Over 350 students took a college calculus final exam. the scores of the students follow a normal distribution. using the information given below, determine the mean and the standard deviation for the students'scores. Give your answers to the nearest tenth of a percent.
1. 32% of the students scored either less than 63.3% or more than 85.9%.
2. Only the top 2.5% of students scored higher than a 97% on the exam.
Mean: standard deviation:
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To determine the mean and standard deviation for the students' scores, we can utilize the properties of the normal distribution and the given information.
1. 32% of the students scored either less than 63.3% or more than 85.9%.
This implies that 68% of the students scored within one standard deviation from the mean, while the remaining 32% (16% on each side) are outside of one standard deviation.
To find the mean and standard deviation, we first need to identify the z-scores corresponding to the given percentages.
- For 16% (one side), the z-score is -1.0, as it represents the distance from the mean to the first percentile.
- Thus, for 32% (both sides), the z-score is -2.0.
Using the z-score formula: z = (x - μ) / σ, where x is the raw score, μ is the mean, and σ is the standard deviation, we can solve for the values.
-2.0 = (63.3 - μ) / σ --> 63.3 - μ = 2.0σ (Equation 1)
-2.0 = (85.9 - μ) / σ --> 85.9 - μ = 2.0σ (Equation 2)
By subtracting Equation 2 from Equation 1, we eliminate the μ term:
63.3 - 85.9 = 2.0σ - 2.0σ
-22.6 = 0
The equation presents an inconsistency, which indicates an error either in the given information or the calculations. Please ensure the given information is accurate, and I'll be happy to help you further.
2. Only the top 2.5% of students scored higher than a 97% on the exam.
In this case, we have a one-sided z-score, as we are interested in the upper tail of the distribution.
To find the z-score, we can use the information about the cumulative probability. From the z-table or by using software:
A z-score of approximately 1.96 corresponds to a cumulative probability of 0.975.
Using the formula: z = (x - μ) / σ, we can solve for the values.
1.96 = (97.0 - μ) / σ
Again, we have one equation with two unknowns (μ and σ). Without more information or additional equations, we cannot determine the mean and standard deviation accurately.