Question 1170491
Let's break down this problem step by step to determine the necessary sample size using proportional allocation.

**1. Calculate the Number of Students in Private and Public Schools**

* Total students (N) = 76,600
* Private school students (Np) = 52% of 76,600 = 0.52 * 76,600 = 39,832
* Public school students (Nu) = 48% of 76,600 = 0.48 * 76,600 = 36,768

**2. Calculate the Number of Female and Male Students in Each School Type**

* **Private Schools:**
    * Female students (Fp) = 55% of 39,832 = 0.55 * 39,832 = 21,907.6 (approximately 21,908)
    * Male students (Mp) = 45% of 39,832 = 0.45 * 39,832 = 17,924.4 (approximately 17,924)
* **Public Schools:**
    * Female students (Fu) = 44% of 36,768 = 0.44 * 36,768 = 16,177.92 (approximately 16,178)
    * Male students (Mu) = 56% of 36,768 = 0.56 * 36,768 = 20,590.08 (approximately 20,590)

**3. Determine the Z-score for the Confidence Level**

* Confidence level (CL) = 95%
* Z-score for 95% confidence level = 1.96 (from the standard normal distribution table)

**4. Calculate the Sample Size for the Entire District**

We will use the formula for sample size calculation with a proportion:

n = (Z^2 * p * (1 - p)) / e^2

Where:

* n = sample size
* Z = Z-score (1.96)
* p = level of variability (0.40)
* e = sampling error (0.05)

n = (1.96^2 * 0.40 * 0.60) / 0.05^2
n = (3.8416 * 0.24) / 0.0025
n = 0.921984 / 0.0025
n = 368.7936 (approximately 369)

**5. Adjust for Response Rate**

We need to adjust the sample size for the response rate of 70% (0.70).

Adjusted sample size = n / response rate
Adjusted sample size = 369 / 0.70
Adjusted sample size = 527.14 (approximately 528)

**6. Allocate the Sample Size Proportionally**

We need to proportionally allocate the adjusted sample size (528) among the four groups:

* **Private Female (Fp):**
    * Proportion = 21,908 / 76,600 ≈ 0.286
    * Sample size = 0.286 * 528 ≈ 151
* **Private Male (Mp):**
    * Proportion = 17,924 / 76,600 ≈ 0.234
    * Sample size = 0.234 * 528 ≈ 123
* **Public Female (Fu):**
    * Proportion = 16,178 / 76,600 ≈ 0.211
    * Sample size = 0.211 * 528 ≈ 111
* **Public Male (Mu):**
    * Proportion = 20,590 / 76,600 ≈ 0.269
    * Sample size = 0.269 * 528 ≈ 142

**Results:**

* **Private Female:** 151 students
* **Private Male:** 123 students
* **Public Female:** 111 students
* **Public Male:** 142 students

**Summary**

To conduct the investigation with a 95% confidence level, 5% sampling error, 40% variability, and a 70% response rate, you would need a sample size of approximately 528 students, allocated as follows:

* 151 female students from private schools.
* 123 male students from private schools.
* 111 female students from public schools.
* 142 male students from public schools.