Question

some nations require their students to pass an exam before earning their primary school degrees or diplomas. a certain nation gives students an exam whose scores are normally distributed with a mean of \[41\] points and a standard deviation of \[9\] points. suppose we select \[2\] of these testers at random, and define the random variable \[d\] as the difference between their scores. we can assume that their scores are independent. find the probability that their scores differ by more than \[15\] points. you may round your answer to two decimal places.

Answer 1

To find the probability that the scores of the two testers differ by more than 15 points, we need to calculate the probability that the absolute difference between their scores is greater than 15. By finding the z-score for 15 and using a standard normal distribution table, we can determine the probability of z being greater than 0.8333, which is approximately 0.2023. By doubling this probability, we can find that the probability that the scores of the two testers differ by more than 15 points is approximately 0.4046.

Step-by-step explanation:

To find the probability that the scores of the two testers differ by more than 15 points, we need to calculate the probability that the absolute difference between their scores is greater than 15.

1. Let X be the random variable representing the difference between the scores of the two testers.
2. We know that X follows a normal distribution with a mean of 0 (since the mean difference is 0) and a standard deviation of 9+9=18 (the sum of the standard deviations of each tester's score).
3. We want to find P(|X| > 15), which is the probability that the absolute difference |X| is greater than 15.
4. First, we can find the z-score for 15 using the formula z = (x - mean) / standard deviation. Here, x = 15, mean = 0, and standard deviation = 18.
5. Calculating z: z = (15 - 0) / 18 = 15/18 = 0.8333 (rounded to 4 decimal places).
6. Next, we can find the probability of z being greater than 0.8333 using a standard normal distribution table or a statistical calculator. The probability is approximately 0.2023 (rounded to 4 decimal places).
7. Since we want the probability of |X| > 15, we need to double the probability of z being greater than 0.8333, as we're considering both sides of the distribution (greater than 15 and less than -15).
8. Therefore, the probability that the scores of the two testers differ by more than 15 points is approximately 2 * 0.2023 = 0.4046 (rounded to 4 decimal places).