The out-of-state tuitions (in dollars) for random samples of...

The out-of-state tuitions (in dollars) for random samples of both public and private four-year colleges in a New England state are listed. Find the 90% confidence interval for the difference in the means. Assume the variables are normally distributed and the variances are unequal. Use a graphing calculator. Round the answers to two decimal places. Private 17,300 16,590 23,400 13,495 13,600 12,500 19,024 14,291 14,150 Public 16,100 7,050 9,758 7,871 7,050 15,820 6,450 Source: New York Times Almanac. Use ?1 for the mean out-of-state tuition of private colleges. $ < ?1 - ?2 < $

Transcript

00:01 So we're looking at some data for private and public school out -of -state tuition.

00:06 And what we want to do today is make a 90 % confidence interval for mu1 minus mu2, where mu1 is the private school tuition, or group 1 is the private school tuition, and public is 2.

00:25 All right.

00:26 So our formula will be as follows.

00:29 X bar 1 minus x bar 2, so it's the point estimate for the difference.

00:32 So you take the sample means and take the difference of them.

00:34 Plus minus t alpha over 2, knowing the degrees of freedom, multiplied by...

00:41 Now this next term is called the standard error.

00:44 This is going to be s squared, the sample variance of group 1, divided by n1, plus s2 squared, the sample variance of group 2, divided by n2.

01:00 Now this is the formula we use, the standard error formula, because we're told to assume the variances are unequal.

01:06 So sigma 1 squared is not equal to sigma 2 squared.

01:11 And because they're unequal, we use this formula for the standard error of our confidence interval.

01:18 And we're told the data is normal, so we can go ahead and use our t distribution.

01:21 We should state that.

01:25 Great.

01:25 So let's go ahead and get our differences here.

01:30 The t score, well, the alpha is going to be 0 .01.

01:35 It's going to be 0 .10, because 1 minus the alpha gives us 0 .90, or that 90%.

01:46 And then the degrees of freedom we have to figure out, and the formula we need for it.

01:56 And something to note about this is that rather than look at it as individual parts and say, oh my gosh, that is so cumbersome, something i want you to notice is that this term here is right here.

02:13 And right here, look at that.

02:13 It's the same thing.

02:14 And then this term and this term are the same as is this term.

02:24 So just as you're going through it, really just you can think about it as like larger pieces.

02:27 Think about this formula in terms of larger pieces, and then you'll see some similarities here.

02:32 All right.

02:32 So with that said, we're going to go ahead and do some calculations here to get these.

02:36 And once you get the degrees of freedom, we'll be able to determine the t score.

02:44 And we'll go ahead and do the sample means first.

02:48 So let's go ahead and do this.

02:50 So the sample means are 16 ,038 .888 for the private and public, 10 ,014 .14.

03:04 And i use the average function and get the standard deviations, 3 ,474 for the private, 4 ,196 for the public.

03:14 Standard deviation.

03:16 Square those to get the variances.

03:19 There's the sample sizes, n and 9 and 7.

03:25 S squared over n.

03:27 This is what i mean if you look at the terms in our equation here as like, instead of looking at them as individual, like instead of like 1, 2, 3, 4, 5, 6, 7, 8, 9 terms, think about it as just 1, 2, 3, 4 terms, right? you just have those terms, just these s squared over n terms...

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