1. The problem involves selecting samples from a population of 200 students divided into four departments and analyzing social media usage data. 2. To select a simple random sample of 40 students: - Assign each student a unique number from 1 to 200. - Use a random number generator or draw lots to select 40 unique numbers. - The students corresponding to these numbers form the simple random sample. 3. To select a stratified sample of 40 students: - Divide the population into strata based on departments: Organizational Studies (60), Marketing (50), HRM (40), Finance (50). - Calculate the sample size for each stratum proportional to its population size: $$\text{Sample size}_i = \frac{\text{Population}_i}{200} \times 40$$ Organizational Studies: $\frac{60}{200} \times 40 = 12$ Marketing: $\frac{50}{200} \times 40 = 10$ HRM: $\frac{40}{200} \times 40 = 8$ Finance: $\frac{50}{200} \times 40 = 10$ - Randomly select the calculated number of students from each department. 4. To select a systematic sample of 40 students: - Calculate the sampling interval $k = \frac{200}{40} = 5$. - Randomly select a starting number between 1 and 5, say $r$. - Select every $k^{th}$ student starting from $r$: $r, r+5, r+10, ..., r+195$. 5. Constructing a frequency distribution table for the given 40 data points with class intervals 1 up to 2, 2 up to 3, etc.: - Classes: 1-2, 2-3, 3-4, 4-5, 5-6, 6-7, 7-8 - Count frequencies: 1-2: 7 2-3: 9 3-4: 10 4-5: 8 5-6: 4 6-7: 5 7-8: 3 6. Using the frequency table: i. Mean estimation: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ where $x_i$ is the midpoint of each class. Midpoints: 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5 Calculate: $$\bar{x} = \frac{7\times1.5 + 9\times2.5 + 10\times3.5 + 8\times4.5 + 4\times5.5 + 5\times6.5 + 3\times7.5}{40} = \frac{10.5 + 22.5 + 35 + 36 + 22 + 32.5 + 22.5}{40} = \frac{181}{40} = 4.525$$ ii. Median estimation: - Total frequency $N=40$, median class is where cumulative frequency reaches $\frac{N}{2} = 20$. - Cumulative frequencies: 7, 16, 26, ... median class is 3-4 hours. - Median formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ where $L=3$, $F=16$, $f_m=10$, $h=1$ $$= 3 + \frac{20-16}{10} = 3 + 0.4 = 3.4$$ iii. Mode estimation: - Mode class is the class with highest frequency: 3-4 hours (frequency 10). - Mode formula: $$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ where $L=3$, $f_1=10$, $f_0=9$ (previous class), $f_2=8$ (next class), $h=1$ $$= 3 + \frac{10-9}{2\times10 - 9 - 8} = 3 + \frac{1}{3} = 3.33$$ iv. Standard deviation estimation: - Calculate variance: $$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N}$$ Compute each term: $(1.5 - 4.525)^2 \times 7 = 63.07$ $(2.5 - 4.525)^2 \times 9 = 36.72$ $(3.5 - 4.525)^2 \times 10 = 10.50$ $(4.5 - 4.525)^2 \times 8 = 0.005$ $(5.5 - 4.525)^2 \times 4 = 3.81$ $(6.5 - 4.525)^2 \times 5 = 19.75$ $(7.5 - 4.525)^2 \times 3 = 26.85$ Sum = 160.7 $$\sigma = \sqrt{\frac{160.7}{40}} = \sqrt{4.0175} = 2.004$$ 7. Variable categorization: i. Daily social media usage (in hours): Quantitative, Ratio scale ii. Preferred social media platforms: Qualitative, Nominal scale iii. Purpose of using social media: Qualitative, Nominal scale iv. Monthly mobile data usage (in GB): Quantitative, Ratio scale v. Level of dependency on social media: Qualitative, Ordinal scale