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)?
To solve these probability questions, we will use the normal distribution and the concept of the standard error of the mean. The formula for calculating the standard error of the mean is:
Standard error of the mean (SE) = population standard deviation / square root of the sample size
a. Let's calculate the probability using the given information:
Population mean (μ) = 502
Sample size (n) = 90
Population standard deviation (σ) = 100
Desired range = within 10 points of the mean, so the range is from 492 to 512
First, we need to calculate the standard error of the mean using the formula:
SE = 100 / √90 ≈ 10.560 (rounded to 3 decimal places)
Next, we will use the normal distribution to calculate the probability of the sample mean falling within the desired range. Since the sample size is large (n > 30), we can assume a normal distribution.
Using a standard normal distribution table or a calculator, find the Z-scores for the lower and upper limits:
Lower Limit Z-score = (492 - 502) / 10.560 = -0.947
Upper Limit Z-score = (512 - 502) / 10.560 = 0.947
Now, find the cumulative probability associated with these Z-scores. Subtract the lower limit cumulative probability from the upper limit cumulative probability to get the probability of the sample mean falling within the desired range:
P(-0.947 ≤ Z ≤ 0.947) ≈ P(Z ≤ 0.947) - P(Z ≤ -0.947)
Using a Z-table or calculator, find the cumulative probabilities for these Z-scores. Subtract the smaller cumulative probability from the larger one to get the result.
b. Follow the same steps as in part (a), but with different values:
Population mean (μ) = 515
Sample size (n) = 90
Population standard deviation (σ) = 100
Desired range = within 10 points of the mean, so the range is from 505 to 525
Calculate the standard error of the mean:
SE = 100 / √90 ≈ 10.560 (rounded to 3 decimal places)
Find the Z-scores for the lower and upper limits:
Lower Limit Z-score = (505 - 515) / 10.560 = -0.947
Upper Limit Z-score = (525 - 515) / 10.560 = 0.947
Calculate the probability:
P(-0.947 ≤ Z ≤ 0.947) ≈ P(Z ≤ 0.947) - P(Z ≤ -0.947)
c. For this part, the sample size changes to 100, so we need to recalculate the standard error of the mean:
SE = 100 / √100 = 10
Continue with the same steps as in part (a) and (b) to calculate the probability.
Please note that for precise calculations, you should use a standard normal distribution table or a calculator that provides accurate cumulative probabilities for Z-scores.