Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.R.19

Chapter 4, Problem 4.R.19

In Exercises 19 and 20, find the mean, variance, and standard deviation of the binomial distribution for the given random variable. Interpret the results and determine any unusual values.
About 13% of U.S. drivers are uninsured. You randomly select eight U.S. drivers and ask them whether they are uninsured. The random variable represents the number who are uninsured. (Source: Insurance Research Council)

Verified step by step guidance

Step 1: Identify the parameters of the binomial distribution. The problem states that the probability of success (a driver being uninsured) is p = 0.13, the number of trials (drivers selected) is n = 8, and the random variable X represents the number of uninsured drivers.

Step 2: Calculate the mean (expected value) of the binomial distribution using the formula:

μ = n × p

. Substitute n = 8 and p = 0.13 into the formula.

Step 3: Calculate the variance of the binomial distribution using the formula:

σ_{2} = n × p × (1 - p)

. Substitute n = 8 and p = 0.13 into the formula.

Step 4: Calculate the standard deviation by taking the square root of the variance:

σ = √ σ_{2}

. Use the variance calculated in Step 3.

Step 5: Interpret the results. The mean represents the expected number of uninsured drivers out of 8. The standard deviation measures the typical deviation from the mean. To determine unusual values, use the rule of thumb that values more than 2 standard deviations away from the mean (i.e.,

μ ± 2 σ

) are considered unusual.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the 'success' is defined as a driver being uninsured. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p), which allows for the calculation of probabilities for different outcomes.

Mean, Variance, and Standard Deviation

In statistics, the mean of a binomial distribution is calculated as n*p, where n is the number of trials and p is the probability of success. The variance is given by n*p*(1-p), and the standard deviation is the square root of the variance. These measures provide insights into the central tendency and the spread of the distribution, helping to interpret the likelihood of different outcomes.

Unusual Values

Unusual values in a statistical context typically refer to outcomes that fall outside the expected range, often defined as more than two standard deviations from the mean. In the case of the binomial distribution, identifying unusual values helps in assessing the likelihood of extreme outcomes, such as a significantly higher or lower number of uninsured drivers than expected, which can inform further analysis or decision-making.

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Step 3: Get P-Value

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