Respuesta :
To calculate the probability of the randomly chosen female scoring at least 2 points higher than the randomly selected male, we need to compare the two distributions and find the area under the curve that represents the desired outcome.
Step 1: Find the difference in means between the two distributions:
The average ACT score for females is 19.6, and for males is 19.2. The difference is 19.6 - 19.2 = 0.4.
Step 2: Find the difference in standard deviations between the two distributions:
The standard deviation for females is 2.3, and for males is 4.1. The difference is 4.1 - 2.3 = 1.8.
Step 3: Standardize the difference in means using the pooled standard deviation:
We use the formula z = (X1 - X2) / sqrt((s1^2 / n1) + (s2^2 / n2)), where X1 and X2 are the means, s1 and s2 are the standard deviations, and n1 and n2 are the sample sizes. Since we're comparing two independent samples, the sample size is 1 for each.
z = (0.4) / sqrt((2.3^2 / 1) + (4.1^2 / 1)).
Step 4: Calculate the probability using the standard normal distribution table or calculator:
We want to find the probability that the difference in scores is at least 2 points higher, which corresponds to the probability of z being greater than or equal to 2.
P(z ≥ 2).
Step 5: Look up the probability in the standard normal distribution table or use a calculator:
The probability P(z ≥ 2) is approximately 0.0228.
So, the probability of the randomly chosen female scoring at least 2 points higher than the randomly selected male is approximately 0.023 (rounded to three decimal places).
