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9.05 Chi squared test for independence

Worksheet
Chi squared test for independence
1

For each of the following matrices being examined for independence using a \chi^2 test, state the null hypothesis and the number of degrees of freedom.

a
Test score 1Test score 2Test score 3
Male597167
Female869089
b
MaleFemale
Teacher241271
Nurse46390
Tradie25252
Lawyer154158
c
JuniorSeniorVeteran
Tennis24115147
Rowing765432
Golf45254232
Bowls825355
Swimming15597167
Running18810237
2

At the last Commonwealth Games, 220 spectators were asked which of four events they preferred to watch. Results are displayed in the following table:

\text{Age under } 40\text{Age } 40 \text{ or more}
Hockey3531
Lawn bowls624
Swimming4549
Archery921

In order to see whether age had any influence over favourite sport, researchers conducted a \chi^2 test of independence at a 5\% significance level.

a

State the hypotheses set for the problem.

b

State the number of degrees of freedom.

c

Calculate the test statistic, \chi^2, correct to two decimal places.

d

Find the p-value, correct to five decimal places.

e

Comment on your findings.

3

The following table of data was collected from a survey where people were asked what their favourite movie genre was:

ActionComedyRomanceTotal
Male515011112
Female344254 130
Total859265242

Researchers conduct a \chi^2 test of independence at a 1\% significance level.

a

State the hypotheses set for the problem.

b

State the number of degrees of freedom.

c

Calculate the test statistic, \chi^2, correct to two decimal places.

d

Given that the critical value for a 1\% significance level is 11.34, state whether the null hypothesis should be accepted. Explain your answer.

4

A cleaning company is interested in determining if good results in window cleaning are dependent on the brand of cleaning product, or some other variable such as cleaning technique. They asked their clients to complete a survey after each clean and results are recorded in the table below:

Product AProduct B Product C Total
Poor67316
Average10111334
Good22212568
Excellent12111235
Total505053153

The company conducts a \chi^2 test of independence at a 5\% significance level.

a

State the hypotheses set for the problem.

b

State the number of degrees of freedom.

c

Calculate the test statistic, \chi^2, correct to two decimal places.

d

Find the p-value, correct to two decimal places.

e

State whether the null hypothesis should be accepted. Explain your answer.

5

A generic smart phone retailer is interested in knowing more about their customer base. They conduct a survey asking customers how important the latest camera technology is to them when choosing their phone. Results are recorded in the table below:

\text{Under } 1818 -\lt 2525 - 40\text{Over } 40\text{Total}
Not important2327113293
Slightly important21182354116
Important1021252278
Very important16435518132
Total70109114126419

The company conducts a \chi^2 test of independence at a 5\% significance level.

a

State the hypotheses set for the problem.

b

State the number of degrees of freedom.

c

Find the expected frequency for people over 40 who think camera technology is very important, rounded to the nearest whole.

d

Calculate the test statistic, \chi^2, correct to two decimal places.

e

If the critical value for this level of significance is 16.92, state whether the null hypothesis should be accepted. Explain your answer.

6

The following table shows the results of surveying a number of adults on their number of minutes of exercise per day and measured blood pressure (bp) level:

\lt 30 30 - 60 \gt 60 \text{ Total}
Low bp5123249
Average bp731644
High bp2213540
Total345643133

The company conducts a \chi^2 test of independence at a 5\% significance level.

a

State the hypotheses set for the problem.

b

Use a calculator or otherwise, to construct a table of expected frequencies for the data.

c

State the number of degrees of freedom.

d

Calculate the test statistic, \chi^2, correct to two decimal places.

e

Find the p-value, correct to two decimal places.

f

State whether the null hypothesis should be accepted. Explain your answer.

7

1000 people were questioned as they exited a polling booth at the last election. The results are displayed in the following table:

Party AParty BParty CTotal
Female20015050400
Male25030050600
Total4504501001000

Perform a \chi^2 test for independence at a 5\% significance level to determine whether voting preference of the men seems to be significantly different to that of the women.

When conducting your test be sure to state the following:

  • Hypotheses and significance level.

  • df, \chi^2 and p-value.

  • A comment on whether the hypothesis is rejected or not rejected.

8

A electronics store is interested in knowing more about their customer base. They conduct a survey asking customers how old they were when they bought their first mobile phone and whether that phone came with internet access. Results are recorded in the table below:

\text{Under } 88 - \lt 1212 - 18\text{Over } 18\text{Total}
Internet access144686137
No internet access13357560
Total14395391197

The company wishes to conduct a \chi^2 test of independence at a 1\% significance level to see if there appears to be a link between phone internet access and age.

a

State the hypotheses set for the problem.

b

There is less than 5 observations in the first two columns. Combine the columns together to create a new table of values.

c

State the number of degrees of freedom.

d

Calculate the test statistic, \chi^2, correct to two decimal places.

e

If the critical value for this level of significance is 9.21, state whether the null hypothesis should be accepted. Explain your answer.

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