/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Let \(X\) be \(N(5,10)\). Find \... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 12

Let \(X\) be \(N(5,10)\). Find \(P\left[0.04<(X-5)^{2}<38.4\right]\).

Short Answer

Expert verified
The Probabilty \(P\left[0.04<(X-5)^{2}<38.4\right]\) for \(X\) is approximately 0.05

Step by step solution

01

Understanding the inequality and the transformation

The given inequality, \(0.04<(X-5)^{2}<38.4\), can be understood as the probability that the transformed variable \((X-5)^{2}\) lies between 0.04 and 38.4. To simplify, we can take the square root of all three parts of the inequality considering both positive and negative root as squaring was involved.
02

Simplifying the inequality

\(\sqrt{0.04}<|X-5|<\sqrt{38.4}\), this gives \(0.2<|X-5|<6.2\), which indicates that the difference between \(X\) and 5 is between 0.2 and 6.2.
03

Split the inequality into two

Splitting the modulus sign will result in two inequalities, \(X-5>0.2 \) and \(X-5<-0.2 \), or \(5.2 < X <4.8 \). Since \( X \) has a normal distribution, so does \( X - 5 \), specifically \(N(0,10)\). With this, we can find the corresponding z-scores.
04

Calculate the z-scores

The z-scores can be calculated as follows: \(Z = (X-\mu)/\sigma\) . Using this formula, we can calculate the two z-scores as \(Z1= (5.2-5)/\sqrt{10} = 0.063\) and \(Z2 = (4.8-5)/\sqrt{10} = -0.063 \).
05

Find the probabilities

Now, refer to a standard Z-table to find the probabilities corresponding to the obtained z-scores. So, the probability for \(Z1=0.063\) is \(P(Z<0.063) = 0.525\) and for \(Z2=-0.063\), it is \(P(Z<-0.063) = 0.475\). The required probability will be the absolute difference between these two Probabilities; \(P[0.04<(X-5)^{2}<38.4]\) = \(|P(Z<0.063) - P(Z<-0.063)|\).

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