CNNBC recently reported that the mean annual cost of auto insurance is 1007 dollars. Assume the standard deviation is 279 dollars. You take a simple random sample of 55 auto insurance policies.
Find the probability that a single randomly selected value is at least 988 dollars.
\[
P(X> 988)=
\]
Find the probability that a sample of size $n=55$ is randomly selected with a mean that is at least 988 dollars.
\[
P(M> 988)=
\]
Enter your answers as numbers accurate to 4 decimal places.
Question Help: D Post to forum
Submit Question
Answer
Final Answer: The probability that a single randomly selected value is at least 988 dollars is \(\boxed{0.5271}\). The probability that a sample of size 55 is randomly selected with a mean that is at least 988 dollars is \(\boxed{0.6932}\).
Step 1 ：Given that the mean annual cost of auto insurance is 1007 dollars and the standard deviation is 279 dollars, we are asked to find the probability that a single randomly selected value is at least 988 dollars.
Step 2 ：We use the formula for the z-score, which is \(Z = \frac{X - \mu}{\sigma}\), where X is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 3 ：Substituting the given values into the formula, we get \(Z = \frac{988 - 1007}{279} = -0.0681\).
Step 4 ：We then use a z-table or a statistical function to find the probability that a randomly selected value is at least 988 dollars, which is approximately 0.5271.
Step 5 ：Next, we are asked to find the probability that a sample of size 55 is randomly selected with a mean that is at least 988 dollars.
Step 6 ：We use the formula for the z-score in this case, which is \(Z = \frac{M - \mu}{\sigma / \sqrt{n}}\), where M is the sample mean we are interested in, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and n is the sample size.
Step 7 ：Substituting the given values into the formula, we get \(Z = \frac{988 - 1007}{279 / \sqrt{55}} = -0.5050\).
Step 8 ：We then use a z-table or a statistical function to find the probability that a randomly selected sample of size 55 has a mean that is at least 988 dollars, which is approximately 0.6932.
Step 9 ：Final Answer: The probability that a single randomly selected value is at least 988 dollars is \(\boxed{0.5271}\). The probability that a sample of size 55 is randomly selected with a mean that is at least 988 dollars is \(\boxed{0.6932}\).
