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a. What is the distribution of X? X - N( b. What is the distribution of I? I ~ N( c. What is the probability that one randomly selected auto insurance is more than $1100?

a. What is the distribution of X? X - N( b. What is the distribution of I? I ~ N( c. What is the probability that one randomly selected auto insurance is more than $1100?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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### Understanding Normal Distributions and Probability Calculations **Context:** CNNBC recently reported that the mean annual cost of auto insurance is $1008. The standard deviation is $242, and the cost is normally distributed. A simple random sample of 10 auto insurance policies is taken. The goal is to calculate probabilities related to these costs. **Instructions:** **1. Determine Distributions:** - **a.** Identify the distribution for a single auto insurance cost, \( X \). - \( X \sim N(1008, 242) \) - **b.** Identify the distribution for the sample mean, \( \bar{x} \), when the sample size is 10. - \( \bar{x} \sim N \left(1008, \frac{242}{\sqrt{10}} \right) \) **2. Calculating Probabilities:** - **c.** Find the probability that one randomly selected auto insurance cost is more than $1100. - **d.** Using a simple random sample of 10 auto insurance policies, compute the probability that the average cost is more than $1100. **3. Assumptions:** - **e.** For the probability in part (d), determine if the assumption of normality is necessary. - Options: Yes ○ No ○ **Explanation:** - The use of the normal distribution in part (a) reflects the given cost distribution with a mean ($\mu$) of 1008 and a standard deviation ($\sigma$) of 242. - In part (b), the distribution of the sample mean considers the Central Limit Theorem, which allows us to assume a normal distribution for the sample mean, with a standard deviation (\(\sigma_{\bar{x}}\)) adjusted by the square root of the sample size (\(n=10\)). - Parts (c) and (d) involve calculating probabilities based on these distributions. - Part (e) questions whether the normality assumption is needed for part (d), considering the sample size and distribution characteristics. The normality assumption is critical when the sample size is small unless the population distribution is already normal.
Transcribed Image Text:### Understanding Normal Distributions and Probability Calculations **Context:** CNNBC recently reported that the mean annual cost of auto insurance is $1008. The standard deviation is $242, and the cost is normally distributed. A simple random sample of 10 auto insurance policies is taken. The goal is to calculate probabilities related to these costs. **Instructions:** **1. Determine Distributions:** - **a.** Identify the distribution for a single auto insurance cost, \( X \). - \( X \sim N(1008, 242) \) - **b.** Identify the distribution for the sample mean, \( \bar{x} \), when the sample size is 10. - \( \bar{x} \sim N \left(1008, \frac{242}{\sqrt{10}} \right) \) **2. Calculating Probabilities:** - **c.** Find the probability that one randomly selected auto insurance cost is more than $1100. - **d.** Using a simple random sample of 10 auto insurance policies, compute the probability that the average cost is more than $1100. **3. Assumptions:** - **e.** For the probability in part (d), determine if the assumption of normality is necessary. - Options: Yes ○ No ○ **Explanation:** - The use of the normal distribution in part (a) reflects the given cost distribution with a mean ($\mu$) of 1008 and a standard deviation ($\sigma$) of 242. - In part (b), the distribution of the sample mean considers the Central Limit Theorem, which allows us to assume a normal distribution for the sample mean, with a standard deviation (\(\sigma_{\bar{x}}\)) adjusted by the square root of the sample size (\(n=10\)). - Parts (c) and (d) involve calculating probabilities based on these distributions. - Part (e) questions whether the normality assumption is needed for part (d), considering the sample size and distribution characteristics. The normality assumption is critical when the sample size is small unless the population distribution is already normal.
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