Question 1164884: CNNBC recently reported that the mean annual cost of auto insurance is 1036 dollars. Assume the standard deviation is 268 dollars, and the cost is normally distributed. You take a simple random sample of 35 auto insurance policies. Round your answers to 4 decimal places.
What is the distribution of X?X~ N(________,_______)
What is the distribution of ¯x?¯x~ N(________,__________)
What is the probability that one randomly selected auto insurance is less than $996?
a simple random sample of 35 auto insurance policies, find the probability that the average cost is less than $996.
For part d), is the assumption of normal necessary? Yes or No
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! To solve this, we will use the parameters of the population and the **Central Limit Theorem** to find the probabilities for both an individual policy and a sample mean.
### a) What is the distribution of ?
The distribution of individual auto insurance costs () is given as normally distributed with the population mean and standard deviation.
### b) What is the distribution of ?
For a sample of size , the sampling distribution of the mean () is also normal.
* **Mean ():** Equal to the population mean, **1036**.
* **Standard Error ():** Calculated as .
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### c) What is the probability that one randomly selected auto insurance is less than $996?
We need to find . First, we calculate the -score:
Using a standard normal distribution table or calculator for :
**Probability = 0.4407**
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### d) For a simple random sample of 35 auto insurance policies, find the probability that the average cost is less than $996.
We need to find . We use the standard error () to find the new -score:
Using a standard normal distribution table or calculator for :
**Probability = 0.1886**
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### e) For part d), is the assumption of normal necessary?
**No**.
Even if the original population were not normally distributed, the **Central Limit Theorem** states that the sampling distribution of the mean will be approximately normal because the sample size () is greater than 30.
Would you like me to show you how the probability changes if we increase the sample size to 100?
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