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:
[tex]\hat{p} = p = 0.44[/tex]
[tex]\sigma = \sqrt{ \frac{p(1-p)}{n} } \\ = \sqrt{ \frac{0.44(1-0.44)}{1400} } \\ = 0.013[/tex]
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%
