In a random sample of 400 university students, 300 failed the examination. In another sample of 500 affiliated college students, 300 failed the same examination. Find the standard error (S.E.) of the difference between the proportion of failures:

i) In university departments.

ii) Between university departments and affiliated colleges.

To calculate the standard error (S.E.) of the difference between the proportion of failures in the two samples, we need to break the problem down step by step. 

### Part i) Standard Error of Proportion of Failures in University Departments

In the first sample of 400 university students, 300 failed the examination. From this data, we can calculate the sample proportion of failures (denoted as p₁). 

The sample proportion of failures for university students is:

\[
p_1 = \frac{\text{Number of failures in university}}{\text{Total number of university students}} = \frac{300}{400} = 0.75
\]

Next, we calculate the standard error for the proportion of failures in the university sample using the formula for the standard error of a proportion:

\[
S.E._{p_1} = \sqrt{\frac{p_1(1 - p_1)}{n_1}}
\]

Where:
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