \(1-0.3413 = 0.6587\). Thus approximately 65.87% of vehicle speeds exceeded 57 mph.

Key concepts

Z-scores Calculation

When analyzing statistical data, the concept of Z-scores is pivotal for interpreting the position of a single data point within a distribution. In simple terms, a Z-score indicates how many standard deviations a data point is from the mean. A Z-score of 0 means the data point is exactly at the mean, while a positive or negative value represents a position above or below the mean respectively.

To calculate a Z-score, the formula \(Z = \frac{(X - \mu)}{\sigma}\) is used where \(X\) is the value of interest, \(\mu\) denotes the population mean, and \(\sigma\) is the standard deviation. For example, if you wanted to find out how usual a speed of 27 mph is within our road accident severity study, calculating the Z-score with the given data points yields \(Z_1 = \frac{(27 - 42)}{15} = -1\). This result shows that a vehicle speed of 27 mph is 1 standard deviation below the mean, implying it's less than typical.

Understanding Z-scores can help students visualize the relative position of data points and is a fundamental skill in statistics. It bridges the gap between individual data points and the overall data set distribution, allowing for a deeper comprehension of data analysis.

Normal Distribution

The normal distribution, often referred to as the bell curve because of its distinctive shape, is a continuous probability distribution that is symmetrical about the mean. It's a powerful tool in statistics that allows us to make inferences about populations.

Characterized by its mean (\(\mu\)) and standard deviation (\(\sigma\)), it shows that data near the mean are more frequent in occurrence than data far from the mean. A key property of the normal distribution is that approximately 68% of the data lies within one standard deviation of the mean, 95% within two, and 99.7% within three, which is known as the Empirical Rule.

In our vehicle speed study, since the histogram of speeds was mound-shaped and symmetric, we can confidently apply the normal distribution model. This enables us to use Z-scores and calculate probabilities for specific ranges of speed, which directly correlate to accident severity — an application that showcases how statistical models can provide insights into real-world phenomena.

Histogram Analysis

A histogram is a graphical representation of the distribution of numerical data. It is an estimate of the probability distribution of a continuous variable and was used in our vehicle speed study to show the distribution of different speeds at the time of the accidents.

Histograms are beneficial because they reveal the underlying frequency distribution (shape) of the data set. They're particularly useful for demonstrating the shape of data distribution, like whether it’s skewed or symmetric, as with the approximately normal distribution in the provided vehicle speed data.

For the histogram that displayed the vehicle speeds, the 'mound-shaped' quality suggests the data closely follows a normal distribution, which subsequently justifies our use of Z-scores and normal distribution properties for analysis. Moreover, inspection of a histogram provides intuitive understanding of the data — in this case, most accidents occurred around the average speed, with fewer occurrences of extremely high or low speeds. Students are encouraged to carefully examine such visual data representations to gain insights before proceeding to quantitative analysis.