Question

Angelita and Cynthia like to play a video game against each other that is scored by round. The goal is to score as many points as possible in each round. Here are summary statistics how many points they each score per round:

Mean Standard deviation
Angelita µA = 900 δA = 96
Cynthia µC = 800 δC = 72

Both distributions are approximately normal. Suppose we choose a round at random and calculate the difference between their scores. We can assume that their scores each round are independent.
Find the probability that Angelita's score is higher than Cynthia's. You may round your answer to two decimal places.
P ( Angelita higher) = ______

Answer 1

To find the probability that Angelita's score is higher than Cynthia's, we can compare their scores using their mean and standard deviation. Subtract the mean of Y from the mean of X to get a new random variable Z, and standardize Z using the formula Z = (X - μZ) / σZ. Finally, calculate the probability P(Z > 0) to find the probability that Angelita's score is higher than Cynthia's.

Step-by-step explanation:

To find the probability that Angelita's score is higher than Cynthia's, we need to compare their scores using their mean and standard deviation. Let X be the random variable representing Angelita's score and Y be the random variable representing Cynthia's score. We know that X follows a normal distribution with a mean of 900 and a standard deviation of 96, and Y follows a normal distribution with a mean of 800 and a standard deviation of 72. We want to find P(X > Y).

Since X and Y are independent,

we can subtract the mean of Y from the mean of X to get a new random variable Z = X - Y with a mean of 100 (900 - 800) and a standard deviation of √(96^2 + 72^2) ≈ 119.24.

We can now standardize Z using the formula Z = (X - μZ) / σZ, where μZ is the mean of Z and σZ is the standard deviation of Z. Substituting the values, we get Z = (X - 100) / 119.24.

To find P(X > Y), we need to find P(Z > 0), that is, the probability that Z is greater than zero. By looking up the standardized normal distribution table or using a calculator, we find that P(Z > 0) is approximately 0.5400.

Therefore, the probability that Angelita's score is higher than Cynthia's is approximately 0.5400.