How much more do public colleges and universities charge...

How much more do public colleges and universities charge out-of-state students for tuition per year? A random sample of 19 public colleges and universities listed at www.collegeboard.com yielded the following data. (TABLE CAN'T COPY) a) Create a 90$\%$ confidence interval for the mean difference in cost. Be sure to justify your procedure. b) Interpret your interval in context. c) A national magazine claims that public institutions charge state residents an average of $\$ 7000$ less than out-of-staters for tuition each year. What does your confidence interval indicate about this assertion?

How much more do public colleges and universities charge out-of-state students for tuition per year?
A random sample of 19 public colleges and universities listed at www.collegeboard.com yielded the following data.
(TABLE CAN'T COPY)
a) Create a 90$\%$ confidence interval for the mean difference in cost. Be sure to justify your procedure.
b) Interpret your interval in context.
c) A national magazine claims that public institutions charge state residents an average of $\$ 7000$ less than out-of-staters for tuition each year. What does your confidence interval indicate about this assertion?

Key Concepts

Evaluation of Claims Using Confidence Intervals

Confidence intervals can be used to assess assertions or claims about population parameters. In this scenario, by checking if a value like $7,000 falls within the interval, one can evaluate whether the national magazine’s claim is supported or refuted by the sample data. If the claimed difference lies outside the interval, it may indicate that the opinion expressed in the media is inconsistent with the evidence provided by the study.

Interpretation in Context

Interpreting a confidence interval in context means explaining what the range of values implies about the actual difference in tuition between in-state and out-of-state students. It involves stating that we are, for example, 90% confident that the true mean difference in cost lies within the calculated interval, thereby providing practical insight into cost differences for decision makers and stakeholders.

T Distribution for Small Samples

When the sample size is small (less than 30) and the population standard deviation is unknown, the t distribution is used instead of the normal distribution. The t distribution accounts for the extra uncertainty associated with estimating the standard deviation from the sample, and its degrees of freedom (typically n-1) adjust the shape of the distribution accordingly. This justifies the use of t-based confidence intervals in many practical situations, including this one.

Confidence Intervals

A confidence interval provides a range of values that is believed to contain the true population parameter with a certain level of confidence. In this context, it is used to estimate the difference in tuition costs between in-state and out-of-state students by constructing an interval estimate based on a sample. It involves calculating the sample mean difference and adding and subtracting a margin of error determined by the relevant t-statistic.

Transcript

00:01 In this problem, we are given a bunch of schools and then their tuition rates for residents and non -residents.

00:08 What we want is a 90 % confidence interval for the mean difference between the resident or the residence rate for tuition and non -residence rates for tuition.

00:19 Then we have to justify our procedure, so i've listed out some of our assumptions we're going to check.

00:24 After that, we need to interpret the interval we get for part a, that 90 % confidence interval.

00:29 And then someone makes the claim that $7 ,000, so on average that residents get to pay $7 ,000 less than non -residents in order to go to school.

00:41 So we want to check and make sure that lines up with what we find.

00:46 Let's go ahead and start with our 90 % confidence interval.

00:49 So what i've gone ahead and done is if i go stat edit, i've put all of the resident rates in list one and the non -resident rates in list two.

00:59 And we are looking for the confidence interval for the mean difference.

01:04 So let's put that.

01:05 Let's just find the differences and put them in list three.

01:08 So we're going to go, it's equal to list two minus list one.

01:12 Get those positive differences.

01:15 There they are.

01:16 And let's figure out which procedure we're going to use.

01:19 Are we going to use a two sample t test or, and by that i mean a two sample t interval or just a regular t interval because maybe it's paired.

01:28 And that's the first one i've put here.

01:30 So if we go back to the structure of our table, every school has two measures that go with it, right? the resident rate and the non -resident rate.

01:37 If we mixed those up and put them with different schools, that would mess up our findings.

01:43 And it wouldn't make sense in the table, right? they've got to stay next to each other because they belong to the same school.

01:50 Therefore, this data is paired.

01:53 And so we need to do a, i'm going to put that right here.

01:58 We want a paired t test, which will in turn.

02:01 Be a paired t interval.

02:05 And by that, it's going to be a t interval, right? all right, random.

02:09 So hopefully these schools are randomly dispersed throughout the country and they're not all from the same state.

02:14 Since they are, we're good.

02:15 Independent.

02:16 If one school changes their tuition, that doesn't mean that the other school's tuition are all going to change or even one of them would change, right? they're independent of each other.

02:24 They make their own choices.

02:26 They don't have to rely on one another.

02:28 So those are independent.

02:30 Nearly normal.

02:31 Let's graph that to find out.

02:34 I want to see if our difference, if our differences are nearly normal.

02:38 Let's double check.

02:39 We've got our histogram right here.

02:40 List three is those differences.

02:43 So we're good on that.

02:44 Let's go zoom...