9.05 Chi squared test for independence
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.
| Test score 1 | Test score 2 | Test score 3 | |
|---|---|---|---|
| Male | 59 | 71 | 67 |
| Female | 86 | 90 | 89 |
| Male | Female | |
|---|---|---|
| Teacher | 241 | 271 |
| Nurse | 46 | 390 |
| Tradie | 252 | 52 |
| Lawyer | 154 | 158 |
| Junior | Senior | Veteran | |
|---|---|---|---|
| Tennis | 241 | 151 | 47 |
| Rowing | 76 | 54 | 32 |
| Golf | 45 | 254 | 232 |
| Bowls | 8 | 25 | 355 |
| Swimming | 155 | 97 | 167 |
| Running | 188 | 102 | 37 |
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} | |
|---|---|---|
| Hockey | 35 | 31 |
| Lawn bowls | 6 | 24 |
| Swimming | 45 | 49 |
| Archery | 9 | 21 |
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.
State the hypotheses set for the problem.
State the number of degrees of freedom.
Calculate the test statistic, \chi^2, correct to two decimal places.
Find the p-value, correct to five decimal places.
Comment on your findings.
The following table of data was collected from a survey where people were asked what their favourite movie genre was:
| Action | Comedy | Romance | Total | |
|---|---|---|---|---|
| Male | 51 | 50 | 11 | 112 |
| Female | 34 | 42 | 54 | 130 |
| Total | 85 | 92 | 65 | 242 |
Researchers conduct a \chi^2 test of independence at a 1\% significance level.
State the hypotheses set for the problem.
State the number of degrees of freedom.
Calculate the test statistic, \chi^2, correct to two decimal places.
Given that the critical value for a 1\% significance level is 11.34, state whether the null hypothesis should be accepted. Explain your answer.
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 A | Product B | Product C | Total | |
|---|---|---|---|---|
| Poor | 6 | 7 | 3 | 16 |
| Average | 10 | 11 | 13 | 34 |
| Good | 22 | 21 | 25 | 68 |
| Excellent | 12 | 11 | 12 | 35 |
| Total | 50 | 50 | 53 | 153 |
The company conducts a \chi^2 test of independence at a 5\% significance level.
State the hypotheses set for the problem.
State the number of degrees of freedom.
Calculate the test statistic, \chi^2, correct to two decimal places.
Find the p-value, correct to two decimal places.
State whether the null hypothesis should be accepted. Explain your answer.
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 } 18 | 18 -\lt 25 | 25 - 40 | \text{Over } 40 | \text{Total} | |
|---|---|---|---|---|---|
| Not important | 23 | 27 | 11 | 32 | 93 |
| Slightly important | 21 | 18 | 23 | 54 | 116 |
| Important | 10 | 21 | 25 | 22 | 78 |
| Very important | 16 | 43 | 55 | 18 | 132 |
| Total | 70 | 109 | 114 | 126 | 419 |
The company conducts a \chi^2 test of independence at a 5\% significance level.
State the hypotheses set for the problem.
State the number of degrees of freedom.
Find the expected frequency for people over 40 who think camera technology is very important, rounded to the nearest whole.
Calculate the test statistic, \chi^2, correct to two decimal places.
If the critical value for this level of significance is 16.92, state whether the null hypothesis should be accepted. Explain your answer.
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 bp | 5 | 12 | 32 | 49 |
| Average bp | 7 | 31 | 6 | 44 |
| High bp | 22 | 13 | 5 | 40 |
| Total | 34 | 56 | 43 | 133 |
The company conducts a \chi^2 test of independence at a 5\% significance level.
State the hypotheses set for the problem.
Use a calculator or otherwise, to construct a table of expected frequencies for the data.
State the number of degrees of freedom.
Calculate the test statistic, \chi^2, correct to two decimal places.
Find the p-value, correct to two decimal places.
State whether the null hypothesis should be accepted. Explain your answer.
1000 people were questioned as they exited a polling booth at the last election. The results are displayed in the following table:
| Party A | Party B | Party C | Total | |
|---|---|---|---|---|
| Female | 200 | 150 | 50 | 400 |
| Male | 250 | 300 | 50 | 600 |
| Total | 450 | 450 | 100 | 1000 |
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.
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 } 8 | 8 - \lt 12 | 12 - 18 | \text{Over } 18 | \text{Total} | |
|---|---|---|---|---|---|
| Internet access | 1 | 4 | 46 | 86 | 137 |
| No internet access | 13 | 35 | 7 | 5 | 60 |
| Total | 14 | 39 | 53 | 91 | 197 |
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.
State the hypotheses set for the problem.
There is less than 5 observations in the first two columns. Combine the columns together to create a new table of values.
State the number of degrees of freedom.
Calculate the test statistic, \chi^2, correct to two decimal places.
If the critical value for this level of significance is 9.21, state whether the null hypothesis should be accepted. Explain your answer.