Question

A television CEO believes viewership of the evening news does not depend on age. She collects a random sample of 2000 television viewers across four different age groups and asks whether or not they watch the evening news. The results are as follows:

Watch              | 18 years old             19 to 35                36 to 54            55 years old

Evening News | or less                       years old              years old             or more

_____________________________________________________________________

Yes                  |           70                        96                          112                  146

                        |

No                   |           430                      404                         388                  354

                        |

_____________________________________________________________________

Test to see whether watching the evening news and age grouping are independent at the 0.05 level using the Chi-Square test. Conduct this test by hand and using the Chi-Square table.

Answer 1

Step-1:

Null Hypothesis Ho: Watching the evening news and age grouping are INDEPENDENT

Alternative Hypothesis Ha: Watching the evening news and age grouping are DEPENDENT

Step-2: Given data is " Observed frequency data " ( O )

For given contigency table we calculate Row and Column Totals as below;

	18 years old or less	19 to 35 years old	36 to 54 years old	55 years old or more	Colum Total (CT)
Yes	70	96	112	146	424
No	430	404	388	354	1576
Row Total (RT)	500	500	500	500	2000

Step-3:

Now, we calculate " Expected frequency data " ( E ) for each cell of the step-2 table by following formula;

Expected frequency data ( E ) = ( Row_Total x Column_Total ) / Grand_Total

Expected frequency (E)	18 years old or less	19 to 35 years old	36 to 54 years old	55 years old or more	Total
Education-only	106.00	106.00	106.00	106.00	424
Deep-cleaning routine	394.00	394.00	394.00	394.00	1576
Total	500	500	500	500	2000

For first cell E calculation :

E = ( 500 x 424 ) / 2000 = 106.00

Similarly, for secodn cell ( 2,1) :

E = (500 x 1576 ) / 2000 = 394.00

Step-4:

Now, we will calculate Chi-square for each cell using the below formula;

Chi-square calculations	18 years old or less	19 to 35 years old	36 to 54 years old	55 years old or more	Total
Education-only	12.23	0.94	0.34	15.09	28.60
Deep-cleaning routine	3.29	0.25	0.09	4.06	7.70
Total	15.52	1.20	0.43	19.16	36.30

For first cell ( 1,1 ), the calculation as below;

( 70 - 106 ) / 106 = 12.23

Similarly, for each cell Chi-square is calculated.

Step-5:

Chi-square statistic = 36.30 ( Grand total of step-4 table )

Now, we will find Chi-square critical value from the below table;

Degrees of freedom (df) = ( Rows-1) x ( Coulmns-1) = (2-1) x (4-1) = 3

df = 3

Given alpha = 0.05

Chi-square critical value = 7.815 ( Refer below table )

Step-6: ( Final)

Since Chi square statistic = 36.30 > Chi-square critical=7.815

Decision:

Reject Ho

Conclusion:

Watching the evening news and age grouping are DEPENDENT at 5% level of significace.

### End of answer