Question

Suppose that 44% of all Americans approve of the job the president is doing. The most recent Gallup poll consisted of a random sample of 1400 American adults.
A) What is the mean of the sampling distribution?
B) What is the standard deviation of the sampling distribution?
C) Describe the normal approximation for this sampling distribution.
D) What is the probability that the Gallup poll will come up with a proportion within 3 percentage points of the true 44%?

Answer 1

This is a typical binomial distribution since we are dealing with only two possible outcomes. For this distribution we have:
n = population = 1400
p = probability of positive answer = 0.44 

A) The mean of a binomial distribution is
μ = np
  = 1400 × 0.44
  = 616

Hence, μ = 616

B) The standard deviation of a binomial distribution is
σ = √[np(1 - p)]
   = √[1400×0.44×(1-0.44)]
   = √(1400×0.44×0.56)
   = 18.6

Hence, σ = 18.6

C) A binomial distribution B(n, p) can be approximated by a normal distribution N(μ, σ) only when:
np ≥ 10 (we have 616 > 10)
np(1 - p) ≥ 10 (we have 344.96 > 10)

Hence, the normal approximation is N(616, 18.6)

D) We need to calculate the probability of the sample population to be within 3% of the true mean.

In order to do so, as a first thing, we need to calculate the sample standard deviation, which will be given by the formula:
$\sigma = \sqrt{ \frac{p(1-p)}{n} }$
= √[0.44 × (1 - 0.44) ÷ 1400]
= √0.000176
= 0.013

Now, we need to calculate the z-score associated with the values we want: since we want 3% from the mean, it means between 41% and 47%.
The z-score can be calculated by the formula:
z = (Y - p) / σ
z₁ = (0.41 - 0.44) / 0.013 = -2.31
z₂ = (0.47 - 0.44) / 0.013 = 2.31

Therefore, we can write
P(0.41 ≤ Y ≤ 0.47) = P(-2.31 ≤ z ≤ 2.31)
                              = P(z ≤ 2.31) - P(z ≤ -2.31)
                              = P(z ≤ 2.31) - [1 - P(z ≤ 2.31)]
                              = 2×P(z ≤ 2.31) - 1

From a standard normal distribution table, we find:
P(z ≤ 2.31) = 0.9896

Therefore:
2×P(z ≤ 2.31) - 1 = 2 × 0.9896 - 1
                            = 0.9792

Hence, the probability to find a proportion within 3% of the real mean is 97.92%