SOLUTION: ACT scores have a mean of 21.3 and 8 percent of the scores are above 27. The scores have a distribution that is approximately normal. Find the standard deviation. Round your answer
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Question 1170771: ACT scores have a mean of 21.3 and 8 percent of the scores are above 27. The scores have a distribution that is approximately normal. Find the standard deviation. Round your answer to the nearest tenth Answer by CPhill(1959) (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step using the properties of the normal distribution.
**1. Understand the Problem**
* We are given:
* Mean (μ) = 21.3
* 8% of scores are above 27.
* The distribution is approximately normal.
* We need to find the standard deviation (σ).
**2. Convert Percentage to Z-score**
* If 8% of scores are above 27, then 92% of scores are below 27.
* We need to find the z-score that corresponds to the 92nd percentile (0.92).
* Using a z-table or calculator, we find the z-score that corresponds to 0.92 is approximately 1.41.
**3. Use the Z-score Formula**
* The z-score formula is:
* z = (x - μ) / σ
* Where:
* z is the z-score
* x is the data point (27)
* μ is the mean (21.3)
* σ is the standard deviation (what we want to find)
* Plug in the values:
* 1.41 = (27 - 21.3) / σ
**4. Solve for Standard Deviation (σ)**
* 1.41 = 5.7 / σ
* σ = 5.7 / 1.41
* σ ≈ 4.04255
**5. Round to the Nearest Tenth**
* σ ≈ 4.0
**Answer**
The standard deviation is approximately 4.0.
