During the first 13 weeks of the television season, the Saturday evening 8:00 P.M. to 9:00 P.M. audience proportions were recorded as ABC 29%, CBS 28%, NBC 25%, and independents 18%. A sample of 300 homes two weeks after a Saturday night schedule revision yielded the following viewing audience data: ABC 95 homes, CBS 70 homes, NBC 89 homes, and independents 46 homes. Test with = .05 to determine whether the viewing audience proportions changed.
Round your answers to two decimal places.
÷2 = ?
p-value is between what? and what?
To test whether the viewing audience proportions have changed, we can use the chi-square test of independence. Here are the steps to calculate the chi-square statistic and determine the p-value:
Step 1: Set up the hypotheses:
Null hypothesis (H0): The viewing audience proportions have not changed.
Alternative hypothesis (Ha): The viewing audience proportions have changed.
Step 2: Calculate the expected frequencies:
To calculate the expected frequencies, we use the proportions from the first 13 weeks of the television season. We multiply these proportions by the total number of homes in the sample (300) to get the expected frequencies for each category. Here are the calculations:
Expected frequency for ABC: 300 * 0.29 = 87
Expected frequency for CBS: 300 * 0.28 = 84
Expected frequency for NBC: 300 * 0.25 = 75
Expected frequency for independents: 300 * 0.18 = 54
Step 3: Calculate the chi-square statistic:
The chi-square statistic measures the difference between the observed and expected frequencies. We calculate it using the formula:
χ^2 = Σ ((O - E)^2) / E
where O is the observed frequency and E is the expected frequency. We sum this expression across all categories. Using the given data, we get:
χ^2 = ((95 - 87)^2 / 87) + ((70 - 84)^2 / 84) + ((89 - 75)^2 / 75) + ((46 - 54)^2 / 54)
Calculate this expression to find the value of the chi-square statistic.
Step 4: Determine the degrees of freedom:
The degrees of freedom (df) for a chi-square test of independence is (R-1) * (C-1), where R is the number of rows and C is the number of columns in the contingency table. In this case, there are 4 rows (ABC, CBS, NBC, independents) and 1 column (before and after revision). Therefore, the degrees of freedom are (4-1) * (1-1) = 3 * 0 = 0.
Step 5: Calculate the p-value:
To determine the p-value, we compare the chi-square statistic to the chi-square distribution with the appropriate degrees of freedom. Since the degrees of freedom are 0, we cannot use the chi-square distribution table. However, we can calculate an approximation of the p-value using statistical software or online calculators.
Once you have obtained the p-value, compare it to the significance level (α), which is given as 0.05 in this case. If the p-value is less than α, we reject the null hypothesis and conclude that the viewing audience proportions have changed. If the p-value is greater than or equal to α, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a change in the viewing audience proportions.
Remember to round your answers to two decimal places.
Note: Since the degrees of freedom are 0 in this case, it implies that there was no random sampling involved, and the proportions cannot be tested for independence. Please verify if the given information is correct, as having 0 degrees of freedom is not a typical scenario for a chi-square test of independence.