An article in the New England Journal of Medicine, discussed...

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites. $\chi^{2}$ test statistic $=$ ________

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis without breaking any constraints. In the context of the chi-square test on a contingency table, they are calculated based on the number of categories in each dimension (typically as (number of rows - 1) multiplied by (number of columns - 1)). This value is crucial for determining the critical value of the chi-square distribution against which the test statistic is compared.

Expected Frequencies

Expected frequencies are the theoretical counts in each cell of a contingency table under the assumption that there is no association between the variables. They are calculated using the marginal totals of the table, and they serve as a baseline for comparing the observed frequencies to determine if any observed differences are statistically significant.

Statistical Hypothesis Testing

Statistical hypothesis testing is a process used to make inferences or draw conclusions about a population based on sample data. In tests like the chi-square test, a null hypothesis (typically stating that there is no association between variables) is tested against an alternative hypothesis. The result of the test helps determine whether to reject the null hypothesis in favor of the alternative hypothesis based on how extreme the observed data are compared to what would be expected if the null hypothesis were true.

Chi-Square Test

The chi-square test is a statistical method used to determine whether there is a significant association between two categorical variables. It compares the observed frequencies in each category to the frequencies expected if there were no relationship between the variables, helping researchers decide if any differences found are likely due to chance or reflect a true underlying association.

Contingency Table

A contingency table is a matrix used to display the frequency distribution of variables. It organizes data into rows and columns, where each cell represents the count of observations for a particular combination of categories. This tool is essential for performing tests like the chi-square test to assess the relationship between different categorical variables.

Transcript

00:01 We want to find the kai square test statistic for the data provided on smoking levels per day.

00:07 And we have 1 to 10 per day or 11 to 20 or 21 to 30 or 31 plus.

00:19 And the information is broken down based on ethnicity.

00:24 So we have african american, we have native hawaiian, latino, japanese -american, white, and we were provided data for each.

00:56 So as we read through that problem, we found that there were 9 ,886 african -americans in the 1 -10 category, 2 ,745 native hawaiians, 12 ,831 latino, 8 ,31 latino, 8 ,310 .775.

01:19 78 japanese american and 7 ,650 white.

01:26 In the 11 to 20 category, we had 6 ,514 african american, 3 ,062 native hawaiian, 4 ,932 latino, 10 ,680 japanese american, and 9 ,87 white.

01:54 In the 21 to 30 category, we had 1 ,671 african -american, 1 ,419, native hawaiian, 1 ,4006 latino, 4 ,715 japanese -american, and 6 ,062 white.

02:18 And in the 31 -plus per day, 759 were african -american, 780.

02:25 Were native hawaiian, 800 were latino, 2 ,305 were japanese -american, and 3 ,970 were white.

02:42 And in order to run a kai square test, we also need to find the totals of each row and column, so that it would help us find our expected values.

02:53 So our totals for the african american were 18 ,830.

03:01 The total for the native hawaiian was 8 ,014.

03:07 Total for latino was 19 ,969.

03:12 The total for japanese american was 26 ,078.

03:18 The total for white was 27 ,559.

03:25 The total for 1 through 10 was 41 ,490.

03:32 The total for 11 through 20 was 35 ,665.

03:41 The total for 21 to 30 was 15 ,273.

03:49 And the total for 31 plus was 8 ,622.

03:54 And the grand total involved in this survey was 100 ,450.

04:04 So these are all our observed counts.

04:15 And to find a kai square test statistic, we needed the expected counts as well.

04:22 So we're going to recreate this chart, and we're going to fill in the expected values for 1 through 10, 11 through 20, 21 through 30, and 31 plus.

04:50 Now, in order to find the expected values, you are going to take the row total, multiplied by the column total, and then you're going to divide by the total in the survey.

05:14 So for this cell, it's row one column one.

05:27 So we're going to take the row total and the column total.

05:32 We're going to multiply those together and then divide by the overall total.

05:36 So i'm going to bring in the calculator.

05:40 So we're taking the row total 41490, multiplying it by the column total, 18830, and dividing it by the overall total, 100450, and dividing it by the overall total, 100450.

05:53 And we should round to two decimal places.

05:57 So as a decimal, that would be 777 .57.

06:08 And we're going to keep going.

06:11 So to find this next one, we're going to take the same row, but this time it's this column.

06:25 So i'm going to bring the calculator in.

06:27 The row was 41490, multiplied by the column 80144, divided by the 100 ,000, 450, and i'm getting 3311 .1.

06:48 And you're going to keep that pattern going to complete the remaining numbers for expected values...

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