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?
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"
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.
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
The correct answer is D. I and II only.
Why?
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.
To determine which statements are true, we need to analyze the given table of results:
The table shows three rows and three columns. Let's represent the table using letters:
Let D represent the variable "Democrat."
Let R represent the variable "Republican."
Let B represent the variable "Boys."
Let G represent the variable "Girls."
The table can be represented as:
| | D | R |
|---|------|------|
| B | x | kx |
| G | y | ky |
Statement I: "The variables gender and party preference are independent."
To check whether the variables gender and party preference are independent, we need to calculate the expected values for each cell. The expected value for each cell is calculated by multiplying the row total by the column total and dividing by the grand total.
Expected value for cell (B,D) = (x+y)(x+kx)/((x+y)+(kx+ky))
Expected value for cell (B,R) = (x+y)(kx+ky)/((x+y)+(kx+ky))
Expected value for cell (G,D) = (x+y)(x+kx)/((x+y)+(kx+ky))
Expected value for cell (G,R) = (x+y)(kx+ky)/((x+y)+(kx+ky))
If the expected values for each cell match the observed values, then the variables are independent.
Statement II: "For these data, X^2 must be 0."
To calculate X^2 (chi-squared), we need to calculate the sum of ((observed value - expected value)^2) / expected value for each cell. If X^2 = 0, it means the observed frequencies match the expected frequencies perfectly, indicating independence.
Statement III: "It cannot be determined from these data whether the variables gender and party preference are related."
This statement is true if the expected values for each cell do not match the observed values, and if the calculated X^2 is not equal to 0.
To determine which statements are true, we need the values of x, y, and k from the given table. Please provide these values to proceed with the analysis.
To determine which statements are true, we need to analyze the table provided and understand the relationship between gender and party preference for the high-school boys and girls.
Let's break down the information in the table:
- The first column represents the variable "Gender."
- The second column represents the variable "Party preference", specifically for Democrats.
- The third column represents the variable "Party preference", specifically for Republicans.
According to the table, we see that:
- The number of boys who prefer the Democrat party is represented by "x."
- The number of girls who prefer the Democrat party is represented by "y."
- The number of boys who prefer the Republican party is represented by "kx."
- The number of girls who prefer the Republican party is represented by "ky."
Now, let's evaluate each statement:
I. The variables gender and party preference are independent.
Two variables are independent if the occurrence of one variable does not affect the occurrence of the other variable. In this case, we should check if the distribution of party preferences is the same for boys and girls.
Looking at the table, if the variables were independent, we would expect the ratio of boys to girls preferring the Democrat party to be the same as the ratio of boys to girls preferring the Republican party. However, we cannot determine this from the given data. So, we cannot conclude that the variables are independent. Therefore, Statement I is not necessarily true.
II. For these data, X2 must be 0.
The χ2 (chi-square) test is used to analyze the independence between variables. Based on the statement, X2 refers to the chi-square statistic.
To calculate the chi-square statistic, we need to compare the observed frequencies of the data with the expected frequencies if the variables were independent.
Since we cannot determine the expected frequencies from the given data, we cannot calculate the chi-square statistic. Therefore, we cannot conclude that the X2 statistic must be 0. Statement II is false.
III. It cannot be determined from these data whether the variables gender and party preference are related.
As discussed earlier, we cannot determine whether the variables gender and party preference are related based on the given data. There is not enough information to make a conclusion about the relationship between the variables. Therefore, Statement III is true.
In summary:
- Statement I is not necessarily true.
- Statement II is false.
- Statement III is true.