The SAT is an entrance exam used by most colleges and universities to make admissions decisions. 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, https://reports.collegeboard.org/pdf/2019-puerto-rico-sat-suite-assessments-annual-report.pdf).

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

1. Find the proportion of Puerto Rico high school students in the class of 2019 who had SAT total scores above 1400.

2. Find the 90th percentile of the SAT total scores of Puerto Rico high school students in the class of 2019.

3. 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.

The proportion of Puerto Rico high school students in the class of 2019 who had SAT total scores above 1400 is approximately 0.14%. The 90th percentile of the SAT total scores of Puerto Rico high school students in the class of 2019 is approximately 1160.76. 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 1.36%.

Explanation:

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 and use the standard normal distribution table or a calculator.

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.

For x = 1400:

z = (1400 - 944) / 180 = 2.978

Step 2: Use the standard normal distribution table or a calculator to find the proportion corresponding to the z-score.

The proportion of Puerto Rico high school students in the class of 2019 who had SAT total scores above 1400 is approximately 0.0014 or 0.14%.

To find the 90th percentile of the SAT total scores of Puerto Rico high school students in the class of 2019, we can use the z-score formula:

z = invNorm(p)

where p is the desired percentile as a decimal.

For p = 0.9:

z = invNorm(0.9) ≈ 1.282

Step 3: Calculate the score using the formula:

x = mean + (z * standard deviation)

x = 944 + (1.282 * 180) ≈ 1160.76

The 90th percentile of the SAT total scores of Puerto Rico high school students in the class of 2019 is approximately 1160.76.

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 can use the binomial distribution 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.

In this case, n = 10, k = 1, and p = 0.0014 (probability of a student having a score above 1400).

Using the formula:

P(X = 1) = (10 choose 1) * (0.0014)^1 * (1 - 0.0014)^(10 - 1)

P(X = 1) ≈ 0.0136 or 1.36%

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