I need to find the chi square distribution. Thanks
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 = ?
6.87
To find the chi-square statistic (x^2) for this problem, we need to follow a series of steps. Let's break it down:
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: Set the significance level (alpha).
In this case, the significance level is given as = 0.05.
Step 3: Create the observed and expected frequency tables.
- Create an observed frequency table using the given data:
- ABC: 95 homes
- CBS: 70 homes
- NBC: 89 homes
- Independents: 46 homes
- To create the expected frequency table, we need to calculate the expected number of homes for each TV network based on the original proportions (ABC: 29%, CBS: 28%, NBC: 25%, Independents: 18%) and the total sample size of 300 homes.
- ABC: (0.29 * 300) = 87 homes
- CBS: (0.28 * 300) = 84 homes
- NBC: (0.25 * 300) = 75 homes
- Independents: (0.18 * 300) = 54 homes
Step 4: Calculate the chi-square (x^2) statistic.
- To calculate the chi-square statistic, we will use the formula:
x^2 = Σ [(observed frequency - expected frequency)^2 / expected frequency]
where Σ denotes summing up the values for each category.
- Using the values from our observed and expected frequency tables, we can now calculate the chi-square statistic:
x^2 = [(95 - 87)^2 / 87] + [(70 - 84)^2 / 84] + [(89 - 75)^2 / 75] + [(46 - 54)^2 / 54]
Step 5: Determine the degrees of freedom (df).
- The degrees of freedom is calculated as (number of categories - 1). In this case, there are 4 categories (ABC, CBS, NBC, Independents), so the degrees of freedom is 4 - 1 = 3.
Step 6: Compare the chi-square statistic to the critical value.
- To determine whether the viewing audience proportions have changed, we need to compare the calculated chi-square statistic to the critical value from the chi-square distribution table based on the given significance level and degrees of freedom. If the calculated chi-square value is greater than the critical value, we reject the null hypothesis.
Step 7: Make a conclusion.
- If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis and conclude that the viewing audience proportions have changed. Otherwise, if the calculated chi-square statistic is less than or equal to the critical value, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a change in the viewing audience proportions.
To find the chi-square distribution critical value for a given alpha and degrees of freedom, you can refer to a chi-square distribution table or use statistical software.
Remember to round your answer to two decimal places as specified in the question.