Suppose we have a population of scores with a mean (ì) of 200 and a standard deviation (ó) of 10. Assume that the distribution is normal. Provide answers to the following questions:

What score would cut off the top 5 percent of scores?
What score would cut off the bottom 5 percent of scores?
What score would cut off the top 2.5 percent of scores?
What score would cut off the bottom 2.5 percent of scores?

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the Z scores related to the above proportions. Insert the values into the equation and calculate.

To answer these questions, we need to use the concept of z-scores. A z-score measures the number of standard deviations a data point is from the mean.

To find the z-score for a given percentile, we can use the standard normal distribution table or a statistical calculator.

1. To find the score that cuts off the top 5 percent of scores:
The top 5 percent corresponds to the 95th percentile.
Using the standard normal distribution table, the z-score corresponding to the 95th percentile is approximately 1.645.
We can calculate the score using the formula:
score = mean + (z-score * standard deviation)
score = 200 + (1.645 * 10)
score = 216.45 (rounded to 2 decimal places)

2. To find the score that cuts off the bottom 5 percent of scores:
The bottom 5 percent corresponds to the 5th percentile.
Using the standard normal distribution table, the z-score corresponding to the 5th percentile is approximately -1.645 (the negative value indicates it is below the mean).
Calculating the score using the formula:
score = mean + (z-score * standard deviation)
score = 200 + (-1.645 * 10)
score = 183.55 (rounded to 2 decimal places)

3. To find the score that cuts off the top 2.5 percent of scores:
The top 2.5 percent corresponds to the 97.5th percentile.
Using the standard normal distribution table, the z-score corresponding to the 97.5th percentile is approximately 1.96.
Calculating the score using the formula:
score = mean + (z-score * standard deviation)
score = 200 + (1.96 * 10)
score = 219.6 (rounded to 2 decimal places)

4. To find the score that cuts off the bottom 2.5 percent of scores:
The bottom 2.5 percent corresponds to the 2.5th percentile.
Using the standard normal distribution table, the z-score corresponding to the 2.5th percentile is approximately -1.96 (the negative value indicates it is below the mean).
Calculating the score using the formula:
score = mean + (z-score * standard deviation)
score = 200 + (-1.96 * 10)
score = 180.4 (rounded to 2 decimal places)

Therefore, the answers to the questions are as follows:
1. The score that cuts off the top 5 percent of scores is approximately 216.45.
2. The score that cuts off the bottom 5 percent of scores is approximately 183.55.
3. The score that cuts off the top 2.5 percent of scores is approximately 219.6.
4. The score that cuts off the bottom 2.5 percent of scores is approximately 180.4.