In a volunteer group, adults 21 and older volunteer from one to nine hours each week to spend time with a disabled senior citizen. The program recruits among community college students, four-year college students, and nonstudents. In the Table below is a sample of the adult volunteers and the number of hours they volunteer per week. Type of Volunteer 1–3 Hours 4–6 Hours 7–9 Hours Row Total Community College Students 111 96 48 255 Four-Year College Students 96 133 61 290 Nonstudents 91 150 53 294 Column Total 298 379 162 839 Table: Number of Hours Worked Per Week by Volunteer Type (Observed) The table contains observed (O) values (data). Is the number of hours volunteered independent of the type of volunteer? The observed table and the question at the end of the problem, "Is the number of hours volunteered independent of the type of volunteer?" tell you this is a test of independence. The two factors are number of hours volunteered and type of volunteer. H0: The number of hours volunteered is independent of the type of volunteer. Ha: The number of hours volunteered is dependent on the type of volunteer. The expected results are in the table below. The table contains expected (E) values (data). Type of Volunteer 1-3 Hours 4-6 Hours 7-9 Hours Community College Students 90.57 115.19 49.24 Four-Year College Students 103.00 131.00 56.00 Nonstudents 104.42 132.81 56.77 Table Number of Hours Worked Per Week by Volunteer Type (Expected) For example, the calculation for the expected frequency for the top left cell is: E=(row total)(column total)/total number surveyed=(255)(298)/839=90.57 Calculate the test statistic: χ2 = 12.99 (calculator or computer) df = (3 columns – 1)*(3 rows – 1) = (2)(2) = 4 Graph: Figure 11.7 Probability statement: p-value=P(χ2 > 12.99) = 0.0113 Compare α and the p-value: Since no α is given, assume α = 0.05. p-value = 0.0113. α > p-value. Make a decision: Since α > p-value, reject H0. This means that the factors are not independent. Conclusion: At a 5% level of significance, from the data, there is sufficient evidence to conclude that the number of hours volunteered, and the type of volunteer are dependent on one another. Create a table below that shows the contribution of each category to the total Chi-square for this study. The table contains the contribution to total Chi-square for each category. Type of Volunteer 1-3 Hours 4-6 Hours 7-9 Hours Community College Students Four-Year College Students Nonstudents Which category, if any, appears to contribute most to the total Chi-square? Which contribute the least?
In a volunteer group, adults 21 and older volunteer from one to nine hours each week to spend time with a disabled senior citizen. The program recruits among community college students, four-year college students, and nonstudents. In the Table below is a sample of the adult volunteers and the number of hours they volunteer per week.
|
Type of Volunteer |
1–3 Hours |
4–6 Hours |
7–9 Hours |
Row Total |
|
Community College Students |
111 |
96 |
48 |
255 |
|
Four-Year College Students |
96 |
133 |
61 |
290 |
|
Nonstudents |
91 |
150 |
53 |
294 |
|
Column Total |
298 |
379 |
162 |
839 |
Table: Number of Hours Worked Per Week by Volunteer Type (Observed) The table contains observed (O) values (data).
Is the number of hours volunteered independent of the type of volunteer?
The observed table and the question at the end of the problem, "Is the number of hours volunteered independent of the type of volunteer?" tell you this is a test of independence. The two factors are number of hours volunteered and type of volunteer.
H0: The number of hours volunteered is independent of the type of volunteer.
Ha: The number of hours volunteered is dependent on the type of volunteer.
The expected results are in the table below.
|
The table contains expected (E) values (data). |
|||
|
Type of Volunteer |
1-3 Hours |
4-6 Hours |
7-9 Hours |
|
Community College Students |
90.57 |
115.19 |
49.24 |
|
Four-Year College Students |
103.00 |
131.00 |
56.00 |
|
Nonstudents |
104.42 |
132.81 |
56.77 |
Table Number of Hours Worked Per Week by Volunteer Type (Expected)
For example, the calculation for the expected frequency for the top left cell is:
E=(row total)(column total)/total number surveyed=(255)(298)/839=90.57
Calculate the test statistic: χ2 = 12.99 (calculator or computer)
df = (3 columns – 1)*(3 rows – 1) = (2)(2) = 4
Graph:
Figure 11.7
Probability statement: p-value=P(χ2 > 12.99) = 0.0113
Compare α and the p-value: Since no α is given, assume α = 0.05. p-value = 0.0113. α > p-value.
Make a decision: Since α > p-value, reject H0. This means that the factors are not independent.
Conclusion: At a 5% level of significance, from the data, there is sufficient evidence to conclude that the number of hours volunteered, and the type of volunteer are dependent on one another.
Create a table below that shows the contribution of each category to the total Chi-square for this study.
|
The table contains the contribution to total Chi-square for each category. |
|||
|
Type of Volunteer |
1-3 Hours |
4-6 Hours |
7-9 Hours |
|
Community College Students |
|
|
|
|
Four-Year College Students |
|
|
|
|
Nonstudents |
|
|
|
Which category, if any, appears to contribute most to the total Chi-square? Which contribute the least?
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