The following table shows site type and type of pottery for a random sample of 628 sherds at a l

The following table shows site type and type of pottery for...

Key Concepts

Chi-Square Test of Independence

The chi-square test of independence is a statistical method used to assess whether there is a significant association between two categorical variables. It involves comparing observed frequencies in a contingency table to expected frequencies calculated under the assumption that the variables are independent, thereby testing whether any deviations from the expected values are due to chance.

Contingency Table

A contingency table is a matrix used to display the distribution of variables and to organize data into rows and columns according to two categorical variables. This table serves as the basis for calculating the observed frequencies that are compared against the expected frequencies in the chi-square test of independence.

Expected Frequencies

Expected frequencies are the counts predicted for each cell in a contingency table, assuming that there is no association between the variables. They are calculated using the formula: (row total * column total) / overall total, and are essential for quantifying the differences used to compute the chi-square statistic.

Significance Level

The significance level, denoted as alpha, is the threshold used to determine whether the observed results are statistically significant. In this context, a significance level of 0.01 implies that there is only a 1% chance of incorrectly rejecting the null hypothesis, thereby making the test more conservative and ensuring strong evidence is required to claim an association.

Hypothesis Testing

Hypothesis testing is the process of evaluating two competing hypotheses—the null hypothesis, which in this case states that the two categorical variables are independent, and the alternative hypothesis, which suggests that there is an association between them. The chi-square test compares observed data with expected data under the null hypothesis to decide whether to reject it.

Degrees of Freedom

Degrees of freedom represent the number of independent pieces of information available for estimating parameters and are used to determine the critical value in the chi-square distribution. For a contingency table, degrees of freedom are calculated as (number of rows - 1) multiplied by (number of columns - 1), and they influence the shape of the chi-square distribution used in hypothesis testing.

Transcript

00:01 Okay, so in this question what we have is site types and pottery types and what exactly is the question the question is we have to use a kai square test to determine if the site type and a pottery type are independent at 0 .01 level of significance so our alpha is okay, let's use the black color.

00:26 Okay, so the alpha is 0 .01 okay, okay, so we have a table with us.

00:36 We have site type.

00:43 Over here we have messa top.

00:49 Then we have cliff talus.

00:51 Let me just write this as ct.

00:53 Then we have canyon benz.

00:55 Let me just write this as c .b.

00:59 Okay.

01:00 Then we have pottery types.

01:02 We have mesa verre, black on white.

01:07 Then we have mac elmo.

01:09 We have mac elmo.

01:14 Elmo then we have mancos mancos okay and over here again we will have the row and the column totals let's just try to fill this table up 75 81 92 61 708 68 then we have 53 we have 62 we have 66 okay what are the totals this is 248 if i'm not wrong yes this is 199 this should be 181 or here we have 189 then we have 213 and then we have 2 to 6 and the grand total is 628 all right now again we first need to find the expected value it is going to be row total multiplied by column total upon the grand total this is going to be done for every cell okay we are going to do this for every cell so i'm not going to write the labels now i will just draw the columns we have three columns and i guess the number of rows is also three yes this is a three by 3.

02:54 All right.

02:55 So, let us just try to find the value of the first cell.

02:59 The first send is messa top and mesa verde.

03:02 Okay.

03:03 So this is going to be row total, which is just a one, which is 189, multiplied by the column total, which is 248, and i divide this by the grand total, that is 628, 628, which is 74 .639 so i can just write this as 74 .64.

03:27 Okay, similarly i'm going to do for all of them.

03:31 For the next one we have 189 into 199 divided by 628.

03:37 This is 59 .9 .59 .9 .9.

03:44 Then we have 189 into 181 divided by 68.

03:49 54 .45 .55.

04:01 Then we have 213 into 2448 divided by 68, 84 .11, 84 .11.

04:13 Then we have 213 into 129, divided by 628, 84 .11.

04:22 67 .5 then we have 2 .13 into 181 divided by 628 this is 61 .39 61 .39 then we have 226 into 248 divided by 628 this is 89 .89 .84 89 .24 then we have 226 into 148 divided by 628 this is 89 .24 89 .24 then we have 226 into 19 divided by 628 this is 71 .61 71 .61 and then we have 226 multiplied by 181 divided by 628 the 65 .13.

05:12 65 .13.

05:16 Okay.

05:18 Now these are my expected values.

05:20 These are, this is the table of my expected values.

05:23 Now how do i find the kai square value for every cell? the formula for that is o minus e whole square upon e.

05:36 I am going to do this for every cell and in the end i'm going to sum them up and this is going to give me my kai square statistic.

05:44 Okay.

05:46 So let's say that i draw a table over here, right? again, there will be three columns.

05:55 There will be three rows.

05:59 Okay just two three columns three rows all right i'm just going to apply this formula over here observed minus expected whole square upon expected so that's why i need both of these tables over here so this is going to be okay this is going to be 75 minus 74 64 into 0 .36 because i'm squaring it and divided by 74 .64 so this is 0 .64.

06:39 So this is 0.

06:40 0 .0017 or i can just write this is 0 .002 this is going to be 0 .002 okay that will go on to the next one this is going to be 61 minus 59 .9 square this and divided by 59 .9 this is 0 .02 this is 0 .02 okay these values are very close 53 this is going to be 53 minus 54 .45.

07:23 I square this.

07:27 Okay.

07:30 Oh, just a moment.

07:31 So this is 53 minus 54 .45.

07:43 Okay, into minus...

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