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Chapter 5: Problem 30

John Beale of Stanford, California, recorded the speeds of cars driving past his house, where the speed limit read 20 mph. The mean of 100 readings was 23.84 mph, with a standard deviation of 3.56 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 mph or one going 10 mph?

Short Answer

Expert verified
In part a, a car going under the speed limit would be approximately 1.08 standard deviations below the mean. In part b, a car going 34 mph is more unusual, since it is approximately 2.86 standard deviations away from the mean, compared to the car going 10 mph which is approximately 3.89 standard deviations below the mean. This means the car going 10 mph is more unusual.

Step by step solution

01

Calculate the Z-score for Speed Under the Limit

For part a, the speed limit is 20 mph, the mean speed is 23.84 mph, and the standard deviation is 3.56 mph. Substitute these numbers into the Z-score formula: \[Z = (20 - 23.84) / 3.56\] .
02

Calculate the Z-score for 34 mph and 10 mph

For part b, calculate the Z-scores for 34 mph and 10 mph, substituting these values for X in turn: \[Z1 = (34 - 23.84) / 3.56, Z2 = (10 - 23.84) / 3.56\] .
03

Compare the Absolute Values of Z-scores

The sign of the Z-score indicates whether the value is above the mean (positive) or below the mean (negative). When we want to find out which value is more unusual, we are interested in the value that deviates more from the mean, regardless of the direction. Therefore, we compare the absolute values of the Z-scores: |Z1| and |Z2|. The larger the absolute value, the more unusual the speed.

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Key Concepts

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

Standard Deviation
Standard deviation is a measure of how spread out the numbers in a set of data are. It tells us how much the values in the dataset deviate from the mean. When we say the standard deviation is 3.56 mph, it means that most of the car speeds John recorded were within 3.56 mph of the average speed, which was 23.84 mph.
  • If the standard deviation is small, it indicates that the data points tend to be very close to the mean.
  • If it is large, the data points are more spread out over a wide range of values.
Understanding standard deviation helps us determine how typical or unusual a particular data point is when compared to the average.
Mean and Average
The mean, often referred to as the average, is calculated by adding up all the numbers in a dataset and dividing by the total number of values. It's a central value that gives us a general sense of all the data combined. In this exercise, the mean speed of 100 car readings is 23.84 mph.
  • To find the mean, simply sum all recorded speeds and divide by 100.
  • The mean provides a "typical" value around which the data is centered.
Finding the mean is essential as it serves as a reference point for calculating other statistical measures, like the standard deviation and Z-scores.
Statistical Deviation
Statistical deviation refers to the difference between each data point and the mean. In essence, it describes how far each value in a dataset is from the average. Calculating deviations is crucial for understanding the variability in data.
  • A small deviation indicates data points that are similar in value.
  • A large deviation indicates more diversity in the dataset.
This concept is foundational when calculating the standard deviation, as the standard deviation is closely related to the average of all absolute deviations from the mean.
Normal Distribution
Normal distribution, sometimes called a bell curve, is a common pattern for data points where most values cluster around the mean, and fewer fall towards the extremes. In a normal distribution:
  • Approximately 68% of the data points lie within one standard deviation of the mean.
  • About 95% fall within two standard deviations.
  • Almost 99.7% are within three standard deviations.
Understanding normal distribution is valuable in this problem because it allows us to use the Z-score to evaluate how far a car's speed is from the mean, relative to other speeds. It helps identify normal and unusual readings by assessing how many standard deviations a particular speed is from the mean.

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Most popular questions from this chapter

Two companies market new batteries targeted at owners of personal music players. DuraTunes claims a mean battery life of 11 hours, while RockReady advertises 12 hours. a. Explain why you would also like to know the standard deviations of the battery lifespans before deciding which brand to buy. b. Suppose those standard deviations are 2 hours for DuraTunes and 1.5 hours for RockReady. You are headed for 8 hours at the beach. Which battery is most likely to last all day? Explain. c. If your beach trip is all weekend, and you probably will have the music on for 16 hours, which battery is most likely to last? Explain.

The mean household income in the U.S. in 2014 was about \(\$ 72,641\) and the standard deviation was about \(\$ 85,000\). (The median income was \(\$ 51,939 .\) ) If we used the Normal model for these incomes, a. What would be the household income of the top \(1 \% ?\) b. How confident are you in the answer in part a? C. Why might the Normal model not be a good one for incomes?

Some IQ tests are standardized to a Normal model, with a mean of 100 and a standard deviation of 15 a. Draw the model for these IQ scores. Clearly label it, showing what the \(68-95-99.7\) Rule predicts. b. In what interval would you expect the central \(95 \%\) of IQ scores to be found? c. About what percent of people should have IQ scores above \(115 ?\) d. About what percent of people should have IQ scores between 70 and \(85 ?\) e. About what percent of people should have IQ scores above \(130 ?\)

For the car speed data in Exercise 30 ?, recall that the mean speed recorded was \(23.84 \mathrm{mph},\) with a standard deviation of \(3.56 \mathrm{mph}\). To see how many cars are speeding, John subtracts 20 mph from all speeds. a. What is the mean speed now? What is the new standard deviation? b. His friend in Berlin wants to study the speeds, so John converts all the original miles-perhour readings to kilometers per hour by multiplying all speeds by 1.609 (km per mile). What is the mean now? What is the new standard deviation?

Exercise 10 proposes modeling IQ scores with \(N(100,15)\). What IQ would you consider to be unusually high? Explain.
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