Re-worded from the "Study Questions" part of Chapter 5 from this site: "Student Study Site for Essentials of Social Statistics for a Diverse Society"
Please show the steps for each question
"When the Scholastic Aptitude Test (SAT) - now known as the Scholastic Assessment Test - was first developed, the idea was that each section of the test would have a mean of 500 and a standard deviation of 100. While the SAT is continually adjusted and readjusted to promote such a distribution, let's assume for this exercise that math scores on the SAT have a mean of 500 and a standard deviation of 100.
1. What percentage of students scored below/above 465?
2. What percentage of students who took the SAT scored between a 465 and 500?
3. What percentage of students scored above 595 on the math portion of the SAT?
4. What is the percentile does a score of 465 correspond to?
5. What percentage of students who took the SAT posted scores between a z-score of .35 and a raw score of 595?"
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
Sure! Let's break down each question and go through the steps to find the answers:
1. What percentage of students scored below/above 465?
To find the percentage of students who scored below 465, we need to calculate the area under the curve to the left of the value 465. We can use a standard normal distribution table or a statistical calculator.
The steps to calculate this are:
- Convert the raw score of 465 to a z-score using the formula: z = (x - μ) / σ, where x is the raw score, μ is the mean, and σ is the standard deviation.
- Once you have the z-score, use the standard normal distribution table or calculator to find the proportion of the area to the left of that z-score.
- Multiply the proportion by 100 to get the percentage.
To find the percentage of students who scored above 465, subtract the percentage you found above from 100%.
2. What percentage of students who took the SAT scored between 465 and 500?
To find the percentage of students who scored between 465 and 500, we need to calculate the area under the curve between these two values. Again, we can use a standard normal distribution table or a statistical calculator.
The steps here are similar:
- Convert both raw scores (465 and 500) to z-scores using the formula mentioned above.
- Use the standard normal distribution table or calculator to find the proportion of the area between these two z-scores.
- Multiply the proportion by 100 to get the percentage.
3. What percentage of students scored above 595 on the math portion of the SAT?
To find the percentage of students who scored above 595, we can follow a similar process as in question 1.
- Convert the raw score of 595 to a z-score using the formula mentioned earlier.
- Use the standard normal distribution table or calculator to find the proportion of the area to the right of that z-score.
- Multiply the proportion by 100 to get the percentage.
4. What percentile does a score of 465 correspond to?
To determine the percentile corresponding to a score of 465, we need to find the proportion of scores that fall below that score.
- Convert the raw score of 465 to a z-score using the formula mentioned earlier.
- Use the standard normal distribution table or calculator to find the proportion of the area to the left of that z-score.
- Multiply the proportion by 100 to get the percentile.
5. What percentage of students who took the SAT posted scores between a z-score of 0.35 and a raw score of 595?
To find the percentage of students who scored between a z-score of 0.35 and a raw score of 595, we need to find the area under the curve between these two points.
- Convert the raw score of 595 to a z-score using the formula mentioned earlier.
- Use the standard normal distribution table or calculator to find the proportion of the area to the left of the z-score of 595.
- Subtract the proportion you found for a z-score of 0.35 from the proportion you found for a z-score of 595.
- Multiply the resulting proportion by 100 to get the percentage.
By following these steps, you should be able to find the answers to each of the questions.