To analyze whether the age distribution of students matches the expected distribution, we'll need to conduct a chi-square goodness-of-fit test. Here are the key steps and formulas used to calculate each part of this question:

Step-by-Step Solution

1. Degrees of Freedom

The degrees of freedom (df) for a chi-square test are given by: $\text{df} = k - 1$ where $k$ is the number of age groups.

In this problem, there are 4 age groups (1: "<19", 2: "19-21", 3: "22-24", 4: ">24"), so: $\text{df} = 4 - 1 = 3$ Thus, the degrees of freedom is 3.

2. Chi-square Test Statistic Calculation

To calculate the chi-square test statistic, we use: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ where $O_i$ is the observed frequency for each age group, and $E_i$ is the expected frequency based on the percentages provided.

If you have the observed frequencies for each group, I can compute the chi-square test statistic for you.

3. p-value

The p-value is determined by the chi-square test statistic and the degrees of freedom. Once we have the test statistic, we can refer to a chi-square distribution table or use statistical software to find the p-value.

Could you provide the observed frequencies of students in each age group? This will allow us to calculate the test statistic and p-value accurately.

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Additional Questions and Exploration

1. How would changing the sample size impact the test statistic and p-value?
2. What alternative statistical tests could we use if the sample size were small?
3. What does it mean if our p-value is lower than the significance level?
4. How can we visually represent the observed vs. expected distributions?
5. How would the degrees of freedom change if an additional age group was added?

Tip: For a chi-square test, ensure observed frequencies are large enough (typically at least 5 per category) to meet test assumptions.