A research organization has collected the following data on household size and telephone ownership for 200 U.S. households. At the 0.05 level, are the two variables independent? Based on the chi-square table, what is the most accurate statement that can be made about the p-value for the test?

Telephones Owned ¡Ü 1 2 ¡Ý 3 Total

Persons ¡Ü 2 49 18 13 80
In the 3-4 40 27 21 88
Household ¡Ý 5 11 13 8 32
100 58 42 200

For three independent samples, each with n= 100, the respective sample proportions are 0.30, 0.35, and 0.25. Use the 0.05 level in testing whether the three population proportions could be the same.

For three independent samples, each with n= 100, the respective sample proportions are 0.30, 0.35, and 0.25. Use the 0.05 level in testing whether the three population proportions could be the same.

Here the number of degrees of freedom is (3-1)=2. Hence the continuity correction of 0.5 is not required.

The observations are 30,35 and 25, and the theoretical values are therefore (30+35+25)/3=30
Χ²
=(30-30)²/30+(35-30)²/30+(25-30)²/30
=0+0.833+0.833
=1.667
At α=0.05, . prob.=0.95 and DF=2, Χ²=5.99
Since Χ²=1.667<5.99, we conclude that the hypothesis that the three populations are the same is not rejected.

To determine if the two variables, household size and telephone ownership, are independent, you can perform a chi-square test of independence.

The first step is to set up a contingency table using the data provided:

Telephones Owned Total
¡Ü 1 2 ¡Ý 3
Persons ¡Ü 2 49 18 13 80
In the 3-4 Household 40 27 21 88
¡Ý 5 11 13 8 32
Total 100 58 42 200

Next, you can calculate the expected frequencies for each cell assuming the variables are independent. The expected frequency for each cell is calculated by multiplying the corresponding row total and column total and dividing by the total number of observations (200 in this case).

After calculating the expected frequencies, you can perform the chi-square test by using the formula:

χ² = ∑((Oij - Eij)² / Eij)

where Oij is the observed frequency and Eij is the expected frequency for each cell.

By calculating the chi-square test statistic and comparing it to the critical value from the chi-square table at a significance level of 0.05, you can determine if the variables are independent. If the calculated chi-square value is greater than the critical value, it suggests that the variables are dependent.

The degrees of freedom for this test can be calculated as (number of rows - 1) * (number of columns - 1) = (3 - 1) * (3 - 1) = 4.

Now, based on the chi-square table or using statistical software, you can find the critical value for a chi-square test with 4 degrees of freedom and a significance level of 0.05.

Once you have the critical value and the calculated chi-square value, you can compare them. If the calculated chi-square value is greater than the critical value, you would reject the null hypothesis and conclude that the two variables are dependent. Otherwise, if the calculated chi-square value is less than or equal to the critical value, you would fail to reject the null hypothesis and conclude that the two variables are independent.

Regarding the most accurate statement about the p-value for the test, you would need to calculate the p-value using the chi-square test statistic and the degrees of freedom. The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed test statistic, assuming the null hypothesis is true. You can compare this p-value to your chosen significance level (0.05 in this case) to make a decision about the null hypothesis. However, the exact p-value cannot be determined just by looking at the chi-square table. It would require additional calculations or the use of statistical software.