Internet Viewing A researcher wishes to estimate the average number of minutes per day a person

Internet Viewing A researcher wishes to estimate the average...

Key Concepts

Critical Value (z-score)

The critical value, often expressed as a z-score in the context of a normal distribution, is the number of standard deviations a data point is from the mean corresponding to a given confidence level. In sample size calculations, it is used to determine the boundary above and below the sample mean which forms the confidence interval. Specifically, at a 90% confidence level, the appropriate z-score is used to ensure that 90% of the distribution falls within these bounds.

Sample Size Calculation

Sample size calculation for estimating a population mean involves determining the number of observations needed to achieve a desired margin of error at a specific confidence level, given the population standard deviation. This is typically done using the formula n = (z * ? / E)², where z is the critical value for the desired confidence level, ? is the population standard deviation, and E is the margin of error. This calculation is fundamental to ensuring that the estimate is both reliable and precise.

Confidence Interval for the Mean

A confidence interval for the mean is a range of values derived from sample data that is likely to contain the population mean. It provides a measure of uncertainty around the point estimate (the sample mean) by incorporating the variability in the data and the confidence level, which represents the probability that this interval contains the true population parameter.

Margin of Error

The margin of error represents the maximum expected difference between the true population mean and the estimated sample mean. It quantifies the precision of the estimate; a smaller margin of error indicates a more precise estimate. When designing studies, it is used to determine how large a sample size is necessary to achieve the desired level of accuracy.

Transcript

00:01 So we have this problem where we're trying to find out a sample size.

00:06 We don't find out how many people we need to sample with some given information.

00:10 We know that they want to be 90 % confident.

00:16 We also know that the person's okay with the sample mean being within 10 of the population mean.

00:22 So this is a margin of error.

00:26 So we're looking at a margin of error of 10 minutes.

00:29 And they've told us that the population standard deviation is 42.

00:38 Now, when we do margin of error problems, margin of error, we have a formula for it, is the critical value times the standard deviation of the sample statistic.

00:51 So we're looking at margin of error equals a critical value.

00:57 Because we're in a mean, my critical value is normally always going to be a t star, and then the standard deviation of the sample statistic is the sample standard deviation divided by square root of n.

01:11 But we know some things here, and we got a couple issues here, we know that t star, in order to find that, we need to know degrees of freedom.

01:23 Well, in order to know degrees of freedom, you need no sample size.

01:26 And lo and behold, we're trying to find sample size here.

01:29 So we can't use a t -star.

01:31 This is the one case that i tell my students that i know we're in a mean.

01:36 I know we always do t -stars and means, but we're going to do a z -star in this case.

01:42 So let me write in some information that i know here.

01:44 My margin of error is 10.

01:47 I'm going to change that t -star to a z -star now because we can't find a t -star.

01:52 My sample standard deviation, well, they gave me a population standard deviation, so whenever they give me that, i'm going to use it.

01:59 So it's going to be a 42 divided by square root of n.

02:03 So again, the n is what we're trying to find.

02:05 But we still got that z star value.

02:08 And how to heck do we find a z star? so i tell my kids, oh, well, z star means we're sort of in a normal distribution here.

02:17 And we're at a 90%.

02:18 So that means that area there is 90%.

02:21 So that means i got 10 % left over.

02:23 Well, i got two tails...

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