Assume simple random sampling. Calculate the variance of the sample mean, ^2 for each of the fol

Assume simple random sampling. Calculate the variance of the...

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Key Concepts

Simple Random Sampling

Simple random sampling is a basic sampling technique where each member of the population has an equal chance of being selected. This concept is fundamental to ensuring that the sample is representative of the entire population, which helps in making unbiased estimations about population parameters.

Variance of the Sample Mean

The variance of the sample mean reflects the spread or variability of the sample means around the population mean. It is crucial for assessing the precision of the sample mean as an estimator of the population mean. In statistical inference, a smaller variance indicates a more reliable estimate.

Finite Population Correction (FPC)

The finite population correction accounts for the fact that the sample is drawn without replacement from a finite population. It reduces the variance of the sample mean compared to what would be expected from an infinite (or very large) population. The FPC factor is computed as (N - n) / (N - 1), where N is the population size and n is the sample size, and it adjusts the variance calculation accordingly.

Transcript

00:01 So we're calculating a variance estimate, and we know that in part a, and we know we do that by taking the sample standard deviation or the variance divided by n times this finite population factor.

00:21 And so this will be our formula.

00:25 So we know that in the first case, the population is 1 ,200, and the sample size is 80.

00:34 And we know that if we take 5 % of this 1 ,200, 12 % times 0 .05, that that is only 60.

00:45 So this is larger than that value, and that's why we need to use this factor.

00:50 And we have the sample standard deviation that's given to us as 10.

00:54 So we want to get our estimate here.

00:56 So let's plug in what we know into our formula.

01:02 And so we have s squared, so we'll have 100 over the sample size, which is 80, and now times that correction factor.

01:10 So that 1 ,200 minus the sample size divided by the one less than the population.

01:19 So this value comes out to be 10 divided by 80 times.

01:26 That in parentheses my let's see that will be 1120 divided by and that will be 1199.

01:37 And so it comes out to be 0 .116 and it would round to 8 as my estimate.

01:46 Now for part b, let's do another color.

01:51 We have 1425 as the population size and the sample size is 90...

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