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Chapter 7: Problem 123

Suppose that \(16 \%\) of all drivers in a certain city are uninsured. Consider a random sample of 200 drivers. a. What is the mean value of the number who are uninsured, and what is the standard deviation of the number who are uninsured? b. What is the (approximate) probability that between 25 and 40 (inclusive) drivers in the sample were uninsured? c. If you learned that more than 50 among the 200 drivers were uninsured, would you doubt the \(16 \%\) figure? Explain.

Short Answer

Expert verified
a. The mean value is 32 drivers and the standard deviation is approximately 5.6 drivers. b. The probability of having between 25 and 40 uninsured drivers is quite high according to the empirical rule. c. More than 50 uninsured drivers among the 200 would cast doubt on the 16% figure since it deviates significantly from the expected mean, falling beyond 3 standard deviations away

Step by step solution

01

Calculate the mean

The mean is calculated by multiplying the total number of drivers by the percentage of uninsured drivers, i.e., \(200 \times 0.16 = 32\). This means that, on average, 32 of the randomly sampled drivers are expected to be uninsured.
02

Calculate the standard deviation

The standard deviation can be calculated using the formula: \( \sqrt{n \times p \times (1-p)} \), where \(n\) is the size of the sample, \(p\) is the probability of success. So, inputting the given values, we get: \( \sqrt{200 \times 0.16 \times (1 - 0.16)} = 5.6 \). Therefore, the standard deviation, i.e., the degree of variation from the average number of uninsured drivers is roughly 5.6 people.
03

Calculate the probability of having between 25 and 40 uninsured drivers

Typically, we could use a normal distribution to approximate the binomial distribution and calculate the probability. However, the given exercise asks for an approximate probability, implying we can use the empirical rule or 68-95-99.7 rule, which says in a normal distribution, almost all value falls within 3 standard deviations of mean. Since 25 and 40 are within about 1 and 2 standard deviation from the mean respectively, it can be determined that the probability is quite high. A detailed probability calculation would require understanding of Z-score and standard normal distribution.
04

Analyze the number of uninsured drivers

If it is learned that more than 50 among the 200 drivers were uninsured, which is more than 3 standard deviations away from the mean, it would indeed cast doubt on the initial 16% figure as it is quite unlikely (less than 0.3% chance according to the Empirical Rule) to obtain such a result from a sample if the true proportion of uninsured was really 16%

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Key Concepts

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

Mean and Standard Deviation
The mean and standard deviation are crucial concepts in probability and statistics, especially when dealing with binomial distributions. In this problem, the mean gives us the average number of uninsured drivers expected in the city. To find the mean (\(\mu\)), you simply multiply the sample size by the probability of each driver being uninsured. Therefore, for a sample of 200 drivers with 16% uninsured, the mean is \(200 \times 0.16 = 32\). This means we expect, on average, 32 uninsured drivers.
For standard deviation (\(\sigma\)), the formula \(\sqrt{n \times p \times (1-p)}\) is used, where \(n\) is the sample size and \(p\) is the probability of success (an uninsured driver in these terms). Substituting \(n = 200\) and \(p = 0.16\), we get \(\sqrt{200 \times 0.16 \times (1 - 0.16)} = 5.6\). This means, on average, the number of uninsured drivers will deviate by about 5.6 from the mean value of 32.
Probability Calculation
Calculating probability in a binomial setting often requires approximations, especially with larger sample sizes. Here, we need the probability that between 25 and 40 drivers are uninsured. Binomial distributions can become complex, so a normal approximation is helpful.
For approximation, use the mean and standard deviation calculated earlier. The scenario asks for the probability of falling between 25 and 40 uninsured drivers. Both values fall within a reasonable range of the mean. When calculating exact probabilities, converting numbers to Z-scores is standard, but a broader perspective using the empirical rule gives an insight. This is not perfect but hints that a substantial probability exists for the drivers falling in this range.
Empirical Rule
The empirical rule, or 68-95-99.7 rule, aids in understanding probabilities in normal distributions, which can approximate binomial distributions. The rule helps estimate how data falls in a bell curve.
According to the rule:
  • 68% of data falls within one standard deviation of the mean
  • 95% within two standard deviations
  • 99.7% within three standard deviations
For this exercise, knowing that more than 50 drivers were uninsured deviates more than three standard deviations from the mean of 32 uninsured drivers (with \(\sigma = 5.6\)). The rule informs us that such a result is statistically unlikely if only 16% of drivers are uninsured. If more than 50 uninsured, it casts doubt on the 16% figure due to its improbability under normal conditions.

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