Question

Find the size of each of two samples (assume that they are of equal size) needed to estimate the difference between the proportions of boys and girls under 10 years old who are afraid of spiders. Assume the "worst case" scenario for the value of both sample proportions. We want a 98% confidence level and for the error to be smaller than 0.04.

Answer 1

Answer:

a) the sample size (n) = 156.25≅ 156

Step-by-step explanation:

Step1 :-

Given the two sample sizes are equal so $n_{1} =n_{2} = n$

Given the standard error (S.E) = 0.04

The standard error of the proportion of the given  sample size

$S.E = \sqrt{\frac{pq}{n} }$

Step 2:-

here we assume that the proportion of boys and girls are equally likely

p= 1/2 and q= 1/2

$S.E = \sqrt{\frac{p(1-p)}{n} } \leq \frac{\frac{1}{2} }{\sqrt{n} }$

$\sqrt{n} = \frac{\frac{1}{2} }{S.E}$

squaring on both sides, we get

$n = \frac{1}{(2X0.04)^{2} }$

on simplification, we get

n= 156.25 ≅ 156

sample size (n) = 156

verification:-

$S.E = \sqrt{\frac{pq}{n} }= \sqrt{\frac{1}{4X156} } =\sqrt{0.0016}$  

Standard error = 0.04