Here's an enhanced and challenging version of the problem:

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A professor at a university in México claims that Group A performs better than both Group B and Group C in last year’s exam scores. To test this claim, an analysis of exam scores for all three groups is conducted with the following data:

1. Group A:

   • Mean score: $\mu_A = 7.5$
   • Standard deviation: $\sigma_A = 0.7$
   • Number of students: $n_A = 200$

2. Group B:

   • Mean score: $\mu_B = 6.0$
   • Standard deviation: $\sigma_B = 0.8$
   • Number of students: $n_B = 150$

3. Group C:

   • Mean score: $\mu_C = 7.2$
   • Standard deviation: $\sigma_C = 0.6$
   • Number of students: $n_C = 250$

Assume the exam scores are normally distributed, and there is no bias in the data. Using a significance level of $\alpha = 0.05$:

1. Perform pairwise two-sample $t$-tests:

   • Test if Group A performs better than Group B.
   • Test if Group A performs better than Group C.
   • Test if Group B performs better than Group C.

2. Use a one-way ANOVA test to check if there is a statistically significant difference among the mean scores of the three groups.

3. For the $t$-tests, calculate the $T$-values and $p$-values for each pair. State if each pairwise comparison is statistically significant.

4. For the ANOVA test:

   • Compute the F-statistic.
   • State whether there is sufficient evidence to conclude that at least one group differs significantly from the others.

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Do you want a solution or hints for solving this?

Here are 5 additional questions you could explore:

1. How would the results change if the sample sizes for each group were equal?
2. What effect would increasing the standard deviation for one group have on the $t$-test results?
3. What is the impact of increasing the significance level $\alpha$ to 0.1?
4. How would using a non-parametric test like the Mann-Whitney U test affect the analysis?
5. What assumptions must be met for the $t$-test and ANOVA to yield valid results?

Tip: Always verify the normality of data before using parametric tests like $t$-tests and ANOVA!