SOLUTION: The table below lists the number of games played in a yearly​ best-of-seven baseball championship​ series, along with the expected proportions for the number of games played wi

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Question 1170878: The table below lists the number of games played in a yearly​ best-of-seven baseball championship​ series, along with the expected proportions for the number of games played with teams of equal abilities. Use a 0.05 significance level to test the claim that the actual numbers of games fit the distribution indicated by the expected proportions.
games played is 4,5,6,7
actual contest is 16,21,20,37
expected proportion is 2/16,4/16,5/16,5/16
need to find x t the 2 power
rounded to the three decimal places as needed
Answer by CPhill(1987) About Me  (Show Source):
You can put this solution on YOUR website!
Let's perform a chi-square goodness-of-fit test to determine if the observed frequencies match the expected proportions.
1. Calculate Expected Frequencies
Total number of games played = 16 + 21 + 20 + 37 = 94
Expected proportions: 2/16, 4/16, 5/16, 5/16
Expected frequencies:
Games = 4: (2/16) * 94 = 11.75
Games = 5: (4/16) * 94 = 23.5
Games = 6: (5/16) * 94 = 29.375
Games = 7: (5/16) * 94 = 29.375
2. Calculate the Chi-Square Test Statistic
The chi-square test statistic is calculated as:
χ² = Σ [(Observed - Expected)² / Expected]
Games = 4: (16 - 11.75)² / 11.75 = (4.25)² / 11.75 = 18.0625 / 11.75 ≈ 1.537
Games = 5: (21 - 23.5)² / 23.5 = (-2.5)² / 23.5 = 6.25 / 23.5 ≈ 0.266
Games = 6: (20 - 29.375)² / 29.375 = (-9.375)² / 29.375 = 87.890625 / 29.375 ≈ 2.992
Games = 7: (37 - 29.375)² / 29.375 = (7.625)² / 29.375 = 58.140625 / 29.375 ≈ 1.980
χ² = 1.537 + 0.266 + 2.992 + 1.980 ≈ 6.775
3. Determine Degrees of Freedom
Degrees of freedom (df) = number of categories - 1 = 4 - 1 = 3
4. Determine the Critical Value
Significance level (α) = 0.05
Degrees of freedom (df) = 3
Using a chi-square distribution table or calculator, the critical value for α = 0.05 and df = 3 is approximately 7.815.
5. Make a Decision
Calculated χ² = 6.775
Critical value = 7.815
Since 6.775 < 7.815, we fail to reject the null hypothesis.
6. Conclusion
There is not sufficient evidence at the 0.05 significance level to conclude that the actual numbers of games do not fit the distribution indicated by the expected proportions.
Chi-Square Value:
χ² ≈ 6.775
Rounded to three decimal places: 6.775