Final answer:
To test if the loaded die behaves differently than a fair die, a goodness-of-fit test can be used. By calculating the expected frequencies assuming the die is fair and comparing them with the observed frequencies, the chi-square test statistic can be determined. The test statistic value is 5.350.
Step-by-step explanation:
To test whether the loaded die behaves differently than a fair die, we can use a goodness-of-fit test. The null hypothesis is that the outcomes of the loaded die are equally likely, while the alternative hypothesis is that they are not equally likely. We can calculate the expected frequencies for each outcome assuming the die is fair. Then, we can use the chi-square test statistic to determine if the observed frequencies significantly differ from the expected frequencies at a significance level of 0.025.
First, calculate the expected frequencies:
- Sum the observed frequencies: 29 + 31 + 46 + 41 + 26 + 27 = 200
- Calculate the expected frequency for each outcome: 200/6 = 33.33 (rounded to two decimal places)
- Calculate the chi-square test statistic:
Chi-square test statistic formula: Σ [(observed frequency - expected frequency)^2 / expected frequency]
Using the observed frequencies 29, 31, 46, 41, 26, 27 and the expected frequency 33.33, we can calculate:
- For outcome 1: (29 - 33.33)^2 / 33.33 = 0.5036
- For outcome 2: (31 - 33.33)^2 / 33.33 = 0.1481
- For outcome 3: (46 - 33.33)^2 / 33.33 = 1.7105
- For outcome 4: (41 - 33.33)^2 / 33.33 = 1.6949
- For outcome 5: (26 - 33.33)^2 / 33.33 = 0.6072
- For outcome 6: (27 - 33.33)^2 / 33.33 = 0.6856
- Sum all the values: 0.5036 + 0.1481 + 1.7105 + 1.6949 + 0.6072 + 0.6856 = 5.3499
The test statistic is the sum of these values, rounded to three decimals: 5.350.