Question

Use a normal approximation to find the probability of the indicated number of voters. In this​ case, assume that 115 eligible voters aged​ 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged​ 18-24, 22% of them voted.

Probability that exactly 30 voted

The probability that exactly 30 of 115 eligible voters voted is...?

Answer 1

Mean = np = 115 * 0.22 = 25.3

Standard deviation = Sqrt (np(1-p))

= sqrt( 115 * 0.22 * 0.78)

= 4.4423

Using normal approximation,

P( X < x) = P( Z < x - mean / SD)

With continuity correction,

P( X = x) = P(x-0.5 < X < x+0.5)

P( X = 30) = P( 29.5 < X < 30.5)

= P( X < 30.5 ) - P( X < 29.5)

= P (Z < 30.5 - 25.3 / 4.4423) - P (Z < 29.5 - 25.3 / 4.4423)

= P( Z < 1.1706) - P (Z < 0.9455)

= 0.8791 - 0.8278

= 0.0513

Answer 2

To find the probability that exactly 30 out of 115 eligible voters aged 18-24 voted, we can use the normal approximation to the binomial distribution. When the sample size is large (n) and the probability of success (p) is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution with mean (μ) equal to n * p and standard deviation (σ) equal to √(n * p * (1 - p)).

Given that the previous study showed that 22% of eligible voters aged 18-24 voted, we have: n = 115 (sample size) p = 0.22 (probability of success - proportion who voted)

Now, we can calculate the mean and standard deviation for the normal distribution:

Mean (μ) = n * p = 115 * 0.22 = 25.3 (approximately) Standard Deviation (σ) = √(n * p * (1 - p)) = √(115 * 0.22 * 0.78) = √20.162 = 4.49 (approximately)

To find the probability of exactly 30 voters, we need to calculate the probability of getting exactly 30 successes (voters) in a normally distributed variable with mean 25.3 and standard deviation 4.49.

Using the standard normal distribution (Z-score): Z = (X - μ) / σ

where X is the number of voters we are interested in (30 in this case).

Z = (30 - 25.3) / 4.49 ≈ 1.047

Now, we need to find the probability corresponding to the Z-score using a standard normal distribution table or a calculator. The probability of exactly 30 voters can be approximated as follows:

P(X = 30) ≈ P(Z = 1.047)

Using a standard normal distribution table or calculator, the probability P(Z = 1.047) is approximately 0.8525.

Therefore, the probability that exactly 30 out of 115 eligible voters aged 18-24 voted is approximately 0.8525 or 85.25%.