Question

Thirty percent of all automobiles undergoing an emissions inspection at a certain inspection station fail the inspection. a. Among 15 randomly selected cars, what is the probability that at most 5 fail the inspection? b. Among 15 randomly selected cars, what is the probability that between 5 and 10 (inclusive) fail the inspection? c. Among 25 randomly selected cars, what is the mean value of the number that pass inspection, and what is the standard deviation? d. What is the probability that among 25 randomly selected cars, the number that pass is within 1 standard deviation of the mean value?

Short answer

a. The probability that at most 5 cars fail inspection out of 15 chosen randomly is calculated in Step 1. \n b. The probability that between 5 and 10 cars fail inspection out of 15 randomly chosen cars is calculated in Step 2. \n c. The mean value of the number of cars that pass inspection and the standard deviation for a sample size of 25 are calculated in Step 3. \n d. The probability that the number of cars that pass is within 1 standard deviation of the mean in a sample of 25 randomly chosen cars is calculated in Step 4.

Step-by-step solution

Calculate probability of at most 5 failures in 15 trials

In this step, calculate the binomial probability \(\sum_ {k=0}^{5} C_ {15,k} (0.30)^k (0.70) ^{15-k}\), where \(C_{n, k}\) is a binomial coefficient.

Calculate probability of having between 5 and 10 failures in 15 trials

In this step, calculate the binomial probability \(\sum_ {k=5}^{10} C_ {15,k} (0.30)^k (0.70) ^{15-k}\)

Calculate the mean and standard deviation for 25 trials

The mean of a binomial distribution is given by \(\mu = np\) and standard deviation by \(\sigma = \sqrt{np(1-p)}\). Here, \(n = 25\) and \(p=0.30\). We need the number of cars that pass the inspection, not fail. So, replace \(p\) with \(1-p=0.70\) to get mean and standard deviation.

Find the probability of number of cars that pass within 1 standard deviation of the mean

This step requires calculating the binomial probability \(\sum_ {\mu - \sigma}^{\mu + \sigma} C_{25,k} (0.70)^k (0.30) ^{25-k}\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation from Step 3.