Statistics

SAT scores are normally distributed with a mean of 1000 and a standard deviation of 150. Use this scenario to answer the following questions. What is the probability that a randomly selected student scored higher than 1020? Round to the nearest ten-thousandth (four decimal places). Question Blank 1 of 4 type your answer... What is the probability that a group of 25 randomly selected students have an average SAT score higher than 1020? Round to the nearest ten-thousandth (four decimal places). Question Blank 2 of 4 type your answer... What is the probability that a group of 25 randomly selected students have an average SAT score lower than 950? Round to the nearest ten-thousandth (four decimal places). Question Blank 3 of 4 type your answer... What is the probability that a group of 25 randomly selected students have an average SAT score between 950 and 1020? Round to the nearest ten-thousandth (four decimal places). Question Blank 4 of 4 type your answer... Question at position 3 3 Fill in the Blank 22 points Question at position 3 SAT scores are normally distributed with a mean of 1000 and a standard deviation of 150. Use this scenario to answer the following questions. Find the mean of the sampling distribution of sample means for a sample size of 200. Question Blank 1 of 2 type your answer... Find the standard deviation of the sampling distribution of sample means for a sample size of 200. Round your answer to three decimal places. Question Blank 2 of 2 type your answer...

Step-by-step solution with explanation

Final Answer

Q1: P(X > 1020) = 0.4470 | Q2: P(x̄ > 1020, n=25) = 0.2525 | Q3: P(x̄ < 950, n=25) = 0.0478 | Q4: P(950 < x̄ < 1020, n=25) = 0.6997 | Sampling distribution (n=200): mean = 1000, standard deviation ≈ 10.607

Step-by-step solution

Set up the population parameters

We are told SAT scores are normally distributed with mean 1000 and standard deviation 150. These are our starting values for every part of this problem.

Question 1: P(X > 1020) for one student

We convert 1020 to a z-score using the population mean and standard deviation. Then we find the area to the RIGHT of z = 0.1333 using the standard normal table: .

Question 2: P(x̄ > 1020) for n = 25

For a sample of 25, the standard error is 150 divided by the square root of 25, which equals 30. We get z ≈ 0.6667, and the area to the right is .

Question 3: P(x̄ < 950) for n = 25

We use the same standard error of 30. The z-score for 950 is about −1.6667. The area to the LEFT of z = −1.6667 from the table is .

Question 4: P(950 < x̄ < 1020) for n = 25

We already found the two z-scores: −1.6667 for 950 and 0.6667 for 1020. The probability between them is the area at z = 0.6667 minus the area at z = −1.6667, which equals 0.7475 − 0.0478 = 0.6997.

Question 3 Part 1: Mean of sampling distribution, n = 200

By the Central Limit Theorem, the mean of the sampling distribution of sample means always equals the population mean. So for any sample size, including 200, the mean is 1000.

Question 3 Part 2: Standard deviation of sampling distribution, n = 200

The standard deviation of the sampling distribution (called the standard error) is the population standard deviation divided by the square root of the sample size. , so .

Understanding this problem

Learning Insight

The Central Limit Theorem says that as sample size grows, the sampling distribution of the mean becomes narrower and more normal. This is WHY a group of 25 students has much less spread in their average score than a single student — averaging cancels out extreme values.

Quick Tip

Standard error = . Bigger sample → smaller standard error → z-scores get bigger → probabilities in the tails get smaller. If the sample size quadruples, the standard error cuts in half.

Common Mistake

Students often use the population standard deviation (150) instead of the standard error () when working with sample means. Always divide by first before computing any z-score for a sample average.