A random sample of two hundred fifteen adults is selected from a certain population of
registered voters. Assume the population is 20% democrats. Use a normal approximation
to estimate
(a) the probability the sample contains exactly 48 democrats, and
(b) the probability the sample contains more than 40 democrats.
To estimate the probabilities in this situation, we can use the normal approximation to the binomial distribution. Let's break down the steps to find the desired probabilities:
(a) To estimate the probability that the sample contains exactly 48 Democrats, we need to calculate the z-score and then find the corresponding probability.
1. Find the mean (μ) and standard deviation (σ) of the binomial distribution:
μ = n * p
σ = √(n * p * q)
Here, n is the sample size (215), p is the proportion of Democrats in the population (0.20), and q is the complement of p (1 - 0.20 = 0.80).
μ = 215 * 0.20 = 43
σ = √(215 * 0.20 * 0.80) ≈ 4.71
2. Calculate the z-score using the formula:
z = (x - μ) / σ
Here, x is the number of Democrats in the sample (48).
z = (48 - 43) / 4.71 ≈ 1.06
3. Find the probability corresponding to the z-score using a standard normal distribution table or a calculator. The probability is the area under the curve to the left of the z-score.
P(Z < 1.06) ≈ 0.8554
So, the estimate of the probability that the sample contains exactly 48 Democrats is approximately 0.8554 or 85.54%.
(b) To estimate the probability that the sample contains more than 40 Democrats, we need to calculate the z-score and find the corresponding probability.
1. Calculate the mean (μ) and standard deviation (σ) as we did in step 1.
μ = 43
σ ≈ 4.71
2. Calculate the z-score using the formula:
z = (x - μ) / σ
Here, x is the number of Democrats in the sample (40).
z = (40 - 43) / 4.71 ≈ -0.64
3. Find the probability corresponding to the z-score. Since we want the probability of having more than 40 Democrats, we need to calculate the complement of the probability to the left of the z-score:
P(Z > -0.64) = 1 - P(Z < -0.64)
Using a standard normal distribution table or a calculator, we find that P(Z < -0.64) ≈ 0.2611.
Therefore, P(Z > -0.64) ≈ 1 - 0.2611 ≈ 0.7389 or 73.89%.
So, the estimate of the probability that the sample contains more than 40 Democrats is approximately 0.7389 or 73.89%.