A group of high-school boys and girls were asked to state their political party preferences. The results are shown in the table (k is the same constant in both places it occurs). Which of the following statements are true?

Question

I. The variables gender and party preference are independent.
II. For these data, X2 must be 0.
III. It cannot be determined from these data whether the variables gender and party preference are related.

The table has three rows and three columns, with the cells of the first row saying, in order, separated by commas: "Gender", "Democrat", and "Republican"
The second row's cells say, in order, and separated by commas: "Boys", "x", "kx"
The third row's cells say, in order, and separated by commas: "Girls", "y", "ky"

Answer 1

I. The variables gender and party preference are not independent, as the proportion of Democrats and Republicans is different between boys and girls.
II. X2 cannot be determined without knowing the values of x, y, and k.
III. False, as we have already determined that the variables are not independent in this case.

Answer 2

You must choose one of the following choices:

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Answer 3

The correct answer is D. I and II only.

Answer 4

Why?

Answer 5

Statement I is true because the data shows that there is a difference in the proportion of Democrats and Republicans between boys and girls, indicating a relationship between gender and party preference.

Statement II is false because the value of X2 cannot be determined without knowing the values of x, y, and k. However, we can say that if there is a significant difference between the observed and expected values (calculated using the formula for X2), then it would indicate a relationship between gender and party preference.

Statement III is false because we have already established that the variables are not independent.