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 between ? and ?

To determine whether the viewing audience proportions changed after the schedule revision, we can conduct a Chi-square goodness-of-fit test.

Let's start by setting up the null and alternative hypotheses:

Null Hypothesis (H0): The viewing audience proportions have not changed.
Alternative Hypothesis (Ha): The viewing audience proportions have changed.

Since we have four categories (ABC, CBS, NBC, and independents), we will calculate the Chi-square statistic using the following formula:

χ^2 = Σ ((Observed frequency - Expected frequency)^2 / Expected frequency)

To calculate the expected frequency for each category, we need to multiply the overall sample size (300 homes) by the original proportions.

Expected frequencies:
ABC: (0.29)*(300) = 87
CBS: (0.28)*(300) = 84
NBC: (0.25)*(300) = 75
Independents: (0.18)*(300) = 54

Now, let's calculate the χ^2 statistic:

χ^2 = ((95-87)^2/87) + ((70-84)^2/84) + ((89-75)^2/75) + ((46-54)^2/54)

Simplifying the calculation:

χ^2 = (64/87) + (196/84) + (196/75) + (64/54)

χ^2 ≈ 0.735 + 2.333 + 2.613 + 1.185 ≈ 6.866

a. χ^2 = 6.866

To find the p-value, we need to consult the Chi-square distribution table with (k-1) degrees of freedom (where k is the number of categories). In this case, we have 4 categories, so df = 4-1 = 3.

Using the Chi-square distribution table or a statistical software, we find that the p-value associated with a χ^2 statistic of 6.866 and 3 degrees of freedom is between 0.076 and 0.149.

b. The p-value is between 0.076 and 0.149.

Since the p-value (0.076 to 0.149) is greater than the significance level of 0.05, we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the viewing audience proportions have changed after the schedule revision.