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 alpha of.05 to determine whether the viewing audience proportions changed.
Round your answers to two decimal places.
a. χ2 = ?
b. p-value is betwe
To determine if the viewing audience proportions changed after the schedule revision, we can use a chi-square test of independence. Here's how you can calculate the chi-square statistic and the p-value for this test:
Step 1: State the null and alternative hypotheses:
- Null Hypothesis (H0): The viewing audience proportions are the same after the schedule revision.
- Alternative Hypothesis (Ha): The viewing audience proportions have changed after the schedule revision.
Step 2: Calculate the expected frequencies:
To perform the chi-square test, we need to calculate the expected frequencies under the assumption that the proportions are the same. To do this, multiply the overall sample size (300) by the proportions recorded during the first 13 weeks.
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. It is calculated using the formula:
χ2 = Σ ((Observed frequency - Expected frequency)^2 / Expected frequency)
Here's how you can calculate it:
χ2 = ((95 - 87)^2 / 87) + ((70 - 84)^2 / 84) + ((89 - 75)^2 / 75) + ((46 - 54)^2 / 54)
= (64 / 87) + (196 / 84) + (196 / 75) + (64 / 54)
= 0.735 + 2.333 + 2.613 + 1.185
≈ 6.866
So, the chi-square statistic (χ2) is approximately 6.866.
Step 4: Determine the degrees of freedom:
The degrees of freedom (df) for this test is equal to the number of categories minus 1. In this case, there are 4 categories (ABC, CBS, NBC, independents), so df = 4 - 1 = 3.
Step 5: Look up the critical value:
To find the critical value for a chi-square test with a significance level of α = 0.05 and 3 degrees of freedom, consult a chi-square distribution table or use a statistical software. The critical value for α = 0.05 and df = 3 is approximately 7.815.
Step 6: Calculate the p-value:
The p-value represents the probability of observing a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true. To calculate the p-value, you can use statistical software or a chi-square distribution table. In this case, the p-value is between 0.05 and 0.02.
So, the answers to the questions are:
a. χ2 = 6.866 (rounded to two decimal places)
b. The p-value is between 0.05 and 0.02