In May 2018 , The College Board reported the following mean...

In May 2018 , The College Board reported the following mean scores for two parts of the Scholastic Aptitude Test (SAT): Evidence-Based Reading and Writing $$ 533$$ Mathematics $$527$$ Assume that the population standard deviation on each part of the test is $\sigma=100$. a. What is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 533 on the Evidence-Based Reading and Writing part of the test? b. What is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 527 on the Mathematics part of the test? c. Comment on the differences between the values computed in parts (a) and (b).

In May 2018 , The College Board reported the following mean scores for two parts of the Scholastic Aptitude Test (SAT):
Evidence-Based Reading and Writing
$$ 533$$
Mathematics
$$527$$
Assume that the population standard deviation on each part of the test is $\sigma=100$.
a. What is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 533 on the Evidence-Based Reading and Writing part of the test?
b. What is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 527 on the Mathematics part of the test?
c. Comment on the differences between the values computed in parts (a) and (b).

Key Concepts

Probability Calculation using the Normal Distribution

Calculating probabilities using the normal distribution involves converting an observed value into a Z-score and then using the standard normal distribution table or software to determine the probability that a value falls within a specified range. This method is applied when dealing with sample means, especially under the assumption of the Central Limit Theorem.

Standard Error

Standard error is a measure of the dispersion of the sample mean around the population mean in the sampling distribution. It is calculated as the population standard deviation divided by the square root of the sample size. Understanding the standard error is crucial for estimating the reliability of the sample mean as an estimate of the population mean.

Z-score

A Z-score represents the number of standard deviations a data point or sample mean is from the population mean. In the context of sampling distributions, it is used to standardize the observation so that probabilities can be computed using the standard normal distribution. The Z-score is calculated by subtracting the population mean from the sample mean and then dividing by the standard error.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the population distribution, the sampling distribution of the sample mean will approximate a normal distribution as the sample size increases, typically when the sample size is 30 or greater. This concept is essential for calculating probabilities related to sample means when the population standard deviation is known.

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution of all possible sample means from a population. Its standard deviation, known as the standard error, decreases as the sample size increases. This distribution is central to statistical inference and is used to determine the probability that a sample mean falls within a certain range around the population mean.

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