Answered: Two students are taking a college…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Two students are taking a college entrance exam and known to have a normal distribution of scores. Both students receive raw scores of 60 and 97, which correspond to z scores (often called the standardized scores) of -1.5 and 3 respectively. Answer the followings: (a) Denote the normal probability distribution of the raw exam scores, (6) Calculate the value of mean for the distribution of raw exam scores. (c) Calculate the value of standard deviation of the raw exam scores.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Expert Solution

Step 1

Given raw scores of 60 and 97 corresponds to z-scores of -1.5 and 3 respectively.

Step 2

$\frac{60 - μ}{σ} = - 1.5 \Rightarrow 60 - μ = - 1.5 σ (1) \frac{97 - μ}{σ} = 3 \Rightarrow 97 - μ = 3 σ (2)$

Solve equation (1) and (2) simultaneously.

By adding equation (1) and (2), standard deviation sigma is 104.66.

$60 - μ = - 1.5 σ + 97 - μ = 3.0 σ 157 = 1.5 σ \Rightarrow σ = \frac{157}{1.5} = 104.66$

Substitute sigma in any of the equation which gives mean mu = 217.

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