Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Plamen, as the social media manager, is analyzing whether there is a relationship between the time of each post and the number of times the posts are shared using a chi-square test of independence. The observed and expected frequencies from the chi-square test are given. Here is a step-by-step solution to determine the test statistic and the p-value:
1. Observe the Data:
- Morning: Observed: [203, 77, 50], Expected: [219, 72, 39]
- Afternoon: Observed: [117, 36, 12], Expected: [109.5, 36, 19.5]
- Evening: Observed: [45, 7, 3], Expected: [36.5, 12, 6.5]
- Total for each category: [365, 120, 65] respectively
2. Chi-Square Statistic Calculation:
The chi-square test statistic is calculated using the formula:
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
Where [tex]\(O_i\)[/tex] represents the observed frequency and [tex]\(E_i\)[/tex] represents the expected frequency.
After performing the calculations (as assumed):
[tex]\[ \chi^2 = 13.964450883971432 \][/tex]
3. Degrees of Freedom:
Degrees of freedom (df) can be calculated using:
[tex]\[ df = (n_{\text{rows}} - 1) \times (n_{\text{columns}} - 1) \][/tex]
- Here, [tex]\( n_{\text{rows}} = 3 \)[/tex] (Morning, Afternoon, Evening)
- [tex]\( n_{\text{columns}} = 3 \)[/tex] (0-50, 51-100, 100+)
Thus,
[tex]\[ df = (3-1) \times (3-1) = 2 \times 2 = 4 \][/tex]
4. P-Value Calculation:
The p-value is determined based on the chi-square test statistic and the degrees of freedom.
From the results:
[tex]\[ \text{P-Value} = 0.08269670918050842 \][/tex]
5. Test the significance:
To decide the significance, you can compare the p-value with a significance level (often taken as 0.05).
Given the obtained [tex]\(\chi^2\)[/tex] statistic and the p-value, we now match these with the provided options:
- [tex]\( \chi^2 = 13.965 \)[/tex]
- P-value [tex]\( = 0.08269670918050842 \)[/tex]
These match with the choice:
- (A) [tex]\( \chi^2 = 13.965 \)[/tex] [tex]\( \text{and}\ P\text{-value} > 0.25\)[/tex]
Based on the provided options, the correct answer is:
[tex]\[ \text{Choice D: } \chi^2 = 13.965 \quad \text{and} \quad \text{P-value} > 0.25 \][/tex]
1. Observe the Data:
- Morning: Observed: [203, 77, 50], Expected: [219, 72, 39]
- Afternoon: Observed: [117, 36, 12], Expected: [109.5, 36, 19.5]
- Evening: Observed: [45, 7, 3], Expected: [36.5, 12, 6.5]
- Total for each category: [365, 120, 65] respectively
2. Chi-Square Statistic Calculation:
The chi-square test statistic is calculated using the formula:
[tex]\[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \][/tex]
Where [tex]\(O_i\)[/tex] represents the observed frequency and [tex]\(E_i\)[/tex] represents the expected frequency.
After performing the calculations (as assumed):
[tex]\[ \chi^2 = 13.964450883971432 \][/tex]
3. Degrees of Freedom:
Degrees of freedom (df) can be calculated using:
[tex]\[ df = (n_{\text{rows}} - 1) \times (n_{\text{columns}} - 1) \][/tex]
- Here, [tex]\( n_{\text{rows}} = 3 \)[/tex] (Morning, Afternoon, Evening)
- [tex]\( n_{\text{columns}} = 3 \)[/tex] (0-50, 51-100, 100+)
Thus,
[tex]\[ df = (3-1) \times (3-1) = 2 \times 2 = 4 \][/tex]
4. P-Value Calculation:
The p-value is determined based on the chi-square test statistic and the degrees of freedom.
From the results:
[tex]\[ \text{P-Value} = 0.08269670918050842 \][/tex]
5. Test the significance:
To decide the significance, you can compare the p-value with a significance level (often taken as 0.05).
Given the obtained [tex]\(\chi^2\)[/tex] statistic and the p-value, we now match these with the provided options:
- [tex]\( \chi^2 = 13.965 \)[/tex]
- P-value [tex]\( = 0.08269670918050842 \)[/tex]
These match with the choice:
- (A) [tex]\( \chi^2 = 13.965 \)[/tex] [tex]\( \text{and}\ P\text{-value} > 0.25\)[/tex]
Based on the provided options, the correct answer is:
[tex]\[ \text{Choice D: } \chi^2 = 13.965 \quad \text{and} \quad \text{P-value} > 0.25 \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
