Question 1174088
Alright, let's work through this problem using hypothetical CGPA data for 5 students.

**1. Hypothetical CGPA Data**

Let's assume the CGPA of 5 students are:

* Student 1: 3.2
* Student 2: 3.5
* Student 3: 3.8
* Student 4: 3.0
* Student 5: 3.7

**2. Population Calculations**

* Population (N): {3.2, 3.5, 3.8, 3.0, 3.7}
* Population Mean (μ): (3.2 + 3.5 + 3.8 + 3.0 + 3.7) / 5 = 17.2 / 5 = 3.44

**3. Sampling Distribution (n=2)**

* We need to find all possible samples of size 2 (n=2) from this population.
* The samples are:
    * (3.2, 3.5)
    * (3.2, 3.8)
    * (3.2, 3.0)
    * (3.2, 3.7)
    * (3.5, 3.8)
    * (3.5, 3.0)
    * (3.5, 3.7)
    * (3.8, 3.0)
    * (3.8, 3.7)
    * (3.0, 3.7)

**4. Calculate Sample Means**

* Now, calculate the mean of each sample:
    * (3.2 + 3.5) / 2 = 3.35
    * (3.2 + 3.8) / 2 = 3.5
    * (3.2 + 3.0) / 2 = 3.1
    * (3.2 + 3.7) / 2 = 3.45
    * (3.5 + 3.8) / 2 = 3.65
    * (3.5 + 3.0) / 2 = 3.25
    * (3.5 + 3.7) / 2 = 3.6
    * (3.8 + 3.0) / 2 = 3.4
    * (3.8 + 3.7) / 2 = 3.75
    * (3.0 + 3.7) / 2 = 3.35

**5. Sampling Distribution of the Means**

* The sampling distribution of the means is:
    * {3.35, 3.5, 3.1, 3.45, 3.65, 3.25, 3.6, 3.4, 3.75, 3.35}

**(a) Develop the sampling distribution of N and n**

* N = {3.2, 3.5, 3.8, 3.0, 3.7}
* Sampling Distribution of n = {3.35, 3.5, 3.1, 3.45, 3.65, 3.25, 3.6, 3.4, 3.75, 3.35}

**(b) Prove that mean of the means of samples is equal to mean of population**

* Mean of the sample means (μ_x̄):
    * (3.35 + 3.5 + 3.1 + 3.45 + 3.65 + 3.25 + 3.6 + 3.4 + 3.75 + 3.35) / 10 = 34.4 / 10 = 3.44
* Population Mean (μ): 3.44

* μ_x̄ = μ = 3.44

**Therefore, the mean of the sample means (3.44) is equal to the population mean (3.44).**