Question

Household Income day A 2016 poll asked people how often they used the Internet. The data collected is included in the table to the right. Are Internet use and household income independent? Complete parts a through d below. Less than $25,000 $25,000 to $49,999 $50,000 to $84,999 $85,000 to $124,999 $125,000 or more How often do you use the Internet? At least once a Once a Every week but not every week or day less 105 46 21 147 35 17 211 27 11 213 25 7 161 17 a) Under the usual null hypothesis, what are the expected values? Expected Values At least once but Once a week Household Income Every day not every day or less Less than $25,000 $25,000 to $49,999 $50,000 to $84,999 $85,000 to $124,999 $125,000 or more (Round to one decimal place as needed.)

Answer 1

Solution:

Here, we have to use chi square test for independence of two categorical variables.

Null hypothesis: H₀: Two categorical variables internet use and household income are independent.

Alternative hypothesis: H_a: Two categorical variables internet use and household income are dependent.

We assume level of significance = α = 0.05

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

We are given

Number of rows = r = 5

Number of columns = c = 3

Degrees of freedom = df = (r – 1)*(c – 1) = 4*2 = 8

α = 0.05

Critical value = 15.50731

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	How often do you use the Internet?
Household Income	Every day	At least once a week but not every day	Once a week or less	Total
Less than $25,000	105	46	21	172
$25,000 to $49,999	147	35	17	199
$50,000 to $84,999	211	27	11	249
$85,000 to $124,999	213	25	7	245
$125,000 or more	161	17	1	179
Total	837	150	57	1044

Expected Frequencies
	How often do you use the Internet?
Household Income	Every day	At least once a week but not every day	Once a week or less	Total
Less than $25,000	137.8966	24.71264	9.390805	172
$25,000 to $49,999	159.5431	28.59195	10.86494	199
$50,000 to $84,999	199.6293	35.77586	13.59483	249
$85,000 to $124,999	196.4224	35.20115	13.37644	245
$125,000 or more	143.5086	25.71839	9.772989	179
Total	837	150	57	1044

Calculations
(O - E)
-32.8966	21.28736	11.6092
-12.5431	6.408046	6.135057
11.37069	-8.77586	-2.59483
16.57759	-10.2011	-6.37644
17.49138	-8.71839	-8.77299

(O - E)^2/E
7.84779	18.33683	14.35164
0.986125	1.436175	3.464255
0.647663	2.15273	0.495271
1.399109	2.956251	3.039595
2.131916	2.955486	7.875311

Test Statistic = Chi square = ∑[(O – E)^2/E] = 70.07614

χ² statistic = 70.07614

P-value = 0.0000

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

There is not sufficient evidence to conclude that two categorical variables internet use and household income are independent.

Part a)

Answer:

The required expected values are given as below:

Expected Frequencies
	How often do you use the Internet?
Household Income	Every day	At least once a week but not every day	Once a week or less	Total
Less than $25,000	137.9	24.7	9.4	172
$25,000 to $49,999	159.5	28.6	10.9	199
$50,000 to $84,999	199.6	35.8	13.6	249
$85,000 to $124,999	196.4	35.2	13.4	245
$125,000 or more	143.5	25.7	9.8	179
Total	837	150	57	1044

Part b)

Answer:

χ² statistic = 70.07614