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Q.1

Expert-verifiedFound in: Page 407

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 $P(X=5)$

(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)$P(X=5)=0.1$

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

(c) Mean=$3.1$

Standard deviation=$1.14$

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 $P(X=5)$

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

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" $P(X=5)$ can be calculated as:

localid="1649746565386" $P(X=5)=x$

localid="1649746571448" $=0.1$

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 $2$

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

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

$=0.1+0.2$

$=0.3$

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

$P(1$ or $2$$)=P(1$ or $2$$)\times P(1$or $2)$$)$

$=0.3\times 0.3$

$=0.09$

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 .

The mean can be calculated as:

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

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

localid="1649746701130" $=3.1$

The standard deviation is calculated as follows:

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

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

localid="1649746712012" $=1.1358$

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