More on insurance An insurance company knows that in the...

More on insurance An insurance company knows that in the entire population of homeowners, the mean annual loss from fire is $\mu=\$ 250$ and the standard deviation of the loss is $\sigma=\$ 300$ . The distribution of losses is strongly right-skewed: many policies have $\$ 0$ loss, but a few have large losses. If the company sells $10,000$ policies, can it safely be its rates on the assumption that its average loss will be no greater than $\$ 275 ?$ Follow the four-step process.

More on insurance An insurance company knows that in the entire population of homeowners, the mean annual loss from fire is $\mu=\$ 250$ and the standard deviation of the loss is $\sigma=\$ 300$ . The
distribution of losses is strongly right-skewed: many policies have $\$ 0$ loss, but a few have large losses. If the company sells $10,000$ policies, can it safely be its rates on the assumption that its average loss will be no greater than $\$ 275 ?$ Follow the four-step process.

Key Concepts

Skewed Distributions and Their Impact

Skewed distributions have an asymmetry in the data, often with a long tail on one side. Even though individual observations from a skewed distribution might not be normally distributed, the average of many observations can still be approximately normal due to the Central Limit Theorem. This is important in risk analysis and pricing in insurance, as it allows actuaries to use parametric methods based on normality when dealing with aggregate data, despite the underlying individual data being skewed.

Central Limit Theorem

The Central Limit Theorem is a fundamental principle in statistics which states that, regardless of the shape of the population distribution, the distribution of the sample means will approach a normal distribution as the sample size becomes large. This concept is crucial when analyzing averages over many policies, because even if individual losses are strongly skewed, the average loss over a large number of policies will follow an approximately normal distribution.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean refers to the probability distribution of all possible sample means from a population. When you take a large number of samples (or have one large sample), the distribution of these means will have a center at the population mean and a spread (standard error) that is equal to the population standard deviation divided by the square root of the sample size. This helps in assessing probabilities related to the sample mean, such as determining the likelihood of exceeding a certain threshold.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It quantifies the amount of variability in the sample mean estimate from sample to sample and is calculated by dividing the population standard deviation by the square root of the sample size. A smaller standard error implies that the sample mean is a more precise estimate of the population mean, making it easier to assess risk in contexts such as insurance.

Transcript

00:01 This problem deals with an insurance company that looks at the population of its homeowners that have a mean annual loss from fire.

00:10 And we know that the distribution curve is strongly right skewed, and it has a mean of 250, and a standard deviation.

00:29 So i'll put one standard deviation on each side.

00:33 The standard deviation, that distance right there, of 300.

00:43 And this is going to be the mean annual loss from fire.

01:01 So the company wants to know if it sells 10 ,000 policies, can it safely say that the average loss will be no greater than 275 per household? so if that's the case, we know that our sample size is 10 ,000, and i'm looking for the probability that the sample mean is going to be greater than or equal to $275.

01:41 So if i draw a graph of the sampling means, i know that the mean is going to be the same as the mean of the population.

01:55 So the mean is going to be 250.

01:58 I can calculate the standard deviation.

02:02 That's going to be the population standard deviation over the square root of the sample size.

02:08 So that's going to be 300 over the square root of 10 ,000.

02:17 So that's 300, and the square root of 10 ,000 is 100.

02:23 So the standard deviation is going to be 3.

02:28 So the standard deviation is much smaller here.

02:32 So the standard deviation is going to be three, which is this distance right here.

02:43 So i'm looking for $275.

02:47 So that's $253...