**Hypothesis Testing: Proportions**

To determine if there is a significant difference between the proportions of smokers in the college and the university, we can conduct a hypothesis test. The null hypothesis (H0) assumes that there is no significant difference, while the alternative hypothesis (Ha) assumes that there is a significant difference.

**H0:** The proportion of smokers in the college is equal to the proportion of smokers in the university.
**Ha:** The proportion of smokers in the college is not equal to the proportion of smokers in the university.

We can calculate the proportions of smokers in the college and the university and compare them using a hypothesis test.

**Calculating Proportions:**

In the college:
Number of smokers = 2000
Total number of students = 1000
Proportion of smokers in the college = Number of smokers / Total number of students = 2000 / 1000 = 2

In the university:
Number of smokers = 1000
Total number of students = 20000
Proportion of smokers in the university = Number of smokers / Total number of students = 1000 / 20000 = 0.05

**Conducting the Hypothesis Test:**

We can use a two-sample z-test to compare the proportions of smokers in the college and the university. The formula for the z-test statistic is:

z = (p1 - p2) / √(p * (1 - p) * (1/n1 + 1/n2))

Where:
p1 = Proportion of smokers in the college
p2 = Proportion of smokers in the university
p = (n1 * p1 + n2 * p2) / (n1 + n2)
n1 = Total number of students in the college
n2 = Total number of students in the university

**Calculating the z-test statistic:**

Using the given values:
p1 = 2
p2 = 0.05
n1 = 1000
n2 = 20000

p = (1000 * 2 + 20000 * 0.05) / (1000 + 20000) = 0.0615

z = (2 - 0.05) / √(0.0615 * (1 - 0.0615) * (1/1000 + 1/20000)) ≈ 24.98

**Interpreting the Results:**

To determine the significance of the test, we compare the calculated z-test statistic with the critical value at a chosen significance level. Let's assume a significance level of 0.05 (5%).

For a two-tailed test at a 5% significance level, the critical z-value is approximately ±1.96.

Since the calculated z-value (24.98) is much larger than the critical z-value (1.96), we can reject the null hypothesis. This means that there is a significant difference between the proportions of smokers in the college and the university.

**Conclusion:**

Based on the results of the hypothesis test, we can conclude that there is a significant difference between the proportions of smokers in the college and the university. The college has a higher proportion of smokers compared to the university.