The College Board reported the following mean scores for the three parts of the Scholastic Aptitude Test (SAT) (The World Almanac, 2009):

Assume that the population standard deviation on each part of the test is = 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 502 on the Critical Reading part of the test (to 4 decimals)?

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 515 on the Mathematics part of the test (to 4 decimals)?

c. What is the probability a sample of 100 test takers will provide a sample mean test score within 10 of the population mean of 494 on the writing part of the test (to 4 decimals)?

a. Z = (score-mean)/SEm

SEm = SD/√n

Solve Z for scores of 502±10.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between mean and Z scores.

b and c. Use similar process.

To solve these questions, we will use the formula for the z-score:

z = (X - μ) / (σ / √n)

Where:
X = sample mean
μ = population mean
σ = population standard deviation
n = sample size

For parts (a) and (b), we need to find the probability that the sample mean falls within 10 points of the population mean. We can calculate this by finding the area under the normal distribution curve between two z-scores.

a. Critical Reading part:
X = 502
μ = 502
σ = 100
n = 90
Margin of error = 10

First, let's find the z-scores for the upper and lower limits of the sample mean range:

Lower limit:
z_lower = (502 - 502) / (100 / √90) = 0

Upper limit:
z_upper = (502 + 10 - 502) / (100 / √90) = 0.9487 (rounded to 4 decimals)

To find the probability, we need to find the area under the normal distribution between these two z-scores. We can use a standard normal distribution table or a calculator to find this area. For example, using a calculator, the probability is approximately 0.1729.

b. Mathematics part:
X = 515
μ = 515
σ = 100
n = 90
Margin of error = 10

Lower limit:
z_lower = (515 - 515) / (100 / √90) = 0

Upper limit:
z_upper = (515 + 10 - 515) / (100 / √90) = 0.9487 (rounded to 4 decimals)

Using a calculator, the probability is also approximately 0.1729.

c. Writing part:
X = 494
μ = 494
σ = 100
n = 100
Margin of error = 10

Lower limit:
z_lower = (494 - 494) / (100 / √100) = 0

Upper limit:
z_upper = (494 + 10 - 494) / (100 / √100) = 1

Using a calculator, the probability is approximately 0.6826.

Please note that these calculations assume that the sample means are normally distributed.

To answer these questions, we can use the concept of the standard normal distribution. The standard normal distribution, also known as the z-distribution, is a theoretical distribution with a mean of 0 and a standard deviation of 1.

To calculate the probability, we first need to calculate the z-score, which measures how many standard deviations a data point is away from the mean.

a. For the Critical Reading part of the test:
The population mean is 502, and the standard deviation is 100. We want to find the probability of a sample mean test score within 10 points of the population mean. This means we are looking for the probability of the sample mean falling between 492 and 512.

To calculate the z-scores for these values, we use the formula:
z = (X - μ) / σ
where X is the sample mean, μ is the population mean, and σ is the standard deviation.

For the lower limit z-score:
z_lower = (492 - 502) / 100 = -0.1

For the upper limit z-score:
z_upper = (512 - 502) / 100 = 0.1

Next, we can use a standard normal distribution table or a statistical calculator to find the probability associated with these z-scores. The probability of the sample mean falling within this range is the difference between the two probabilities.

b. For the Mathematics part of the test:
The population mean is 515, and the standard deviation is 100. We want to find the probability of a sample mean test score within 10 points of the population mean (from 505 to 525).

Calculate the z-scores for these values using the same formula as above.

For the lower limit z-score:
z_lower = (505 - 515) / 100 = -0.1

For the upper limit z-score:
z_upper = (525 - 515) / 100 = 0.1

Calculate the probability using a standard normal distribution table or a statistical calculator.

c. For the Writing part of the test:
The population mean is 494, and the standard deviation is 100. We want to find the probability of a sample mean test score within 10 points of the population mean (from 484 to 504).

Calculate the z-scores for these values using the same formula as above.

For the lower limit z-score:
z_lower = (484 - 494) / 100 = -0.1

For the upper limit z-score:
z_upper = (504 - 494) / 100 = 0.1

Calculate the probability using a standard normal distribution table or a statistical calculator.

Note: When using a standard normal distribution table, you need to find the probability associated with the z-scores and then calculate the difference between the two probabilities. Alternatively, you can use a statistical calculator or software to calculate the probabilities directly.