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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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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.

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 + ...

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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