Use the population of ages 56 49 58 46 of the four us...

Use the population of ages {56, 49, 58, 46} of the four U.S. presidents (Lincoln, Garfield, McKinley, Kennedy) when they were assassinated in office. Assume that random samples of size n = 2 are selected with replacement. 1. List the 16 different samples. For example, the samples for age 56 would be 56, 56 56, 49 56, 58 56, 46. 2. After listing all 16 samples, find the mean of each sample, then construct a table representing the sampling distribution of the sample mean. In the table, combine values of the sample mean that are the same. 3. Compare the mean of the population {56, 49, 58, 46} to the mean of the sampling distribution of the sample mean. 4. Do the sample means target the value of the population mean? In general, do sample means make good estimators of population means? Why or why not?

Transcript

00:01 So for this problem, i'll first create a grid where we'll, or a table rather, where we'll have our sample and our sample mean.

00:10 So first sample would be 56 and 56.

00:14 We'd have 56 and 49, 56 and 58, 56 and 46 and 46.

00:22 And we could have 49 and 49, 49 and 58, 49 and 49 and 49 and 59 and 59 and 59 and 59 and 58, 49 and, and 40, pardon me.

00:38 So that's 49, 49, 49, 58, and 49 46.

00:43 We could have 58, 858, 8 .58.

00:54 And i just realized that with the way that i was going through the procedure here, i'm undercounting because i was assuming that basically the ordering didn't matter, but for it to work out for the proper number of different matches, the order should matter.

01:14 So we could have 49, 56, 49, 49, 49, 48, 49, 46.

01:20 And then i'll need to add in, actually, i'll just erase these and add in the few rows.

01:25 Then we could have 58, 56, 58, 49, 58, 58, 8, 46, and 46, and 46, 46, 46, 46, 49, and 46, 56, 46, 46, 49, 4658, 658 and 40, oops see daisy, why is it not i just want another one another row there we go 46 46 as the last of the 16 so obviously for the ones where we have doubled up values it's just going to be equal to that value then we'll do all right so i just calculated these out ahead of time here and i'll just report the results obviously, it's just taking the average of two numbers over and over again.

02:24 So it's going to be 52 .5, 57, and 51, and 49, 56 is 52 .5, 49, 49, obviously, 49, and 58.

02:41 It's going to be 53 .5, then we get 47 .5.

02:47 And we have 57, 53 .5, 58, 52, 51, 47 .5, 52, and lastly, 46.

03:00 So now, having figured out all of our different samples, we can create our sampling distribution where we have the sample mean, then the frequency, and also we can get the probability from that.

03:14 So lowest sample mean is 46, which occurs with frequency 1, so probability is 1 over 16.

03:21 Actually, i'll just leave it as a fraction...

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