College students may be interested in whether or not their majors have any effect on starting sa

College students may be interested in whether or not their...

Key Concepts

Contingency Tables

Contingency tables are used to summarize the relationship between two categorical variables by displaying the frequency counts of their occurrences in a matrix format, which facilitates further statistical analysis.

Chi-Square Test of Independence

The chi-square test of independence assesses whether there is a significant association between two categorical variables by comparing the observed frequencies in their contingency table to the frequencies expected under the assumption of independence.

Null and Alternative Hypotheses

In the context of a chi-square test of independence, the null hypothesis states that the two categorical variables are independent, while the alternative hypothesis posits that there is some form of association between them.

Expected Frequencies

Expected frequencies are calculated for each cell of a contingency table under the assumption that the categorical variables are independent; these values are used to compare with the observed frequencies and form the basis of the chi-square statistic.

Degrees of Freedom

Degrees of freedom, which for a contingency table are determined by the formula (number of rows - 1) multiplied by (number of columns - 1), indicate the number of values that can vary freely in the calculation of the chi-square statistic.

p-value

The p-value represents the probability of observing a chi-square statistic as extreme as, or more extreme than, the value calculated from the data if the null hypothesis of independence is true, thereby guiding the decision to either reject or fail to reject the null hypothesis.

Transcript

00:01 In this problem, we want to conduct a test of independence to see if starting salaries are independent of college majors.

00:11 So we were given some data, and the majors included english, engineering, nursing, business, and psychology.

00:50 And the starting salaries are less than 50 ,000, between 50 ,000, and 68, 999 actually, and 69 ,000 plus.

01:35 And we know that five english majors earned less than 50 ,000, 10 engineering majors, 10 nursing majors, 10 business majors, and 20 psychology majors.

01:54 We know that there were 20 english majors that earned between 50 ,000 and 68 -999, 30 were in engineering, 15 were in nursing, 20 were in business, and 30 were in psychology.

02:14 And for the 69 ,000 plus, we had five english majors, 60 engineering majors, 15 nursing majors, 30 business majors, and 20 psychology majors.

02:37 So those are the observed values.

02:40 So we're going to say observed is in green.

02:48 We now need to calculate the expected values, and any time the information is in chart format, we need to know the totals of each row and each column.

03:05 Actually, i'm going to take and move this down just a little bit that we could get those totals.

03:18 Okay, so our totals, i'm going to put in orange.

03:24 So we had 55 people surveyed in the less than $50 ,000 starting salary, 115 in the middle category, and 130 in the 69k plus, for a total of 300 individuals involved in this survey.

03:46 30 of them were english majors, 100 of them were engineering majors, 40 were nursing majors, 60 were business majors, and 70 were psychology majors.

04:06 So now we're going to determine the expected values.

04:16 And we're going to put that in blue and keep in mind how you find expected.

04:23 Our row total times our column total and we divide it by the total that is surveyed.

04:41 So for us to find the expected value for english less than 50 ,000, we're going to take the 30 multiplied by the 55 and divide by 300.

04:56 And when we do that, we get an expected value of 5 .5.

05:04 To find the english majors that make between 50 and 6899, we are going to take the row total times the column total and divide it by the overall total, and you will get 11 .5.

05:37 To find the english majors making 69k plus, we would take the row.

05:43 Row total times the column total divide by the total surveyed and you will get 13.

05:54 And then we do the same thing for the engineering.

05:57 We're going to do the row total times the column total and divide by the total surveyed.

06:09 And we will get 18 .33.

06:16 We're then going to take the row total times the cop, excuse me, column total, divide by the total surveyed, and we will get 38 .33.

06:31 And we're going to keep that pattern going.

06:36 Row total times column total, divided by the overall total, and we get 43 .33.

06:47 So we're going to write down all the expected values.

06:50 This will be 7 .33, 15 .33, 17 .33.

06:58 17 .33, 11, 23, 26, and then 12 .83, 26 .8, and 26 .83, and 30 .33.

07:26 Okay, so now that we have are observed and are expected, we are ready to write our null hypotheses and begin our test for independence.

07:48 Our null hypothesis will be that starting salaries are independent of college majors.

08:21 And the alternative to that will be that starting salaries are dependent on college majors.

08:46 And our test for independence is going to be a kai square test.

08:52 So therefore, we need to find a kai square test statistic based on our collected data.

09:05 And to find that kai square test statistic, we are going to total up the observed minus the expected squared divided by the expected.

09:19 So we're going to go back up to our chart, and the easiest way to get that information is to put all, of our data into the graphing calculator.

09:31 So we're going to open up our graph and calculator.

09:38 And we're going to hit our stat button.

09:42 We're going to clear any lists that might have something in them.

09:51 And then we're going to hit stat, edit.

09:54 We are going to put all of the observed or green values into list one.

10:31 Okay, so let's just make sure that we didn't have a typo there.

10:34 Okay.

10:40 And then we're going to put all of the expected values in the same order in list two.

10:46 So we have to make sure that the five from the english less than 50 is paired up with the 5 .5 english less than 50.

10:59 And then we have to make sure that the engineering 10 observed is paired up with the 18 .33 expected.

11:07 And again, just make a quick scan...

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