Barron's reported that the average number of weeks an...

Barron's reported that the average number of weeks an individual is unemployed is 17.5 weeks (Barron's, February $18,2008 ) .$ Assume that for the population of all unemployed individuals the population mean length of unemployment is 17.5 weeks and that the population standard deviation is 4 weeks. Suppose you would like to select a random sample of 50 unemployed individuals for a follow-up study. a. Show the sampling distribution of $\overline{x},$ the sample mean average for a sample of 50 unemployed individuals. b. What is the probability that a simple random sample of 50 unemployed individuals will provide a sample mean within 1 week of the population mean? c. What is the probability that a simple random sample of 50 unemployed individuals will provide a sample mean within 1$/ 2$ week of the population mean?

Barron's reported that the average number of weeks an individual is unemployed is 17.5 weeks (Barron's, February $18,2008 ) .$ Assume that for the population of all unemployed individuals the population mean length of unemployment is 17.5 weeks and that the population standard deviation is 4 weeks. Suppose you would like to select a random sample of 50 unemployed individuals for a follow-up study.
a. Show the sampling distribution of $\overline{x},$ the sample mean average for a sample of 50 unemployed individuals.
b. What is the probability that a simple random sample of 50 unemployed individuals will provide a sample mean within 1 week of the population mean?
c. What is the probability that a simple random sample of 50 unemployed individuals will provide a sample mean within 1$/ 2$ week of the population mean?

Key Concepts

Sampling Distribution of the Sample Mean

This concept refers to the probability distribution of the sample mean obtained from repeatedly taking samples of a fixed size from a population. The sampling distribution has its own mean (which equals the population mean) and standard deviation (known as the standard error). It forms the basis for making inferences about the population mean using sample data.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the shape of the population distribution, the distribution of the sample mean will approximate a normal distribution as the sample size becomes large (typically n ? 30). This principle is essential for estimating probabilities concerning the sample mean, even when the underlying population distribution is not normal.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size. This value quantifies the variability in the sample mean estimates and is crucial for constructing confidence intervals and performing hypothesis tests.

Standard Normal Distribution (Z-Score)

The Z-score is a standardized value that indicates how many standard errors an observation (or sample mean) is from the population mean. By converting a sample mean to its corresponding Z-score, one can use the standard normal distribution to calculate the probabilities associated with specific deviations from the mean.

Transcript

00:02 All right, so this question asks us about the sampling distribution on the mean time of unemployment.

00:09 So the first part wants us to sketch the sampling distribution if we have a sample of 50 employees.

00:20 So that's just going to look like we have our mean of 17 .5 weeks right here.

00:32 And it's just going to be a bell curve because n equals 50.

00:38 Let's line this up a little better.

00:44 And this is normal approximately with a mean of 17 .5 weeks and a standard deviation of four weeks.

00:56 And remember it's going to be a bell curve because the central limit theorem says if your sample size is big enough, everything starts to look normal in the sampling distribution, which in this case our sample size is 50, which is more than enough to guarantee normality in this case.

01:15 Then part b wants the probability that a sample of 50 is within one week of the mean.

01:34 So if the mean is 17 .5, within one week means the probability that it's between 16 .5 and 18 .5, right? one week above, one week below.

01:57 And to do this, we need to know the standard error, because we can't use the population standard deviation because we're working with a sample.

02:07 We're not just looking at one individual employee.

02:11 We're looking at multiple.

02:13 So we're looking for standard error, which is the population sigma over the square root of the sample size, which in this case is 4 divided by square root, of 50, which works out to 0 .567.

02:40 So now we have everything we need.

02:44 So the probability that it's within one week...

Please give Ace some feedback