25. The results for a mandatory competency test for high school sophomores has a normal distribution with a mean of 400 and a standard deviation of 100 points. a.) Suppose that the top 3% of students receive a scholarship. What is the minimum score required to receive this scholarship? b.) The bottom 1.5% of the students must go to summer school. What is the minimum score you would need to stay out of summer school? c.) What are the two scores that will define the middle 50% of the test scores?
25. The results for a mandatory competency test for high school sophomores has a normal distribution with a mean of 400 and a standard deviation of 100 points. a.) Suppose that the top 3% of students receive a scholarship. What is the minimum score required to receive this scholarship? b.) The bottom 1.5% of the students must go to summer school. What is the minimum score you would need to stay out of summer school? c.) What are the two scores that will define the middle 50% of the test scores?
25. The results for a mandatory competency test for high school sophomores has a normal distribution with a mean of 400 and a standard deviation of 100 points. a.) Suppose that the top 3% of students receive a scholarship. What is the minimum score required to receive this scholarship? b.) The bottom 1.5% of the students must go to summer school. What is the minimum score you would need to stay out of summer school? c.) What are the two scores that will define the middle 50% of the test scores?
25. The results for a mandatory competency test for high school sophomores has a normal distribution with a mean of 400 and a standard deviation of 100 points.
a.) Suppose that the top 3% of students receive a scholarship. What is the minimum score required to receive this scholarship?
b.) The bottom 1.5% of the students must go to summer school. What is the minimum score you would need to stay out of summer school?
c.) What are the two scores that will define the middle 50% of the test scores?
Transcribed Image Text:**Problem 25: Normal Distribution in Test Scores**
The results for a mandatory competency test for high school sophomores follow a normal distribution with a mean of 400 and a standard deviation of 100 points.
**Questions:**
a.) Suppose that the top 3% of students receive a scholarship. What is the minimum score required to receive this scholarship?
b.) The bottom 1.5% of the students must go to summer school. What is the minimum score you would need to stay out of summer school?
c.) What are the two scores that will define the middle 50% of the test scores?
**Explanation:**
- ***Mean (µ):*** 400
- ***Standard Deviation (σ):*** 100
Use these values to calculate z-scores and determine the required scores for the specified percentages outlined in the questions.
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
)=400
)=100
