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?
why don't you know the answer
To answer these questions, we can use the Z-score formula. The Z-score, also known as the standard score, measures the number of standard deviations a particular score is away from the mean. We can then use the Z-score to find the corresponding values in the standard normal distribution table or use statistical software or a calculator.
The formula to calculate the Z-score is:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Given score
μ = Mean
σ = Standard deviation
Let's apply this formula to answer the questions:
1. What score would cut off the top 5 percent of scores?
To find this score, we need to find the Z-score that corresponds to the top 5 percent of the distribution. This corresponds to a cumulative probability of 0.95. So, we need to find the Z-score from the standard normal distribution table (or calculator) that has a cumulative probability of 0.95.
Using the Z-table or calculator, we find that the Z-score for a cumulative probability of 0.95 is approximately 1.645.
Now, we can use the Z-score formula to find the score (X):
Z = (X - μ) / σ
1.645 = (X - 200) / 10
Rearranging the equation to solve for X:
X - 200 = 1.645 * 10
X - 200 = 16.45
X = 200 + 16.45 = 216.45
So, the score that would cut off the top 5 percent of scores is approximately 216.45.
2. What score would cut off the bottom 5 percent of scores?
Using the same logic as above, the Z-score for a cumulative probability of 0.05 (bottom 5 percent) is approximately -1.645.
Z = (X - μ) / σ
-1.645 = (X - 200) / 10
X - 200 = -1.645 * 10
X - 200 = -16.45
X = 200 - 16.45 = 183.55
So, the score that would cut off the bottom 5 percent of scores is approximately 183.55.
3. What score would cut off the top 2.5 percent of scores?
Similar to the previous questions, the Z-score for a cumulative probability of 0.975 (top 2.5 percent) is approximately 1.96.
Z = (X - μ) / σ
1.96 = (X - 200) / 10
X - 200 = 1.96 * 10
X - 200 = 19.6
X = 200 + 19.6 = 219.6
So, the score that would cut 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?
Similarly, the Z-score for a cumulative probability of 0.025 (bottom 2.5 percent) is approximately -1.96.
Z = (X - μ) / σ
-1.96 = (X - 200) / 10
X - 200 = -1.96 * 10
X - 200 = -19.6
X = 200 - 19.6 = 180.4
So, the score that would cut off the bottom 2.5 percent of scores is approximately 180.4.
Note: These calculations assume that the distribution is exactly normal. In practice, there may be slight variations, and the values obtained are approximate.