Question

In a survey 1000 adults were asked whether they used a smartphone, a mobile phone (but not a smartphone) or did not use a mobile phone. The results were as follows: Mobile phone use Age of respondentSmartphone Mobile phone but not smartphone No use Total 16 to 24 133 51 7 191 25 to 34 116 82 201 35 to 54 85 97 188 20 206 55 to 64 39 147 65 or older 10 127 77 214 Total 383 504 113 1000 Using these results the relationship between the age of the respondent and the type of mobile phone they use is to be tested using the x2 test for contingency tables. In this test, what is the expected value for the respondents aged 25 to 34 who use a smartphone? (Give your answer correct to three decimal places.)

Answer 1

Expected Values smartphone mobile Total no_use 383x191 1000 504x191 1000 113x191 1000 16-24 73.153 96.264 21.583 191 113x201 22.713 1000 504x 201 1000 383x 201 1000 25-34 76.983 101.304 201 383x 188 1000 504x188 1000 113x188 1000 35-54 94.752 72.004 21.244 188 383x206 1000 504x206 1000 113x206 1000 55-64 103.824 78.898 23.278 206 383x214 1000 504x214 1000 113x214 1000 81.962 24.182 65 or older 107.856 214 _ Total 383 504 113 1000

formula: (E−O)² / E

Squared Distances smartphone mobile no_use (133-73.153) (7-21.583) (51-96.264)2 16-24 21.283 73.153 9.853 _ 96.264 21,583 48.961 (116-76.983) (3-22,713) 17.109 (82-101.304)2 25-34 76.983 3.678 101.304 22.713 19.775 (85-72.004) (97-94.752)2 94,752 (6-21.244) 35-54 2.346 0.053 10.939 72.004 21.244 147-103.824)2 (39-78.898) 78.898 (20-23.278)2 55-64 0.462 20.176 103.824 23.278 17.955 (127-107.856) 107.856 3.398 (77-24.182) (10-81.962) 65 or older 63.182 24.182 81.962 115.364

1. Null and Alternative Hypotheses

H0​: The two variables are independent

Ha​: The two variables are dependent

2.

df=(5−1)×(3−1)=8

R={χ2:χ2>15.507}

3.

χ2= 48.961+19.775+2.346+20.176+63.182+21.283+3.678+0.053+17.955+3.398+9.853+17.109+10.939+0.462+115.364

χ2=354.535

4.

354.535>15.507(critical value), it is then concluded that the null hypothesis is rejected.

5.

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the .05 significance level.

The corresponding p-value for the test is p = Pr(χ2​>354.535)=0.