Final answer:
The Chapin Social Insight Test evaluates how accurately the subject appraised other people. In the reference population used to develop the test, scores are approximately normally distributed with mean 25 and standard deviation 5. We can calculate the percentage of the population that scores below 18, the percentage of the population that scores between 27 and 40, and the score needed to be in the top 5%.
Step-by-step explanation:
To solve this problem, we can use the Z-score formula: Z = (x - μ) / σ, where Z is the Z-score, x is the value we want to find the percentage of, μ is the mean, and σ is the standard deviation.
a) To find the percentage of the population that scores below 18, we calculate the Z-score for 18: Z = (18 - 25) / 5 = -1.4. Looking up the Z-score in the Z-table, a Z-score of -1.4 corresponds to approximately 0.0808 or 8.08%. Therefore, approximately 8.08% of the population scores below 18 on this test.
b) To find the percentage of the population that scores between 27 and 40, we calculate the Z-scores for 27 and 40: Z1 = (27 - 25) / 5 = 0.4 and Z2 = (40 - 25) / 5 = 3. Therefore, we need to find the area between Z1 and Z2 in the Z-table. The area between Z1 = 0.4 and Z2 = 3 is approximately 0.1525 or 15.25%. Therefore, approximately 15.25% of the population scores between 27 and 40 on this test.
c) To find the score needed to be in the top 5%, we need to find the Z-score that corresponds to the top 5% of the distribution. Looking up the Z-score in the Z-table, a Z-score of approximately 1.645 corresponds to the top 5%. Therefore, the score needed to be in the top 5% is calculated as: Z = (x - 25) / 5 = 1.645. Solving for x, we get x = 1.645 * 5 + 25 = 33.225. Therefore, you need to score at least 33.225 to be in the top 5%.