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Chapter 4: Problem 39

In a study investigating the effect of car speed on accident severity, 5000 reports of fatal automobile accidents were examined, and the vehicle speed at impact was recorded for each one. For these 5000 accidents, the average speed was \(42 \mathrm{mph}\) and the standard deviation was \(15 \mathrm{mph}\). A histogram revealed that the vehicle speed at impact distribution was approximately normal. a. Roughly what proportion of vehicle speeds were between 27 and \(57 \mathrm{mph}\) ? b. Roughly what proportion of vehicle speeds exceeded \(57 \mathrm{mph}\) ?

Short Answer

Expert verified
a. The proportion of vehicle speeds between 27 and 57 mph is approximately 68.26%. b. The proportion of vehicle speeds exceeding 57 mph is approximately 65.87%. These are rough estimates based on Z-scores and standard normal distribution.

Step by step solution

01

Calculate Z-Scores

The Z-score associated with a value is a measure of how many standard deviations the value is away from the mean. For a normal distribution, the Z-score can be calculated using the formula: \(Z = (X - 渭) / 蟽\), where \(X\) is the value, \(渭\) is the mean and \(蟽\) is the standard deviation. Calculate the Z-scores for 27 mph and 57 mph using the provided mean (42 mph) and standard deviation (15 mph). For \(X = 27\) mph, \(Z_1 = (27-42)/15 = -1\). For \(X = 57\) mph, \(Z_2 = (57-42)/15 = 1\).
02

Calculate Proportions

With the z-scores, you can now use a standard normal distribution table to find the proportions. For z-scores of -1 and 1, the proportion in a standard normal distribution is approximately 0.3413 for each. Therefore, the proportion of car speeds between 27 mph and 57 mph is approximately \(2 * 0.3413 = 0.6826\) or 68.26%.
03

Calculate the Proportion for Speeds Exceeding 57 mph

To calculate the proportion of speeds exceeding 57 mph, you'll only need the z-score of 1. Since the total area under a normal distribution is 1, the proportion exceeding this z-score is \(1 - 0.3413 = 0.6587\) or 65.87%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-score
The concept of a Z-score is crucial when dealing with a normal distribution. A Z-score tells us how many standard deviations a particular value is from the mean of the distribution. This is useful because it enables us to understand where a value falls within a distribution. For example, if you have a Z-score of 2, this means that the value lies two standard deviations above the mean. This calculation helps statisticians determine how unusual or typical a certain data point is in a dataset. To find a Z-score, use the formula: \[ Z = \frac{X - \mu}{\sigma} \] Where:
  • \(X\) is the value you are examining
  • \(\mu\) is the mean of the dataset
  • \(\sigma\) is the standard deviation
In the context of the car speed exercise, the Z-score helped us see how speeds of 27 mph and 57 mph relate to the average speed. Thus, a Z-score gives you a way to compare different values within any normal distribution, making this concept extremely valuable.
Standard Deviation
The standard deviation is a measure that explains how much individual data points in a set vary, on average, from the mean. It's a key element in statistics used to quantify the amount of variation or dispersion in a dataset. When the standard deviation is small, it indicates that the data points are clustered closely around the mean. Conversely, a large standard deviation signifies that data points are spread out over a broader range. In the case of our exercise on car accident speeds, we had a standard deviation of 15 mph. This means that most of the speed data points fall within a 15 mph range of the average speed, which is 42 mph. Understanding standard deviation aids in analyzing how variable or consistent the data is, which can be particularly useful when assessing risk factors or performance metrics.
Proportions in Statistics
Proportions in statistics refer to the fraction of data points that fall within a certain range in a data set. Understanding proportions helps to interpret large datasets by translating numbers into percentages or probabilities. For a normal distribution, once you calculate Z-scores, you can find proportions using a standard normal distribution table. This table helps identify the percentage of data points that lie below, above, or between specific Z-scores. In our example, we calculated the proportion of car speeds between 27 mph and 57 mph by using their Z-scores. The table showed that approximately 68.26% of vehicles were driving within this range. Meanwhile, for speeds exceeding 57 mph, the proportion was about 65.87%. Therefore, by converting data into proportions, statisticians can easily communicate data insights and make informed decisions based on statistical evidence.

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Most popular questions from this chapter

The Insurance Institute for Highway Safety (www.iihs.org, June 11,2009 ) published data on repair costs for cars involved in different types of accidents. In one study, seven different 2009 models of mini- and micro-cars were driven at 6 mph straight into a fixed barrier. The following table gives the cost of repairing damage to the bumper for each of the seven models. $$ \begin{array}{lc} \text { Model } & \text { Repair Cost } \\ \hline \text { Smart Fortwo } & \$ 1,480 \\ \text { Chevrolet Aveo } & \$ 1,071 \\ \text { Mini Cooper } & \$ 2,291 \\ \text { Toyota Yaris } & \$ 1,688 \\ \text { Honda Fit } & \$ 1,124 \\ \text { Hyundai Accent } & \$ 3,476 \\ \text { Kia Rio } & \$ 3,701 \\ \hline \end{array} $$ Compute the values of the mean and median. Why are these values so different? Which of the two-mean or median-appears to be better as a description of a typical value for this data set?

In 1997 , a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessey v. Digital Equipment Corporation). The jury awarded about \(\$ 3.5\) million for pain and suffering, but the court then set aside that award as being unreasonable compensation. In making this determination, the court identified a "normative" group of 27 similar cases and specified a reasonable award as one within 2 standard deviations of the mean of the awards in the 27 cases. The 27 award amounts were (in thousands of dollars) \(\begin{array}{rrrrrrrr}37 & 60 & 75 & 115 & 135 & 140 & 149 & 150 \\ 238 & 290 & 340 & 410 & 600 & 750 & 750 & 750 \\\ 1050 & 1100 & 1139 & 1150 & 1200 & 1200 & 1250 & 1576 \\ 1700 & 1825 & 2000 & & & & & \end{array}\) What is the maximum possible amount that could be awarded under the "2-standard deviations rule?"

The Insurance Institute for Highway Safety (www.iihs.org, June 11,2009 ) published data on repair costs for cars involved in different types of accidents. In one study, seven different 2009 models of mini- and micro-cars were driven at 6 mph straight into a fixed barrier. The following table gives the cost of repairing damage to the bumper for each of the seven models: $$ \begin{array}{lc} \text { Model } & \text { Repair Cost } \\ \hline \text { Smart Fortwo } & \$ 1,480 \\ \text { Chevrolet Aveo } & \$ 1,071 \\ \text { Mini Cooper } & \$ 2,291 \\ \text { Toyota Yaris } & \$ 1,688 \\ \text { Honda Fit } & \$ 1,124 \\ \text { Hyundai Accent } & \$ 3,476 \\ \text { Kia Rio } & \$ 3,701 \\ \hline \end{array} $$ a. Compute the values of the variance and standard deviation. The standard deviation is fairly large. What does this tell you about the repair costs? b. The Insurance Institute for Highway Safety (referenced in the previous exercise) also gave bumper repair costs in a study of six models of minivans (December 30,2007 ). Write a few sentences describing how mini- and micro-cars and minivans differ with respect to typical bumper repair cost and bumper repair cost variability. $$ \begin{array}{lr} \text { Model } & \text { Repair Cost } \\ \hline \text { Honda Odyssey } & \$ 1,538 \\ \text { Dodge Grand Caravan } & \$ 1,347 \\ \text { Toyota Sienna } & \$ 840 \\ \text { Chevrolet Uplander } & \$ 1,631 \\ \text { Kia Sedona } & \$ 1,176 \\ \text { Nissan Quest } & \$ 1,603 \\ \hline \end{array} $$

Suppose that 10 patients with meningitis received treatment with large doses of penicillin. Three days later, temperatures were recorded, and the treatment was considered successful if there had been a reduction in a patient's temperature. Denoting success by \(S\) and failure by \(\mathrm{F}\), the 10 observations are \(\begin{array}{llllllllll}\text { S } & \text { S } & \text { F } & \text { S } & \text { S } & \text { S } & \text { F } & \text { F } & \text { S } & \text { S }\end{array}\) a. What is the value of the sample proportion of successes? b. Replace each \(S\) with a 1 and each \(F\) with a 0 . Then calculate \(\bar{x}\) for this numerically coded sample. How does \(\bar{x}\) compare to \(\hat{p}\) ? c. Suppose that it is decided to include 15 more patients in the study. How many of these would have to be S's to give \(\hat{p}=.80\) for the entire sample of 25 patients?

A sample of concrete specimens of a certain type is selected, and the compressive strength of each specimen is determined. The mean and standard deviation are calculated as \(\bar{x}=3000\) and \(s=500\), and the sample histogram is found to be well approximated by a normal curve. a. Approximately what percentage of the sample observations are between 2500 and 3500 ? b. Approximately what percentage of sample observations are outside the interval from 2000 to 4000 ? c. What can be said about the approximate percentage of observations between 2000 and 2500 ? d. Why would you not use Chebyshev's Rule to answer the questions posed in Parts (a)-(c)?
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