Question

5. At a local high school 5000 juniors and seniors recently took an aptitude test. The results of the exam were normally distributed with mean = 450 and o = 50. Calculate the following: a. The PERCENT of students to the nearest tenth of a percent that scored over 425 b. The number of students that scored more than 475 C. The probability of a student selected at random having scored between 400 and 575 6. A statistics instructor recorded the grades of his students on the final exam. The grades are 65, 72, 85, 92, 60, 52, 75, 79, 80, 89, 50, 59, 95, 99, 89, 77, 62, 65, 67, 73, 85, 23, 89, 94, 97 a. Construct a stem-and-leaf display. b. Describe the shape of the distribution. c. Deterinine the mode and median of these scores. d. What percentage of the students passed (at least a 70).

Answer 1

Note to student : As per HOMEWORKLIB POLICY, if a post contains more than one distinct questions, we should solve one question.

Solving the question 5.

5)

Let X be the results
X follows Normal Distribution with mean μ and standard deviation σ
Given
μ = 450           ....... Mean
σ = 50             ....... Standard Deviation

a)
To find Percent of students that scored over 425

that is to find P(X > 425)
P(X > 425) = 1 - P(X ≤ 425)

Using standard normal tables Or Using Excel Function "NORM.DIST", we get

P(X > 425) = 1 - NORM.DIST(425, 450, 50, TRUE)
P(X > 425) = 0.691462

                   = 69.1%

Percent of students that scored over 425 = $\mathbf{69.1\ \%}$

b)

First we find the probability

P(students scored more than 475)

that is to find P(X > 475)
P(X > 475) = 1 - P(X ≤ 475)

Using standard normal tables Or Using Excel Function "NORM.DIST", we get

P(X > 475) = 1 - NORM.DIST(475, 450, 50, TRUE)

P(X > 475) = 0.30854

To get the number of students we multiply by total students which is 5000

Number of students = 5000 * 0.30854

                                = 1542.7

                          = 1543            (rounding up)

Number of students that scored more than 475 = $\mathbf{1543}$

c)

P(student scores between 400 and 575)

that is to find P(400 < X < 575)
P(400 < X < 575) = P(X < 575) - P(X < 400)
Using Excel Function "NORM.DIST", we get
                          = NORM.DIST(575, 450, 50, TRUE) - NORM.DIST(400, 450, 50, TRUE)
P(400 < X < 575) = 0.83514

Probability of a student having scored between 400 and 575 = $\mathbf{0.83514}$

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