Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To find the chi-square test statistic for the chi-square goodness-of-fit test, we follow these steps:
1. Identify the observed frequencies (O): These are given as:
- [tex]\(O_1 = 8\)[/tex]
- [tex]\(O_2 = 12\)[/tex]
- [tex]\(O_3 = 17\)[/tex]
- [tex]\(O_4 = 10\)[/tex]
- [tex]\(O_5 = 11\)[/tex]
- [tex]\(O_6 = 14\)[/tex]
2. Identify the expected frequencies (E): Since we are testing the fairness of the die, each outcome (1 to 6) should ideally occur 72/6 = 12 times (given directly in the table):
- [tex]\(E_1 = 12\)[/tex]
- [tex]\(E_2 = 12\)[/tex]
- [tex]\(E_3 = 12\)[/tex]
- [tex]\(E_4 = 12\)[/tex]
- [tex]\(E_5 = 12\)[/tex]
- [tex]\(E_6 = 12\)[/tex]
3. Apply the chi-square formula:
[tex]\[ \chi_0^2 = \sum_{k=1}^{6} \frac{(O_k - E_k)^2}{E_k} \][/tex]
We calculate each term in the sum separately:
[tex]\[ \frac{(O_1 - E_1)^2}{E_1} = \frac{(8 - 12)^2}{12} = \frac{(-4)^2}{12} = \frac{16}{12} = 1.333 \][/tex]
[tex]\[ \frac{(O_2 - E_2)^2}{E_2} = \frac{(12 - 12)^2}{12} = \frac{0^2}{12} = 0 \][/tex]
[tex]\[ \frac{(O_3 - E_3)^2}{E_3} = \frac{(17 - 12)^2}{12} = \frac{5^2}{12} = \frac{25}{12} = 2.083 \][/tex]
[tex]\[ \frac{(O_4 - E_4)^2}{E_4} = \frac{(10 - 12)^2}{12} = \frac{(-2)^2}{12} = \frac{4}{12} = 0.333 \][/tex]
[tex]\[ \frac{(O_5 - E_5)^2}{E_5} = \frac{(11 - 12)^2}{12} = \frac{(-1)^2}{12} = \frac{1}{12} = 0.083 \][/tex]
[tex]\[ \frac{(O_6 - E_6)^2}{E_6} = \frac{(14 - 12)^2}{12} = \frac{2^2}{12} = \frac{4}{12} = 0.333 \][/tex]
4. Sum the calculated terms:
[tex]\[ \chi_0^2 = 1.333 + 0 + 2.083 + 0.333 + 0.083 + 0.333 = 4.166 \][/tex]
5. Round the final answer to three decimal places:
[tex]\[ \chi_0^2 \approx 4.167 \][/tex]
Therefore, the chi-square test statistic [tex]\(\chi_0^2 = 4.167\)[/tex].
1. Identify the observed frequencies (O): These are given as:
- [tex]\(O_1 = 8\)[/tex]
- [tex]\(O_2 = 12\)[/tex]
- [tex]\(O_3 = 17\)[/tex]
- [tex]\(O_4 = 10\)[/tex]
- [tex]\(O_5 = 11\)[/tex]
- [tex]\(O_6 = 14\)[/tex]
2. Identify the expected frequencies (E): Since we are testing the fairness of the die, each outcome (1 to 6) should ideally occur 72/6 = 12 times (given directly in the table):
- [tex]\(E_1 = 12\)[/tex]
- [tex]\(E_2 = 12\)[/tex]
- [tex]\(E_3 = 12\)[/tex]
- [tex]\(E_4 = 12\)[/tex]
- [tex]\(E_5 = 12\)[/tex]
- [tex]\(E_6 = 12\)[/tex]
3. Apply the chi-square formula:
[tex]\[ \chi_0^2 = \sum_{k=1}^{6} \frac{(O_k - E_k)^2}{E_k} \][/tex]
We calculate each term in the sum separately:
[tex]\[ \frac{(O_1 - E_1)^2}{E_1} = \frac{(8 - 12)^2}{12} = \frac{(-4)^2}{12} = \frac{16}{12} = 1.333 \][/tex]
[tex]\[ \frac{(O_2 - E_2)^2}{E_2} = \frac{(12 - 12)^2}{12} = \frac{0^2}{12} = 0 \][/tex]
[tex]\[ \frac{(O_3 - E_3)^2}{E_3} = \frac{(17 - 12)^2}{12} = \frac{5^2}{12} = \frac{25}{12} = 2.083 \][/tex]
[tex]\[ \frac{(O_4 - E_4)^2}{E_4} = \frac{(10 - 12)^2}{12} = \frac{(-2)^2}{12} = \frac{4}{12} = 0.333 \][/tex]
[tex]\[ \frac{(O_5 - E_5)^2}{E_5} = \frac{(11 - 12)^2}{12} = \frac{(-1)^2}{12} = \frac{1}{12} = 0.083 \][/tex]
[tex]\[ \frac{(O_6 - E_6)^2}{E_6} = \frac{(14 - 12)^2}{12} = \frac{2^2}{12} = \frac{4}{12} = 0.333 \][/tex]
4. Sum the calculated terms:
[tex]\[ \chi_0^2 = 1.333 + 0 + 2.083 + 0.333 + 0.083 + 0.333 = 4.166 \][/tex]
5. Round the final answer to three decimal places:
[tex]\[ \chi_0^2 \approx 4.167 \][/tex]
Therefore, the chi-square test statistic [tex]\(\chi_0^2 = 4.167\)[/tex].
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.