Question 1199957: Suppose that the scores of architects on a particular creativity test are normally distributed with a mean of 285 and a standard deviation of 25. Using a normal curve table, find the top and bottom scores for each of the following middle percentages of architects.
(a)
%55
(b)
%86
(c)
%99
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **1. Understand the Problem**
* We're dealing with a normally distributed variable (creativity test scores) with:
* Mean (μ) = 285
* Standard Deviation (σ) = 25
* We need to find the scores that correspond to specific percentiles (middle 55%, 86%, and 99%).
**2. Approach**
* We'll use the concept of z-scores and the standard normal distribution table (or a calculator with statistical functions) to find the corresponding scores.
**3. Calculate Z-scores**
* **(a) Middle 55%:**
* This means 22.5% of scores are below the lower bound and 22.5% are above the upper bound.
* Find the z-score corresponding to the 22.5th percentile (using a standard normal table or calculator). Let's call it z1.
* Find the z-score corresponding to the 77.5th percentile (using a standard normal table or calculator). Let's call it z2.
* **(b) Middle 86%:**
* This means 7% of scores are below the lower bound and 7% are above the upper bound.
* Find the z-score corresponding to the 7th percentile (z1).
* Find the z-score corresponding to the 93rd percentile (z2).
* **(c) Middle 99%:**
* This means 0.5% of scores are below the lower bound and 0.5% are above the upper bound.
* Find the z-score corresponding to the 0.5th percentile (z1).
* Find the z-score corresponding to the 99.5th percentile (z2).
**4. Convert Z-scores to Raw Scores**
* Use the following formula to convert z-scores to raw scores:
* X = μ + z * σ
* Where:
* X = raw score
* μ = mean
* z = z-score
* σ = standard deviation
**Example (Middle 55%):**
* Let's assume (from a standard normal table or calculator) that:
* z1 (22.5th percentile) ≈ -0.72
* z2 (77.5th percentile) ≈ 0.72
* Calculate the lower bound score:
* X1 = 285 + (-0.72) * 25
* X1 = 285 - 18
* X1 = 267
* Calculate the upper bound score:
* X2 = 285 + 0.72 * 25
* X2 = 285 + 18
* X2 = 303
* Therefore, the middle 55% of architects have scores between 267 and 303.
**Repeat these steps for the middle 86% and middle 99% using the appropriate z-scores from the standard normal table or calculator.**
**Note:**
* You'll need to use a standard normal distribution table or a calculator (like a scientific calculator or statistical software) to find the precise z-scores for each percentile.
* This approach provides you with the general method to solve this type of problem.
I hope this helps! Let me know if you have any further questions.
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