Suppose we have a population of scores with a mean (μ) of 200 and a standard deviation (σ) 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?
To find the scores that cut off certain percentages of the distribution, we can use the concept of z-scores and the standard normal distribution.
The formula to calculate the z-score is:
z = (x - μ) / σ
Here, x is the score, μ is the mean, and σ is the standard deviation.
Now, let's solve each question step by step:
1. What score would cut off the top 5 percent of scores?
To find the score that cuts off the top 5 percent of scores, we need to find the z-score corresponding to the 95th percentile. The 95th percentile corresponds to a z-score of 1.645.
Using the z-score formula, we can rearrange it to solve for x:
x = z * σ + μ
Substituting the values, we have:
x = 1.645 * 10 + 200
x = 16.45 + 200
x ≈ 216.45
Therefore, the score that cuts off the top 5 percent of scores is approximately 216.45.
2. What score would cut off the bottom 5 percent of scores?
Finding the score that cuts off the bottom 5 percent of scores is similar to the previous question. The 5th percentile corresponds to a z-score of -1.645.
Using the z-score formula and substituting the values, we have:
x = -1.645 * 10 + 200
x = -16.45 + 200
x ≈ 183.55
Therefore, the score that cuts off the bottom 5 percent of scores is approximately 183.55.
3. What score would cut off the top 2.5 percent of scores?
Similarly, to find the score that cuts off the top 2.5 percent of scores, we need to find the z-score corresponding to the 97.5th percentile. The 97.5th percentile corresponds to a z-score of 1.96.
Using the z-score formula and substituting the values, we have:
x = 1.96 * 10 + 200
x = 19.6 + 200
x ≈ 219.6
Therefore, the score that cuts off the top 2.5 percent of scores is approximately 219.6.
4. What score would cut off the bottom 2.5 percent of scores?
To find the score that cuts off the bottom 2.5 percent of scores, we use the z-score equivalent to the 2.5th percentile, which is -1.96.
Using the z-score formula and substituting the values, we have:
x = -1.96 * 10 + 200
x = -19.6 + 200
x ≈ 180.4
Therefore, the score that cuts off the bottom 2.5 percent of scores is approximately 180.4.