A study of health care planning undertook a mail survey of hospital administrators in Arkansas,
Louisiana, and east Texas. Research question: At á = .05, is the type of planning independent
of the intensity of competition? (Data are from D. O. McKee, P. R. Varadarajan, and J. Vassar,
“A Taxonomy of Marketing Planning Styles,” Journal of the Academy of Marketing Science 18,
no. 2 [Spring 1990], pp. 131–41.) Planning
Degree of Limited Constituency Comprehensive
Competition Planning Planning Planning Row Total
Low 11 25 33 69
Moderate 19 23 15 57
High 43 25 17 85
Col Total 73 73 65 211
To determine if the type of planning is independent of the intensity of competition, we can conduct a chi-square test of independence. This statistical test will help us determine if there is a significant relationship between the two variables.
First, let's set up the null and alternative hypotheses:
Null Hypothesis (H0): The type of planning and the intensity of competition are independent.
Alternative Hypothesis (Ha): The type of planning and the intensity of competition are not independent.
Next, let's calculate the expected frequencies for each cell in the contingency table:
To calculate the expected frequency for each cell, we can use the formula:
Expected frequency = (Row total x Column total) / Grand total
Using this formula, let's calculate the expected frequencies for the given data:
Planning | Degree of Limited | Constituency | Comprehensive | Row Total
| Competition | Planning | Planning |
--------------------------------------------------------------------------------
Low | 11 | 25 | 33 | 69
Moderate | 19 | 23 | 15 | 57
High | 43 | 25 | 17 | 85
--------------------------------------------------------------------------------
Col Total | 73 | 73 | 65 | 211
Expected Frequency Calculation:
For the cell in the first row and first column:
Expected frequency = (69 * 73) / 211 = 23.81 (rounded to two decimal places)
Similarly, calculate the expected frequencies for the rest of the cells.
Now, we can perform the chi-square test of independence using the observed frequencies and the expected frequencies.
The chi-square test statistic (χ2) can be calculated using the formula:
χ2 = ∑ ((Observed frequency - Expected frequency)^2) / Expected frequency
Sum the values for each cell using this formula, and you will get the chi-square test statistic (χ2).
The degrees of freedom (df) can be calculated using the formula:
df = (Number of rows - 1) * (Number of columns - 1)
In this case, df = (3 - 1) * (3 - 1) = 2 * 2 = 4.
Finally, compare the calculated chi-square test statistic (χ2) with the critical value from the chi-square distribution table using a significance level (α) of 0.05 and the degrees of freedom (df) calculated earlier.
If the calculated χ2 value is greater than the critical value, we reject the null hypothesis (H0) and conclude that the type of planning is dependent on the intensity of competition. If the calculated χ2 value is less than or equal to the critical value, we fail to reject the null hypothesis (H0) and conclude that there is no significant relationship between the two variables.
I hope this helps! Let me know if you have any further questions.
To determine if the type of planning is independent of the intensity of competition, you can conduct a chi-square test of independence. This test will analyze the relationship between two categorical variables, in this case, the type of planning (limited, constituency, comprehensive) and the intensity of competition (low, moderate, high).
Here's how you can calculate the chi-square test statistic and perform the test:
Step 1: Set up the hypothesis:
- Null hypothesis (H0): The type of planning is independent of the intensity of competition.
- Alternative hypothesis (Ha): The type of planning is not independent of the intensity of competition.
Step 2: Calculate the expected frequencies:
- Calculate the expected frequency for each cell in the table. The expected frequency is calculated using the formula: (row total x column total) / grand total.
- For example, the expected frequency for the first cell (low competition, limited planning) would be (69 x 73) / 211 = 23.25.
Step 3: Calculate the chi-square test statistic:
- Calculate the chi-square test statistic using the formula: Σ ((observed frequency - expected frequency)^2) / expected frequency.
- Sum up the values for each cell in the table to calculate the test statistic.
Step 4: Determine the critical value:
- Look up the critical value for the chi-square distribution with degrees of freedom equal to (number of rows - 1) x (number of columns - 1). In this case, the degrees of freedom would be (3-1) x (3-1) = 4.
Step 5: Compare the test statistic to the critical value:
- If the test statistic is greater than the critical value, reject the null hypothesis. If it is smaller, fail to reject the null hypothesis.
Step 6: Interpret the result:
- If the null hypothesis is rejected, it suggests that the type of planning is associated with the intensity of competition.
Performing the calculations with the given data:
| | Limited | Constituency | Comprehensive | Row Total |
|:-------------:|:-------:|:-----------:|:-------------:|:---------:|
| Low | 11 | 25 | 33 | 69 |
| Moderate | 19 | 23 | 15 | 57 |
| High | 43 | 25 | 17 | 85 |
| Column Total | 73 | 73 | 65 | 211 |
Expected frequencies:
- Expected frequency for the first cell (low competition, limited planning): (69 × 73) / 211 = 23.25
Performing the calculations for all the cells, we get the expected frequencies as follows:
| | Limited | Constituency | Comprehensive | Row Total |
|:-------------:|:-------:|:-----------:|:-------------:|:---------:|
| Low | 23.25 | 23.25 | 22.5 | 69 |
| Moderate | 19.4 | 19.4 | 18.2 | 57 |
| High | 30.35 | 30.35 | 28.3 | 85 |
| Column Total | 73 | 73 | 65 | 211 |
Next, we calculate the chi-square test statistic:
Σ((observed frequency - expected frequency)^2) / expected frequency =
((11 - 23.25)^2 / 23.25) + ((25 - 23.25)^2 / 23.25) + ((33 - 22.5)^2 / 22.5) + ((19 - 19.4)^2 / 19.4) + ((23 - 19.4)^2 / 19.4) + ((15 - 18.2)^2 / 18.2) + ((43 - 30.35)^2 / 30.35) + ((25 - 30.35)^2 / 30.35) + ((17 - 28.3)^2 / 28.3) = 9.92
The degrees of freedom for this test are (3 - 1) x (3 - 1) = 4.
Using a chi-square distribution table or a calculator with the appropriate degrees of freedom, we find that the critical value at α = 0.05 is 9.488.
Since the test statistic (9.92) is greater than the critical value (9.488), we reject the null hypothesis.
Therefore, we can conclude that at α = .05, the type of planning is not independent of the intensity of competition.