Question

A sample consisting of 420 men and 775 women was asked various questions pertaining to international affairs. With a 95% level of confidence, find the margin of error associated with the following samples. (Round your answers to one decimal place.)

(a) the male sample ± %

(b) the female sample ± %

(c) the combined sample ± %

Answer 1

Margin of error = ME = z * sqrt {p(1-p)/n}

or Margin of error = Critical value x Standard error of the sample.

Confidence level = 95% .

Then, significance level = alpha = 1 – confidence level = 1 – 0.95 = 0.05 , Alpha/2 = 0.05/2 =0.02

assuming sample proportion is same in all groups( male & female).

lets say sample proportion = p = 0.5

since population proportion is unknown hence we will use t distribution.

a)for male sample ,

sample size = n = 420 ,degree of freedom = df = n-1 = 419

Since sample size is known hence we use t distribution

For one-tail probability of 0.025,

critical value = t0.025 = 1.9762 ( from interpolation)

so, margin of error = 1.9762 * sqrt(0.5*0.5 / 420) = 0.04821 or 4.821% = 4.8 % (rounded to one decimal )

similarly,

b)

for female sample ,

sample size = n = 775 ,degree of freedom = df = n-1 = 774

Since sample size is known hence we use t distribution

For one-tail probability of 0.025,

critical value = t0.025 = 1.984 - (0.022)/900*674 = 1.9675 ( from linear interpolation)

so, margin of error = 1.9675 * sqrt(0.5*0.5 / 775) = 0.03534 or 3.534% = 3.5% (rounded to one decimal )

c)

for combined sample ,

sample size = n = 1195 ,degree of freedom = df = n-1 = 1195

Since sample size is known hence we use t distribution

For one-tail probability of 0.025,

critical value = t0.025 = 1.96 (approximately as it is getting closer to z distribution values)

so, margin of error = 1.96 * sqrt(0.5*0.5 / 1195) = 0.02835 or 2.835% = 2.8% (rounded to one decimal )

note : you may also use z distribution for finding critical values of all three cases here as the sample is getting closer to z distribution and the error between two distributions will be negligible.