Answered: A manufacturer knows that their items…

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

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 14.7 years, and standard deviation of 4.8 years.

If you randomly purchase 7 items, what is the probability that their mean life will be longer than 14 years?

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 that,

$Mean, μ = 14.7 Standard Deviation, σ = 4.8$

The standard deviation for the 7 items is,

$\begin{array}{rcl} σ' & = & \frac{σ}{\sqrt{n}} \\ = & \frac{4.8}{\sqrt{7}} \\ = & \frac{4.8}{2.6457} \\ = & 1.81 \end{array}$

Step 2

Solve for the z-value as,

$\begin{array}{rcl} z - v a l u e & = & \frac{14 - 14.7}{1.81} \\ = & \frac{- 0.7}{1.81} \\ = & - 0.39 \end{array}$

The required probability is,

$\begin{array}{rcl} P (z > - 0.39) & = & 1 - P (z < - 0.39) \\ = & 1 - ϕ (- 0.39) \end{array}$

From the Standard Normal Distribution Table,

$ϕ (- 0.39) = 0.3483$

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A manufacturer knows that their items have a normally distributed lifespan, with a mean of 14.7 years, and standard deviation of 4.8 years. If you randomly purchase 7 items, what is the probability that their mean life will be longer than 14 years