A simple random sample of size n was drawn which provided a...

A simple random sample of size n was drawn, which provided a sample mean of x̄ = 18.5 with a standard deviation of s = 4.6 and a sample size of n = 20. The population is known to be normally distributed. (a) Using a 96% level of confidence, determine the critical value by hand. (b) Calculate the Margin of Error for a sample size of n = 20. (c) Use the results from Parts A & B to construct a 96% confidence interval for μ. (d) Construct a 96% confidence interval for a sample size of n = 50. Compare this result with Parts B & C. What can you say about the effect on the Margin of Error when the sample size increases?

Transcript

00:01 Okay, this problem is all about creating a confidence interval using a sample mean.

00:08 And we're going to first just write down the information they give us.

00:11 They tell us that the sample mean is, so x bar sample mean is 18 .5.

00:21 They also tell us the sample standard deviation of 0 .6 and the sample size of 20.

00:41 The population is known to be normally distributed, so that's helpful because we can use a normal distribution for the sampling distribution.

00:47 Using a 96 % confidence level, determine the critical value by hand.

00:53 Well, what we need to know here is that the degrees of freedom is always the sample size minus one.

01:00 The degrees of freedom would be 19.

01:03 And what we need to do is we need to pull up a t -distribution chart, which i have one right here.

01:10 We've got to go to the degrees of freedom of 19.

01:14 At the bottom of the chart, they show you different confidence levels.

01:18 We're doing 96%.

01:20 Now we cross -reference it.

01:21 We go to the row that says 19 on there.

01:25 We go over until we get to the confidence level of 96.

01:29 And we get a value right here of 2 .205.

01:32 That is called our critical value.

01:35 So that's the number we're going to write down for the critical value.

01:38 It's all about knowing how to read the t -distribution properly with the degrees of freedom.

01:43 And so part a, find the critical value, which is basically degrees of freedom of 19.

01:53 Look at the table, find it.

01:55 The critical value in this case is 2 .205.

02:02 So we have that.

02:04 Then it says part b, calculate the margin of error for the sample size of n equals 20.

02:12 The formula for margin of error, i'm going to abbreviate it as moe.

02:23 Try that again.

02:25 Moe for margin of error is the critical value times the standard error of the sampling distribution, which that ends up being the critical value times the sample standard deviation divided by the square root of n.

02:43 All these formulas can be found on a formula sheet.

02:47 You just need to know which one we're using for each situation.

02:50 This is a situation for when we're dealing with sample means.

02:53 So let's go and plug the numbers in.

02:56 The critical value is the 2 .205.

03:01 The sample standard deviation they had given to us is 4 .6.

03:07 The sample size is 20.

03:10 And then we grab a calculator and we plug this all in and we come up with a value for that.

03:15 The value that i get for my sample standard deviation ends up being 2 .268.

03:23 And so that's called the standard error.

03:31 I'm sorry, that's called the margin of error.

03:33 The standard error is the 4 .6 over the square root of 20.