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 = ?
To find the chi-square value, we need to perform a chi-square test of independence. The formula to calculate chi-square test statistic is:
χ² = Σ [(Observed frequency - Expected frequency)² / Expected frequency]
Here's how we can calculate the chi-square value step-by-step:
Step 1: Set up the null and alternative hypotheses:
Null Hypothesis (H₀): The viewing audience proportions haven't changed.
Alternative Hypothesis (H₁): The viewing audience proportions have changed.
Step 2: Determine the expected frequencies:
To calculate the expected frequencies, we need to assume that the null hypothesis is true and use the original proportions 29%, 28%, 25%, and 18% as the expected proportions. We can calculate the expected frequencies by multiplying these proportions by the total number of homes (300).
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
Step 3: Calculate the chi-square test statistic:
Using the formula mentioned earlier, we calculate the chi-square test statistic:
χ² = [ (95 - 87)² / 87 ] + [ (70 - 84)² / 84 ] + [ (89 - 75)² / 75 ] + [ (46 - 54)² / 54 ]
Step 4: Determine the degrees of freedom:
The degrees of freedom for a chi-square test of independence are calculated using the formula (number of rows - 1) * (number of columns - 1). In this case, we have 4 categories (ABC, CBS, NBC, independents), so the degrees of freedom will be (4 - 1) * (2 - 1) = 3.
Step 5: Find the critical chi-square value:
Since the significance level is not provided in the question, we can assume a significance level of α = 0.05. Using the chi-square distribution table or a calculator, we find the critical chi-square value for a chi-square test with 3 degrees of freedom and a significance level of 0.05 to be approximately 7.81.
Step 6: Compare the calculated chi-square value with the critical chi-square value:
If the calculated chi-square value is greater than the critical chi-square value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
x² = [ (95 - 87)² / 87 ] + [ (70 - 84)² / 84 ] + [ (89 - 75)² / 75 ] + [ (46 - 54)² / 54 ]
≈ 1.92
Step 7: Determine the conclusion:
Since the calculated chi-square value (1.92) is less than the critical chi-square value (7.81), we fail to reject the null hypothesis. Therefore, we can conclude that there is not enough evidence to suggest that the viewing audience proportions have changed.
Please note that the chi-square value has been rounded to two decimal places as requested.
To find the chi-square distribution in this case, you will need to perform a chi-square goodness-of-fit test. This test compares the observed frequencies (the sample data) with the expected frequencies (the population proportions) to determine if there is a significant difference.
Here's how you can calculate the chi-square statistic (x^2):
1. Determine the expected frequency for each category. To do this, multiply the total sample size (300 homes) by the respective proportion for each category.
- Expected frequency for ABC: 300 * 0.29 = 87 homes
- Expected frequency for CBS: 300 * 0.28 = 84 homes
- Expected frequency for NBC: 300 * 0.25 = 75 homes
- Expected frequency for independents: 300 * 0.18 = 54 homes
2. Set up a table with two columns: "Categories" and "Expected/ Observed Frequencies." Fill in the table with the respective values you calculated in step 1. Additionally, include the observed frequencies from the sample data.
```
| Categories | Expected Frequencies | Observed Frequencies |
|----------------|---------------------|---------------------|
| ABC | 87 | 95 |
| CBS | 84 | 70 |
| NBC | 75 | 89 |
| Independents | 54 | 46 |
```
3. Calculate the chi-square statistic (x^2) using the formula:
x^2 = ∑ [(Observed Frequency - Expected Frequency)^2 / Expected Frequency]
For each category, calculate (Observed Frequency - Expected Frequency)^2 / Expected Frequency, then sum them up.
- For ABC: [(95-87)^2 / 87] = 0.920
- For CBS: [(70-84)^2 / 84] = 2.333
- For NBC: [(89-75)^2 / 75] = 2.520
- For independents: [(46-54)^2 / 54] = 0.889
Summing them up, x^2 = 0.920 + 2.333 + 2.520 + 0.889 = 6.662
4. Determine the degrees of freedom (df). For a goodness-of-fit test, df = number of categories - 1. In this case, there are 4 categories, so df = 4 - 1 = 3.
5. Look up the critical value of chi-square for your desired level of significance and df. In this case, you stated that the significance level (α) is 0.05 (5%) with df = 3. Using a chi-square distribution table or a calculator, you can find that the critical value is approximately 7.815.
6. Compare the calculated chi-square statistic (6.662) with the critical value (7.815). If the calculated x^2 is greater than the critical value, you can reject the null hypothesis and conclude that the viewing audience proportions have changed significantly. Otherwise, if the calculated x^2 is less than or equal to the critical value, you fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant change in the viewing audience proportions.
In your case, the calculated chi-square statistic (6.662) is less than the critical value (7.815). Therefore, you fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant change in the viewing audience proportions.
Remember to round your answers to two decimal places.