Answer:
a. n=2401 students
b. n=2377 students
c. B. No, using the additional survey information from part (b) only slightly reduces the sample size.
Step-by-step explanation:
a. The sample size for a sample proportion about the mean is calculated using the formula:
[tex]n=(\frac{z_{\alpha/2}}{E})^2p(1-p)[/tex]
Where p is the proportion and E is the margin of error.
-If nothing is known about the proportion to be studied, we use p=0.5:
[tex]n=(\frac{z_{\alpha/2}}{E})^2p(1-p)\\\\=(1.96/0.02)^20.5(1-0.5)\\\\=2401[/tex]
Hence, the required sample size is 2401
b. If the proportion to be estimated is given, we substitute it for p in the formula.
-Given p=0.55, the required sample size can be calculated as:
[tex]n=(\frac{z_{\alpha/2}}{E})^2p(1-p)\\\\=(1.96/0.02)^20.55(1-0.55)\\\\=2376.99\approx2377[/tex]
Hence, the required sample size for a given proportion of 55% is approximately 2377 students
c. The added information in b had a reducing effect on the sample size:
[tex]\bigtriangleup n=n_a-n_b\\\\=2401-2377\\\\=24[/tex]
-The sample size slightly reduces by 24 students.
Hence, No, using the additional survey information from part (b) only slightly reduces the sample size.
