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 test whether the viewing audience proportions have changed, we can use a chi-squared test for independence. This test will compare the observed frequencies with the expected frequencies under the assumption that the proportions have not changed.

Here's how you can calculate the chi-squared value and the associated p-value:

Step 1: Set up the hypothesis:

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 need to assume that the proportions are still the same as during the first 13 weeks of the television season. We can multiply the proportions from the first 13 weeks by the total number of homes in the sample (300).

Expected frequency for ABC: (29% * 300) = 87 homes
Expected frequency for CBS: (28% * 300) = 84 homes
Expected frequency for NBC: (25% * 300) = 75 homes
Expected frequency for independents: (18% * 300) = 54 homes

Step 3: Calculate the chi-squared statistic:

χ2 = Σ((Observed frequency - Expected frequency)^2 / Expected frequency)

For each category (ABC, CBS, NBC, independents), calculate the squared difference between the observed and expected frequencies, divide it by the expected frequency, and then sum up these values.

χ2 = ((95-87)^2 / 87) + ((70-84)^2 / 84) + ((89-75)^2 / 75) + ((46-54)^2 / 54)

Calculate the final value of χ2.

Step 4: Find the p-value:

To find the p-value associated with the calculated chi-squared value, we need to compare it to the chi-squared distribution with degrees of freedom equal to the number of categories minus 1 (in this case, 4 - 1 = 3).

You can use a chi-squared distribution table or a calculator to find the p-value. Look for the chi-squared value in the table that corresponds to the degrees of freedom and compare it to the significance level (α). The p-value is the probability of observing a chi-squared value as extreme or more extreme than the calculated value.

p-value = (1 - area under the chi-squared distribution curve) for the calculated chi-squared value.

For example, if the calculated chi-squared value is 10.24 and the table value at 3 degrees of freedom for α = 0.05 is 7.815, the p-value would be 0.016.

So, to summarize:

a. The χ2 value can be calculated using the formula: χ2 = ((95-87)^2 / 87) + ((70-84)^2 / 84) + ((89-75)^2 / 75) + ((46-54)^2 / 54)

b. The p-value can be found by comparing the calculated χ2 value to the chi-squared distribution table or using a calculator to find the probability of obtaining a chi-squared value as extreme or more extreme than the calculated value.

Note: Always remember to round your answers to two decimal places as specified in the question.