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 1,400 American adults. a. What is the mean of the sampling distribution? b. What is the standard deviation of the sampling distribution (don't forget to justify the use of the formula)? c. Describe the normal approximation for this sampling distribution (don't forget to justify this) You can simply write as N (mean, standard deviation) d. What is the probability that the Gallup poll will come up with a proportion within three percentage points of the true 44%

Answer 1

Since the possible outcomes of the poll are only two, approve or not, we are dealing with a binomial distribution, where:
n = 1400
p = 44% = 0.44

A) The mean of the sampling distribution is μ = 616
The mean of a binomial distribution can be calculated by the formula:
μ = n · p
   = 1400 · 0.44
   = 616

B) The standard deviation of the sampling distribution is σ = 18.6
The standard deviation of a binomial distribution can be calculated by the formula:
σ = √[n · p · (1 - p)]
   = √[1400 · 0.44 · (1 - 0.44)]
   = √344.96
   = 18.6

C) The normal approximation is N(616, 18.6)

In order to normally approximate a binomial distribution, two conditions must be satisfied:
n · p ≥ 10
n · p · (1 - p) ≥ 10

In our case,
n · p = 616 > 10
n · p · (1 - p) = 344.96 > 10

Since the two conditions are satisfied, the binomial distribution B(n, p) can be approximated with a normal distribution N(μ, σ).
In our case:
B(1400, 0.44) ≈ N(616, 18.6)

D) The probability that the Gallup poll will come up with a proportion within three percentage points of the true 44% is P = 0.98 or 98%

We need to apply the continuity correction to our normal approximation:
P(Y = 0.44) = P(0.44 - 0.03 ≤ Y ≤ 0.44 + 0.03)
                = P(0.41 ≤ Y ≤ 0.47)

In order to calculate this probability, we need to calculate mean and standard deviation of the sample proportion:
$\hat{p} = p = 0.44$

$\sigma = \sqrt{ \frac{p(1-p)}{n} } \\ = \sqrt{ \frac{0.44(1-0.44)}{1400} } \\ = 0.013$

Now, we need to calculate the z-score for each Y-value:
z = (Y - p) / σ

z(Y = 0.41) = (0.41 - 0.44) / 0.013 = -2.31
z(Y = 0.47) = (0.47 - 0.44) / 0.013 = 2.31

Therefore, we can say that
P(0.41 ≤ Y ≤ 0.47) = P(-2.31 ≤ z ≤ 2.31)
                              = P(z ≤ 2.31) - P(z ≤ -2.31)

Looking at a normal standard distribution table, we find
P(z ≤ -2.31) = 0.0104
P(z ≤ 2.31) = 0.9896

Therefore:
P(0.41 ≤ Y ≤ 0.47) = 0.9896 - 0.0104
                              = 0.9792

Hence, the probability of the poll coming up with a proportion within three percent of the true mean is 0.98 which means 98%