You may need to use the appropriate appendix table or...

You may need to use the appropriate appendix table or technology to answer this question. A simple random sample of 40 items resulted in a sample mean of 25. The population standard deviation is σ = 6. (Round your answers to two decimal places.) (a) What is the standard error of the mean, σx? (b) At 95% confidence, what is the margin of error? You may need to use the appropriate appendix table or technology to answer this question. A simple random sample of 60 items resulted in a sample mean of 70. The population standard deviation is σ = 5. (a) Compute the 95% confidence interval for the population mean. (Round your answers to two decimal places.) (b) Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean. (Round your answers to two decimal places.) (c) What is the effect of a larger sample size on the interval estimate? A larger sample size provides a larger margin of error. A larger sample size provides a smaller margin of error. A larger sample size does not change the margin of error.

Transcript

00:01 Alrighty, so we have some questions about common intervals.

00:03 So we're going to find the standard error of the mean given a sample made of 25, population standardation of 6, and the sample size was 40.

00:15 So here's my little spreadsheet printout for it.

00:19 So sigma sub x bar, this is the standard error of the mean.

00:26 That is given, or the formula is the population standard deviation divided by the square of the sample size.

00:34 So that is equal to six divided by a square of 40.

00:37 So you get this number here.

00:39 0 .94, it's six, eight, three, two, nine, they want.

00:47 So we have a confidence interval.

00:50 And the confidence interval formula is x bar plus minus z, alpha over two times the standard error.

01:01 And that's it.

01:04 And we're using the z distribution because the sample size is greater than 30 and by the central limit theorem, as long as our sample size is greater than equal to 30, the distribution of the sample, meaning it'll be approximate normal, so that's what you use scores.

01:22 Well, and we use the z because they have the population standardation.

01:28 So we can assume it's normal because it's sample size is high.

01:31 And we're going to use z because the standard deviation and the population is known.

01:36 All right, so we can go ahead and do this.

01:38 We just need the z score and the alpha is the value that makes the competence interval 100%.

01:43 So that's 0 .05, over 2.

01:46 And the z square that corresponds with that is 1 .96.

01:50 So our competence interval is going to be 25 plus my, i'll write that, we'll write that, 25 plus minus 1 .96 times 0 .0.

02:01 9, 4, 8, 7.

02:04 And then i did this because we're going to, i just do it in general, find the margin of error.

02:14 We're asked to do it here, the margin of error.

02:17 I did the compensate way anyway, because it's just, oh wait, no, sorry, that's laid on that we do that.

02:25 But the margin of error is this, 1 .85 .4.

02:30 And we're not told the, oh, two decimal places, so 1 .86.

02:36 So margin of error.

02:43 All right, here we go.

02:46 And just so you know, i use the full precision that my spreadsheet gave me for the standard error.

02:50 So if that changes this slightly, you don't see this exact value, that's why.

02:57 Plus it's always a good practice to wait until the very end to round, so that's why i didn't want to there.

03:03 So now we're going to do the same kind of thing with a sample size of 60.

03:10 That's our n, sample mean of 70.

03:13 It's x bar.

03:14 Population of standard deviation, five, that's sigma.

03:18 And we wanna make a confidence interval for the population mean, so we're gonna do this whole thing...

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