Question

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

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

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.

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.

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.

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%