At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To conduct a chi-square Goodness-of-Fit test, we follow a standard methodology:
1. Set up the hypotheses:
- Null hypothesis ([tex]\( H_0 \)[/tex]): The die has a uniform distribution.
- Alternative hypothesis ([tex]\( H_a \)[/tex]): The die does not have a uniform distribution.
2. Observed and expected frequencies:
- Observed counts: [tex]\([7, 10, 14, 16, 9, 22]\)[/tex]
- Expected counts: [tex]\([13, 13, 13, 13, 13, 13]\)[/tex]
3. Calculate the chi-square test statistic ([tex]\( \chi^2 \)[/tex]):
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
where [tex]\( O_i \)[/tex] is the observed frequency and [tex]\( E_i \)[/tex] is the expected frequency.
4. Compute the individual components:
- For outcome 1: [tex]\(\frac{(7 - 13)^2}{13} = \frac{36}{13}\)[/tex]
- For outcome 2: [tex]\(\frac{(10 - 13)^2}{13} = \frac{9}{13}\)[/tex]
- For outcome 3: [tex]\(\frac{(14 - 13)^2}{13} = \frac{1}{13}\)[/tex]
- For outcome 4: [tex]\(\frac{(16 - 13)^2}{13} = \frac{9}{13}\)[/tex]
- For outcome 5: [tex]\(\frac{(9 - 13)^2}{13} = \frac{16}{13}\)[/tex]
- For outcome 6: [tex]\(\frac{(22 - 13)^2}{13} = \frac{81}{13}\)[/tex]
5. Sum these components:
[tex]\[ \chi^2 = \frac{36}{13} + \frac{9}{13} + \frac{1}{13} + \frac{9}{13} + \frac{16}{13} + \frac{81}{13} \][/tex]
[tex]\[ \chi^2 = \frac{36 + 9 + 1 + 9 + 16 + 81}{13} = \frac{152}{13} \approx 11.692 \][/tex]
So, the test statistic [tex]\( \chi^2 \)[/tex] rounded to three decimal places is:
[tex]\[ \boxed{11.692} \][/tex]
1. Set up the hypotheses:
- Null hypothesis ([tex]\( H_0 \)[/tex]): The die has a uniform distribution.
- Alternative hypothesis ([tex]\( H_a \)[/tex]): The die does not have a uniform distribution.
2. Observed and expected frequencies:
- Observed counts: [tex]\([7, 10, 14, 16, 9, 22]\)[/tex]
- Expected counts: [tex]\([13, 13, 13, 13, 13, 13]\)[/tex]
3. Calculate the chi-square test statistic ([tex]\( \chi^2 \)[/tex]):
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
where [tex]\( O_i \)[/tex] is the observed frequency and [tex]\( E_i \)[/tex] is the expected frequency.
4. Compute the individual components:
- For outcome 1: [tex]\(\frac{(7 - 13)^2}{13} = \frac{36}{13}\)[/tex]
- For outcome 2: [tex]\(\frac{(10 - 13)^2}{13} = \frac{9}{13}\)[/tex]
- For outcome 3: [tex]\(\frac{(14 - 13)^2}{13} = \frac{1}{13}\)[/tex]
- For outcome 4: [tex]\(\frac{(16 - 13)^2}{13} = \frac{9}{13}\)[/tex]
- For outcome 5: [tex]\(\frac{(9 - 13)^2}{13} = \frac{16}{13}\)[/tex]
- For outcome 6: [tex]\(\frac{(22 - 13)^2}{13} = \frac{81}{13}\)[/tex]
5. Sum these components:
[tex]\[ \chi^2 = \frac{36}{13} + \frac{9}{13} + \frac{1}{13} + \frac{9}{13} + \frac{16}{13} + \frac{81}{13} \][/tex]
[tex]\[ \chi^2 = \frac{36 + 9 + 1 + 9 + 16 + 81}{13} = \frac{152}{13} \approx 11.692 \][/tex]
So, the test statistic [tex]\( \chi^2 \)[/tex] rounded to three decimal places is:
[tex]\[ \boxed{11.692} \][/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.