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Expert-verified Found in: Page 407 ### The Practice of Statistics for AP

Book edition 4th
Author(s) David Moore,Daren Starnes,Dan Yates
Pages 809 pages
ISBN 9781319113339 # Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of 1 (low) to 5 (high). Let X be the pain score for a randomly selected patient. The following table gives part of the probability distribution for X. Value$1$$2$$3$$4$$5$Probability$0.1$$0.2$$0.3$$0.3$?(a) Find $P\left(X=5\right)$(b) If two patients who received artificial knees are chosen at random, what’s the probability that both of them report pain scores of $1$or $2$? Show your work.(c) Compute the mean and standard deviation of $X$. Show your work.

(a)$P\left(X=5\right)=0.1$

(b)$P\left(1\text{or}2\right)=0.09$

(c) Mean=$3.1$

Standard deviation=$1.14$

See the step by step solution

## Part (a) Step 1: Given information

Given in the question that, Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of $1$ (low) to $5$ (high). Let $X$be the pain score for a randomly selected patient.

We need to find $P\left(X=5\right)$

## Part (a) Step 2: Explanation

Given:

the probability distribution is

 Value $1$ $2$ $3$ $4$ $5$ Probability localid="1649746499147" $0.1$ localid="1649746505085" $0.2$ localid="1649746511170" $0.3$ localid="1649746516856" $0.3$ ??

The formulas to compute the mean and standard deviation are:

localid="1649746524407" $\sigma =\sqrt{\sum {x}^{2}×P\left(x\right)-{\left(\sum x×P\left(x\right)\right)}^{2}}$

Let localid="1649746530834" $\mathrm{x}$ be the missing value.

The missing value can be calculated as:

localid="1649746539988" $0.1+0.2+0.3+0.3+x=1$

localid="1649746546770" $x+0.9=1$

localid="1649746553075" $x=0.1$

localid="1649746559408" $P\left(X=5\right)$ can be calculated as:

localid="1649746565386" $P\left(X=5\right)=x$

localid="1649746571448" $=0.1$

## Part (b) Step 1: Given information

Given in the question that, Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of $1$ (low) to $5$ (high). Let$X$be the pain score for a randomly selected patient.

We need to find the probability that both of the patients scores either $1$ or $2$

## Part (b) Step 2: Explanation

The probability that both of the patients scores either$1$or $2$ is computed as:

$P\left(1\text{or}2\right)=P\left(X=1\right)+P\left(X=2\right)$

$=0.1+0.2$

$=0.3$

The probability that both of the patients scores either$1$ or $2$ is computed as:

$P\left(1$ or $2$$\right)=P\left(1$ or $2$$\right)×P\left(1$or $2\right)$$\right)$

$=0.3×0.3$

$=0.09$

## Part (c) Step 1: Given information

Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of $1$(low) to $5$ (high). Let $X$ be the pain score for a randomly selected patient.

We need to compute the mean and standard deviation of X .

## Part (c) Step 2: Explanation

The mean can be calculated as:

$\mathrm{Mean}=\sum x×P\left(x\right)$

$=1\left(0.1\right)+2\left(0.2\right)+3\left(0.3\right)+4\left(0.3\right)+5\left(0.1\right)$

localid="1649746701130" $=3.1$

The standard deviation is calculated as follows:

localid="1649746705865" $\sigma =\sqrt{\sum {x}^{2}×P\left(x\right)-{\left(\sum x×P\left(x\right)\right)}^{2}}$

localid="1649746709064" $=\sqrt{{1}^{2}×0.1+{2}^{2}×0.2+\dots .+{5}^{2}×0.2-\left(3.1{\right)}^{2}}$

localid="1649746712012" $=1.1358$ ### Want to see more solutions like these? 