Answer:
The key takeaway is that the mean falls within the range of 48 and 72, but we cannot determine the exact value without further information about the standard deviation.
Step-by-step explanation:
We are given valuable information about a normal distribution test:
One Standard Deviation Below Mean: A score of 48 falls one standard deviation below the mean. This implies that the mean is higher than 48.
Three Standard Deviations Above Mean: A score of 72 falls three standard deviations above the mean. This implies that the mean is lower than 72.
Since the normal distribution is symmetrical around the mean, these two pieces of information together tell us that the mean must be located between 48 and 72.
Let μ (mu) represent the unknown mean of the test.
Here's what we can infer:
Mean (μ) > 48 (because 48 is one standard deviation below the mean)
Mean (μ) < 72 (because 72 is three standard deviations above the mean)
Finding the Mean:
Unfortunately, with the given information, we cannot pinpoint the exact value of the mean (μ). We only know it lies within the range of 48 and 72.
However, we can make an educated guess if we assume the standard deviation is relatively constant across the distribution. In that case, the distance between the mean and 48 (one standard deviation) would likely be equal to the distance between the mean and 72 (three standard deviations).
Estimated Mean (Optional):
Following this assumption:
Distance below the mean = Distance above the mean (assuming equal standard deviations)
μ - 48 = 72 - μ
Solving for μ:
2μ = 48 + 72
2μ = 120
μ ≈ 60 (estimated mean)
Important Note:
This estimated mean (μ ≈ 60) is based on the assumption of equal standard deviations on both sides of the distribution. Without additional information about the standard deviation, this is just an approximation. The actual mean could lie anywhere between 48 and 72.
