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.

x2 = ?
p-value is between what? and what?

To determine if the viewing audience proportions have changed, we can perform a chi-square test of independence.

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 for each category:
To calculate the expected frequencies, we first need to determine the expected proportions for each category. The expected proportions can be calculated by multiplying the overall proportions by the total number of homes in the sample (300).

Expected proportion for ABC: 0.29 * 300 = 87
Expected proportion for CBS: 0.28 * 300 = 84
Expected proportion for NBC: 0.25 * 300 = 75
Expected proportion for independents: 0.18 * 300 = 54

Step 3: Calculate the chi-square test statistic:
The chi-square test statistic (x2) can be calculated using the formula:
x2 = Σ((Observed frequency - Expected frequency)^2 / Expected frequency)

Using the observed and expected frequencies, we can calculate x2 as follows:

x2 = ((95 - 87)^2 / 87) + ((70 - 84)^2 / 84) + ((89 - 75)^2 / 75) + ((46 - 54)^2 / 54)

Calculating the above expression gives:
x2 = 2.687

Step 4: Find the p-value:
The p-value can be found by comparing the calculated x2 value to the chi-square distribution table with (4-1) * (2-1) = 3 degrees of freedom at the given significance level (α = 0.05).

Looking up the critical value for 3 degrees of freedom and α = 0.05 in the chi-square distribution table gives an x2 value of 7.815.

Since the calculated x2 value (2.687) is less than the critical value (7.815), we fail to reject the null hypothesis.

Step 5: Conclusion:
Based on the test results, we can conclude that there is not enough evidence to suggest a change in the viewing audience proportions.

To answer this question, we need to perform a chi-square goodness of fit test to determine if the viewing audience proportions have changed.

Let's start by calculating the expected frequencies for each category based on the initial proportions. We can do this by multiplying each proportion by the total sample size (300 homes).

Expected frequency for ABC: 0.29 * 300 = 87
Expected frequency for CBS: 0.28 * 300 = 84
Expected frequency for NBC: 0.25 * 300 = 75
Expected frequency for independents: 0.18 * 300 = 54

Next, we'll set up our null and alternative hypotheses:
Null hypothesis (H0): The viewing audience proportions have not changed.
Alternative hypothesis (Ha): The viewing audience proportions have changed.

Now, we can calculate the chi-square test statistic (x2) using the formula:
x2 = Σ((O - E)^2 / E)
where O is the observed frequency and E is the expected frequency for each category.

For ABC: (95 - 87)^2 / 87 = 0.9207
For CBS: (70 - 84)^2 / 84 = 2.6667
For NBC: (89 - 75)^2 / 75 = 3.5733
For independents: (46 - 54)^2 / 54 = 0.8889

Summing up these values, we get:
x2 = 0.9207 + 2.6667 + 3.5733 + 0.8889 = 8.0496

Now, we need to find the p-value associated with this chi-square test statistic. To do this, we need to determine the degrees of freedom (df), which is calculated as (number of categories - 1).

In this case, df = 4 - 1 = 3.

Using the chi-square distribution table or a statistical calculator, we find that the p-value for a chi-square test statistic of 8.0496 with 3 degrees of freedom is between 0.05 and 0.10.

Therefore, the p-value is between 0.05 and 0.10.

Remember that we set our significance level (α) to be 0.05. Since the p-value is greater than α, we fail to reject the null hypothesis.

In conclusion, based on the test results, there is not enough evidence to suggest that the viewing audience proportions have changed.

Bho