Determine the sample size needed for each of the following...

Determine the sample size needed for each of the following situations. a. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=50$ b. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=100$ c. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=200$ d. Compare and comment on your answers to parts a through c.

Determine the sample size needed for each of the following situations.
a. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=50$
b. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=100$
c. $N=1,650 \quad \sigma=500 \quad 1.96 \sigma_{\bar{x}}=200$
d. Compare and comment on your answers to parts a through c.

Key Concepts

Finite Population Correction Factor

When sampling from a finite population without replacement, the standard error is adjusted by the finite population correction factor. This factor, which depends on the sample size relative to the total population size, reduces the standard error when the sample constitutes a significant fraction of the population.

Standard Error of the Mean

The standard error of the mean quantifies the variability of sample means around the true population mean. In its basic form, it is calculated as the population standard deviation divided by the square root of the sample size, reflecting the precision of the mean estimation in repeated sampling.

Margin of Error

The margin of error represents the maximum expected difference between the sample estimate and the true population parameter at a given confidence level. It is directly proportional to the standard error of the estimator and the critical value from the normal (or other) distribution.

Sample Size Determination

This concept involves calculating the number of observations required so that estimation of the parameter (typically the mean) meets a specified level of precision, often expressed through the desired margin of error. The calculation uses known parameters, such as the population standard deviation and confidence level, combined with sample variability formulas.

Confidence Level and z-Score

A confidence level defines the probability that the calculated confidence interval contains the true population parameter and is associated with a z-score (or critical value). For example, a 95% confidence level corresponds to a z-score of approximately 1.96, which is used to scale the standard error in determining the margin of error.

Transcript

00:01 So in this question, we're going to use this formula to help us find what sample size we need.

00:07 And our capital n1 is our population.

00:14 And we can see what the other symbols are.

00:18 And so we have, on the first one, we have that capital n is given to us as 1650.

00:26 In fact, that will be the case for all a, b, and c, and that the sigma is 500.

00:31 And then we're told on this particular example that 1 .96 times the standard air is equal to 50.

00:41 And so we can substitute in the values that we know here.

00:46 And we'd have n is equal to, and we have that 1650 times the 500 squared.

00:57 And then we can see that that will be also a 500 squared right here.

01:02 And we can see that for the purpose of this problem we need to take the 50 divided by the 1 .96 and that's going to end up giving us our value for the standard air and we'll have the n minus 1 is that 1649 times and then we're going to have to square that value that 50 over 1 .96 quantity squared and add that on.

01:36 And so putting that in my calculator, i've got my left parentheses for the top 1650 times the 500 squared.

01:46 And then i'm going to close my parentheses off and then divided by another left parentheses 1649 times.

01:54 And we'll put that quantity of 50 divided by 1 .96 in a parentheses and square it and then add on the 500.

02:03 Squared and close that parentheses off and that tells me my sample size is 311 .76.

02:11 So let's round that to 312.

02:16 And then on part b we're again maintaining these as the same.

02:20 We're changing.

02:22 The only thing that we're changing is that 1 .96 times the standard air is equal to now 100.

02:29 And so that means that our standard error will be 100 divided by to 1 .96...

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