The mean annual cost of automobile insurance is 939(C N B...

The mean annual cost of automobile insurance is $\$ 939(C N B C,$ February $23,2006)$ Assume that the standard deviation is $\sigma=\$ 245 .$ a. What is the probability that a simple random sample of automobile insurance policies will have a sample mean within $\$ 25$ of the population mean for each of the following sample sizes: $30,50,100,$ and 400$?$ b. What is the advantage of a larger sample size when attempting to estimate the population mean?

The mean annual cost of automobile insurance is $\$ 939(C N B C,$ February $23,2006)$ Assume that the standard deviation is $\sigma=\$ 245 .$
a. What is the probability that a simple random sample of automobile insurance policies will have a sample mean within $\$ 25$ of the population mean for each of the following sample sizes: $30,50,100,$ and 400$?$
b. What is the advantage of a larger sample size when attempting to estimate the population mean?

Key Concepts

Central Limit Theorem

This theorem states that the distribution of the sample means will approximate a normal distribution as the sample size becomes larger, regardless of the shape of the population distribution. It is foundational for making inferences about the population mean using sample data, especially when the sample size is sufficiently large.

Sampling Distribution of the Sample Mean

This concept refers to the probability distribution of all possible values of the sample mean from a given population. It provides the framework for understanding how sample means vary from the population mean and is essential for calculating probabilities regarding the sample mean's proximity to the population mean.

Standard Error

Standard error is the standard deviation of the sampling distribution of a statistic, most commonly the sample mean. It quantifies the variability of the sample mean estimate from the true population mean and is computed by dividing the population standard deviation by the square root of the sample size.

Z-score and Probability Calculation

A z-score measures how many standard errors an observation (or sample mean) is away from the population mean. It is used to determine probabilities by converting the distance into a standardized form, which then allows for the use of the standard normal distribution to find the likelihood that the sample mean falls within a specified interval.

Effect of Sample Size on Estimation

Larger sample sizes reduce the standard error, leading to a narrower sampling distribution. This improved precision enhances the reliability of the estimate for the population mean, which is critical for ensuring that the sample mean is close to the true population mean with a higher probability.

Transcript

00:02 All right, this question deals with the price of auto insurance that's normally distributed, and it asks us to compute the probability that we are within $25 of the mean with various sample sizes.

00:15 So the first question wants us to compute this for n equals 30, and we're looking for the probability that we are within $25, which is the same thing as asking, what is the probability that our sample has a mean between $914 on the lower end and $964 on the upper end.

00:58 So, since we have a normal curve and we're dealing with probabilities, we can use normal cdf.

01:09 So normal cdf.

01:16 Our lower bound is 914.

01:18 Our upper bound is 964.

01:25 Our mean is 939.

01:32 And our standard deviation, this is what's going to change, because we're not dealing with one sample.

01:40 We're dealing with a sample size of 30.

01:42 So it's our population standard deviation divided by the square root of our sample size.

01:57 And putting this into your calculator, find that this is 0 .4 -2 -3 -8.

02:14 Then it wants a bigger sample and equal to 50.

02:26 So again, the only thing we're changing is the sample size.

02:30 So we do normal cdf with everything the same except the standard deviation because it's asking the same exact question.

02:49 But since we have a bigger sample, we have to divide by the square root of our sample size because more samples lowers variability...

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