There are two required SAT sections: Math and Evidence-Based Reading and Writing. The SAT total scores of Puerto Rico high school students in the class of 2019 who took the SAT during high school have a mean of 944 and a standard deviation of 180. (Source: "2019 Puerto Rico SAT Suite of Assessments Annual Report," College Board).

Suppose that these SAT total scores follow an approximately normal distribution.

a) (6 pts) Find the proportion of Puerto Rico high school students in the class of 2019 who had SAT total scores above 1400.

b) (6 pts) Find the 90th percentile of the SAT total scores of Puerto Rico high school students in the class of 2019.

c) (6 pts) For a random sample of ten Puerto Rico high school students in the class of 2019, find the probability that exactly one of them had an SAT total score above 1400.

c) The probability that exactly one of the ten random Puerto Rico high school students in the class of 2019 had a SAT total score above 1400 is approximately 1.8%.

Explanation:

Question a:

To find the proportion of Puerto Rico high school students in the class of 2019 who had SAT total scores above 1400, we need to calculate the z-score for 1400 and find the area under the normal curve to the right of that z-score. Step 1: Calculate the z-score using the formula: z = (x - mean) / standard deviation where x is the value we want to find the proportion for. Substituting the given values, we get: z = (1400 - 944) / 180 Calculating this, we find that the z-score is approximately 2.89. Step 2: Find the area under the normal curve to the right of the z-score using a standard normal distribution table or a calculator. The area to the right of 2.89 is approximately 0.0019. Therefore, the proportion of Puerto Rico high school students in the class of 2019 who had SAT total scores above 1400 is approximately 0.0019, or 0.19%.

Question b:

To find the 90th percentile of the SAT total scores of Puerto Rico high school students in the class of 2019, we need to find the z-score that corresponds to the 90th percentile and then convert it back to the original score. Step 1: Find the z-score that corresponds to the 90th percentile using a standard normal distribution table or a calculator. The z-score for the 90th percentile is approximately 1.28. Step 2: Convert the z-score back to the original score using the formula: x = (z * standard deviation) + mean Substituting the given values, we get: x = (1.28 * 180) + 944 Calculating this, we find that the 90th percentile of the SAT total scores is approximately 1159.

Question c:

To find the probability that exactly one of the ten random Puerto Rico high school students in the class of 2019 had a SAT total score above 1400, we need to use the binomial probability formula. Step 1: Calculate the probability of one student having a SAT total score above 1400. We already found in question a that this probability is approximately 0.0019. Step 2: Calculate the probability of exactly one success in ten trials using the binomial probability formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k) where n is the number of trials, k is the number of successes, and p is the probability of success. Substituting the given values, we get: P(X = 1) = (10 choose 1) * (0.0019)^1 * (1 - 0.0019)^(10 - 1) Calculating this, we find that the probability of exactly one student having a SAT total score above 1400 is approximately 0.018. Therefore, the probability that exactly one of the ten random Puerto Rico high school students in the class of 2019 had a SAT total score above 1400 is approximately 0.018, or 1.8%.

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