A researcher would like to determine if the proportion of...

A researcher would like to determine if the proportion of households without health insurance coverage differs with household income. Suppose the following data were collected from 700 randomly selected households. Household Income | Yes | No Less_than_25000 | 54 | 20 25000_to_49999 | 141 | 42 50000_to_74999 | 203 | 36 75000_or_more | 172 | 32 Using ?=0.01, perform a chi-square test to determine if the proportion of households without health insurance differs by income bracket. 6) Determine the correct null and alternative hypotheses 7) What is the test statistic and what is the critical value? (show your work). 8) State and interpret your conclusion(show your work).

Transcript

00:01 Okay, this is a bit of a tedious problem if you don't have technology, but i'm going to show you most of it and then sort of use my technology to finish it up.

00:11 So the first thing you need to do in order to run this kai squared test is what this would be called is you need to find the row totals and the column totals for your table.

00:23 So if i added up these numbers, i got a column total of 570 yeses.

00:28 And if i added up these numbers, i got a total of 130 noes.

00:32 Now, these are a little bit skewed, but 74 is the first row total.

00:36 183 is the second row total.

00:39 239 is the third row total.

00:42 And 204 is the fourth row total.

00:45 And that's a total of 700 people that were, or 700 households that were included in this study.

00:53 The null hypothesis would be the proportion of health.

00:58 Households without health insurance is the same in each income bracket.

01:04 The null hypothesis for a chi -squared test is always these things are equal, if you will.

01:11 The alternative hypothesis is therefore the proportion of households without health insurance is not the same in each income bracket.

01:21 Okay.

01:22 Now, that's your answer to number six, your hypotheses.

01:25 Number seven, what is the test statistic and what is the critical value show your work? okay.

01:34 So now, the test statistic is called kai squared.

01:38 And in order to find kai squared, you take one observed count minus the corresponding expected count, square it divided by the expected count, plus the next observed minus its corresponding expected squared divided by the expected.

01:58 And you just keep adding those up and you'll eventually get kai squared.

02:02 So you see the dilemma here is we've got to calculate these expected counts.

02:08 So i want to scroll down and show you how to calculate your expected counts.

02:14 So in other words, here's my table of the corresponding expected counts.

02:20 So to fill in your table of expected counts, you go up to your table of observed counts and you take the row total times the column total divided by the grand total of the table.

02:36 So for example, you see this first expected count that's in the less than 25 ,000 row and the yes column.

02:46 Where did this 60 .26 come from? it came from taking the less than 25 ,000 row total, which is 74.

03:00 So 74 was the row total times the yes column total, and the yes column total is 570.

03:13 So 74 times 570, divided by the grand total of the table, 770.

03:23 My apologies for those lines so if i take 74 times 570 divided by 700 that shows you where the 60 .26 came from so then for example just one more time where did this 149 came come from that comes from the second row total divided by the first excuse me multiplied by the first column total divided by 700 so the second row total is 183, and the column total is 570.

04:01 So if i take 183 times 570 divided by 700, that's the second row total times the first column total.

04:18 So 183 times 570 divided by 700...

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