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Q.84

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Introductory Statistics

Found in: Page 393

Book edition OER 2018

Author(s) Barbara Illowsky, Susan Dean

Pages 902 pages

ISBN 9781938168208

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Short Answer

An expert witness for a paternity lawsuit testifies that the length of a pregnancy is normally distributed with a mean of $280$ days and a standard deviation of $13$ days. An alleged father was out of the country from $240$ to $306$ days before the birth of the child, so the pregnancy would have been less than $240$ days or more than $306$ days long if he was the father. The birth was uncomplicated, and the child needed no medical intervention. What is the probability that he was NOT the father? What is the probability that he could be the father? Calculate the localid="1653472319552" $z$ -scores first, and then use those to calculate the probability.

His probability of being the father is $0.024$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information

Let variable $X$ has normal distribution with mean localid="1653469397337" $μ$ and variance $σ^{2}$ . Then probability density function is defined by:

$f (x) = \frac{1}{\sqrt{2 π} σ} e^{- \frac{1}{2} \frac{(x - μ)^{2}}{σ^{2}}}, x \in R$

The localid="1653469465424" $z$ statistic for value $x$ is calculated as follows:

$z = \frac{x - μ}{σ}$

If $z$ statistic is positive, $x$ value is $| z |$ standard deviation to the right of the mean, whereas $z$ statistic is negative, the $x$ value is $| z |$ standard deviation to the left of the mean.

Step 2: Explanation:

The length of a pregnancy in this task has a normal distribution with a mean of $μ = 280$ and a standard deviation of $σ = 13$ days. Now we must determine the likelihood that this individual is not a father, followed by the likelihood that he is a father. We must first locate the $z$ statistics for the days $240$ and $306$ days. We have:

localid="1653471405557" $z_{240} = \frac{240 - 280}{13}$

$= - 3.077,$

$z_{306} = \frac{306 - 280}{13}$

$= \frac{306 - 280}{13}$

$= 2$

Step 3: Explanation

We'll utilise these numbers to figure out whether he's the father or not. We'll start by calculating the odds that he isn't the father. Let's define the variable $X$ , which represents the length of pregnancy, and then write $X : N (280, 13)$ . We need to find the likelihood that standardizing variable $X * : N (0, 1)$ is between $z_{240}$ and $z_{306}$ to calculate that he is not the father. Now we have:

$P (z_{240} < X^{*} < z_{306}) = P (- 3.077 < X^{*} < 2)$ &

$= P (- 3.077 < X^{*} < 2)$

$= \int_{- 3.077}^{2} \frac{1}{\sqrt{2 π}} e^{- \frac{u^{2}}{2}} d u$

= N O R M S D I S T (2) - NORMSDIST (- 3.077)

= 0.977 - 0.001

$= 0.976$ .

Step 4: Explanation

The probability that he is not the father is $0.976$ . We must now determine the likelihood that he is a parent. We must calculate the probability that the standardize variable $X *$ is smaller than $z_{240}$ and greater than $z_{306}$ . Now we have:

$P (X^{*} < z_{240} \cup X^{*} > z_{306}) = P (X^{*} < - 3.077 \cup X^{*} > 2)$ $= P (X^{*} < - 3.077 \cup X^{*} > 2)$

$= N O R M S D I S T (- 3.077) + (1 - NORMSDIST(2))$

$= 0.001 + 1 - 0.976$

$= 0.024$

Most popular questions for Math Textbooks

Terri Vogel, an amateur motorcycle racer, averages $129.71$ seconds per $2.5$ mile lap (in a seven-lap race) with a standard deviation of $2.28$ seconds. The distribution of her race times is normally distributed. We are interested in one of her randomly selected laps.

a. In words, define the random variable $X$ .

b. $X ~$ ,

c. Find the percent of her laps that are completed in less than $130$ seconds.

d. The fastest $3 %$ of her laps are under

e. The middle $80 %$ of her laps are from seconds to seconds.

X~N(– $4, 1$ )

What is the median?

Find the $60 t h$ percentile.

Suppose X ~ N(2, 3). What value of x has a z-score of –0.67?

Use the following information to answer the next two exercises: The life of Sunshine CD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts.

$X ~_____(_____,_____)$

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