Contingency tables and tests for independence：練習問題

In a contingency table analysis, if the initial table has 4 rows and 3 columns, and the researcher is performing a \(\chi^2\) test for independence, how many degrees of freedom should be used?

A \( 3 \times 3 \) contingency table is used to test for an association between two categorical variables. During the analysis, the researcher finds that two cells have expected frequencies less than 5. To address this, the researcher combines two adjacent rows into one. Calculate the degrees of freedom for the \( \chi^2 \) test after this modification.

A researcher is testing for an association between two categorical variables using a contingency table. One of the cells has an expected frequency of 3.2. According to standard statistical requirements for the \(\chi^2\) test, what action should the researcher take?

A \(\chi^2\) test for independence is being conducted on the following contingency table showing preferred leisure activity by age group:

Activity | Under 30 | 30 and Over | Total
Sport | 45 | 35 | 80
Reading | 15 | 25 | 40
Total | 60 | 60 | 120

Calculate the expected frequency for the 'Reading' and 'Under 30' cell.

A contingency table has 4 rows and 3 columns. During a \(\chi^2\) test for independence, it is found that the expected frequencies for one specific row are all less than 5. Describe the pooling procedure the researcher must take to ensure the test remains valid.

A contingency table with 3 rows and 4 columns is used to test for independence between two categorical variables. If one row and one column are combined because of low expected frequencies, calculate the new number of degrees of freedom for the \(\chi^2\) test.

A local council wants to determine if there is an association between the residential area (Urban, Suburban, or Rural) and the primary mode of transport used by residents (Car, Public Transport, or Cycling/Walking). A random survey of 450 residents yielded the following results:

Area | Car | Public Transport | Cycling/Walking
Urban | 40 | 85 | 55
Suburban | 95 | 40 | 15
Rural | 80 | 15 | 25

(a) State the null hypothesis for a chi-squared test of independence. (1 point)
(b) Calculate the expected frequency for the 'Suburban' residents who use 'Public Transport'. (2 points)
(c) State the number of degrees of freedom for this test. (1 point)
(d) Explain the condition regarding expected frequencies that must be met for this test to be valid. (1 point)

A researcher is studying the relationship between the type of employment and the level of job satisfaction among 300 randomly selected adults. The observed frequencies are recorded in the contingency table below:

Job Satisfaction | Self-Employed | Public Sector | Private Sector | Total
Satisfied | 45 | 52 | 83 | 180
Not Satisfied | 15 | 48 | 57 | 120
Total | 60 | 100 | 140 | 300

(a) State the null and alternative hypotheses for a chi-squared test of independence. (1 point)
(b) Calculate the expected frequency of people in the 'Public Sector' who are 'Satisfied'. (1 point)
(c) Calculate the contribution to the \(\chi^2\) test statistic for the 'Self-Employed' and 'Satisfied' cell. (1 point)
(d) State the number of degrees of freedom for this test. (1 point)
(e) Given that the total test statistic is \(\chi^2 = 8.16\), determine if there is a significant association between employment type and job satisfaction at the 5% significance level. (The critical value for this test is 5.991). (1 point)