To determine the probability of the treatment group's mean being greater than the control group's mean by 8 points or more, we will analyze the provided frequency table.
The table summarizes the results of 1,000 re-randomizations of the data, with differences of means rounded to the nearest 2 points. Here's a step-by-step solution:
1. Identify relevant frequencies: We need to consider the frequencies where the mean difference is 8 points or more. From the table provided, the relevant differences are 8, 10, and 12.
- For a difference of 8 points, the frequency is 114.
- For a difference of 10 points, the frequency is 57.
- For a difference of 12 points, the frequency is 26.
2. Sum these frequencies: Add the frequencies where the mean difference is 8 points or more.
[tex]\[
114 + 57 + 26 = 197
\][/tex]
3. Total number of re-randomizations: There are 1,000 re-randomizations in total.
4. Calculate the probability: The probability is the ratio of the sum of the relevant frequencies to the total number of re-randomizations.
[tex]\[
\frac{197}{1000} = 0.197
\][/tex]
Thus, the probability that the treatment group's mean is greater than the control group's mean by 8 points or more is [tex]\(0.197\)[/tex] or [tex]\(19.7\%\)[/tex]. This result is not less than the significance level of [tex]\(5\%\)[/tex], so the difference observed could be due to random chance and is not considered statistically significant.
