To analyze this problem, we can define the hypotheses as follows:

Null Hypothesis ($H_0$):

The likelihood of being promoted is independent of the age group. In other words, age does not influence the promotion rates.

$H_0: \text{Promotion rate is the same across all age groups.}$

Alternative Hypothesis ($H_a$):

The likelihood of being promoted is dependent on the age group. In other words, age influences the promotion rates.

$H_a: \text{Promotion rate differs across age groups.}$

This is a chi-square test of independence where we compare observed frequencies with expected frequencies to determine whether there is a statistically significant relationship between age group and promotion rates.

Let me know if you'd like further details or assistance with the calculations! Here are related questions:

1. How do we calculate the expected frequencies for a chi-square test?
2. How do we compute the chi-square statistic?
3. What is the critical value for $\alpha = 0.01$ and how is it used?
4. How is the p-value determined for this test?
5. How does the chi-square test result influence the conclusion about promotion rates and age?

Tip: Always check if the expected frequencies in a chi-square test are all greater than 5 for valid results.