14. What are the degrees of freedom for each of the following situations? a. N=28, C=6, Goodness of fit test b. N=200, K=10, R=2, Test of independence c. N=130, C=5, Test of homogeneity 15. A researcher selects a sample of 40 people to investigate the relationship between brand preference (Brand A or Brand B) and gender (Male or Female). Of the 20 males in the sample, five prefer Brand A. Of the 20 females in the sample, 12 prefer Brand A. Based on this scenario: | Preference | Male | Female | Total | |------------|------|--------|-------| | Brand A | 5 | 12 | | | Brand B | | | | | Total | 20 | 20 | 40 | a. What is the expected frequency for females who prefer Brand A? b. What is the expected frequency for females who prefer Brand B? c. What is the observed frequency for females who prefer Brand B?
Answer (1)
Let's find the degrees of freedom for each situation one by one:
a. Goodness of fit test (N=28, C=6):
For a goodness of fit test, the degrees of freedom (df) can be calculated using the formula:
df = C − 1
Where C is the number of categories.
Thus, df = 6 − 1 = 5 .
b. Test of independence (N=200, K=10, R=2):
For a test of independence, the degrees of freedom are calculated using the formula:
df = ( R − 1 ) × ( C − 1 )
In this scenario, R = 2 and C = 10 . Therefore, df = ( 2 − 1 ) × ( 10 − 1 ) = 9 .
c. Test of homogeneity (N=130, C=5):
Similar to the goodness of fit test, the test of homogeneity uses:
df = C − 1
Thus, df = 5 − 1 = 4 .
In this scenario, we have the following table filled partially:
PreferenceMaleFemaleTotalBrand A512Brand BTotal202040
a. Expected frequency for females who prefer Brand A:
The expected frequency can be calculated using:
Expected Frequency = Grand Total (Row Total) × (Column Total)
The row total for Brand A is 5 + 12 = 17 .
Expected frequency = 40 17 × 20 = 8.5 .
b. Expected frequency for females who prefer Brand B:
The total number who prefer Brand B (row total) is 40 − 17 = 23 .
Expected frequency = 40 23 × 20 = 11.5 .
c. Observed frequency for females who prefer Brand B:
Out of 20 females, 12 prefer Brand A. Therefore, the observed frequency for females who prefer Brand B is 20 − 12 = 8 .
