Answered: UPPOSE THE LIFETIMES OF TV TUBES ARE…

Name: SUPPOSE THE LIFETIMES OF TV TUBES ARE NORMALLY DISTRIBUTED WITH A STAN
Uploaded: 2024-03-20T07:06:16.000Z
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Description: SUPPOSE THE LIFETIMES OF TV TUBES ARE NORMALLY DISTRIBUTED WITH A STANDARD DEVIATION OF 1.1 YEARS. SUPPOSE THAT EXACTLY 20% OF ALL TUBES DIE BEFORE 5 YEARS. FIND MEAN LIFETIME OF TV. TUBES.

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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SUPPOSE THE LIFETIMES OF TV TUBES ARE NORMALLY DISTRIBUTED WITH A STANDARD DEVIATION OF 1.1 YEARS. SUPPOSE THAT EXACTLY 20% OF ALL TUBES DIE BEFORE 5 YEARS. FIND MEAN LIFETIME OF TV. TUBES.

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 + ...

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Step 1

GIVEN DATA : lifetimes of TV tubes are normally distributed with a standard deviation of $1.1 years$

$20 %$ of the tubes die before $5$ years

TO FIND : Mean of the lifetime of T.V tubes

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Z-score is a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean.

$Z = \frac{x - μ}{σ}$

Suppose $X$ the random variable that represent the lifetimes of TV tubes of a population, and for this case we know the distribution for X is given by

$X ~ N (μ, σ)$

It is given that standard deviation is $σ = 1.1$

$X ~ N (μ, 1.1)$

Step 2

From the given data, Before $5$ years $20 %$ tubes die

thus, Probability of tubes before $5$ years $P (X < 5) = \frac{20}{100}$

$P (X < 5) = 0.2$

thus, after $5$ years $P (X > 5) = 1 - P (X < 5)$

$P (X > 5) = 1 - 0.2$

$P (X > 5) = 0.8$

From the $z$ value that satisfy the condition with $0.2$ of the area on the left and $0.8$ of the area on the right it's $z = - 0.842 .$ On this case $P (z < - 0.842) = 0.2$ and $P (z > - 0.842) = 0.8$

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