Question

the patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and standard deviation of 1.8 days. use your graphing calculator to answer the following questions. write your answers in percent form. round your answers to the nearest tenth of a percent.
a) what is the probability of spending less than 8 days in recovery?
b) what is the probability of spending more than 4 days in recovery?
c) what is the probability of spending between 4 days and 8 days in recovery?
question 24
suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 242 feet and a standard deviation of 60 feet.
use your graphing calculator to answer the following questions. write your answers in percent form. round your answers to the nearest tenth of a percent.
a) if one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 240 feet?
p(fewer than 240 feet) =

b) if one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled more than 252 feet?
p(more than 252 feet) =

Explanation:

Part 1: Patient Recovery Time (Normal Distribution, $\mu = 5.3$, $\sigma = 1.8$)

a) Probability of spending less than 8 days in recovery

To find $P(X < 8)$ for a normal distribution, we first calculate the z - score:
The formula for the z - score is $z=\frac{x-\mu}{\sigma}$.
For $x = 8$, $\mu = 5.3$, and $\sigma = 1.8$, we have:
$z=\frac{8 - 5.3}{1.8}=\frac{2.7}{1.8}=1.5$

Using a graphing calculator (or standard normal table) to find the area to the left of $z = 1.5$. The area to the left of $z = 1.5$ is approximately 0.9332. Converting this to a percentage, we get $0.9332\times100 = 93.3\%$ (rounded to the nearest tenth of a percent).

b) Probability of spending more than 4 days in recovery

First, calculate the z - score for $x = 4$:
$z=\frac{4 - 5.3}{1.8}=\frac{- 1.3}{1.8}\approx - 0.7222$

We want $P(X>4)$, which is equal to $1 - P(X\leq4)$. Using the z - score, we find the area to the left of $z=-0.7222$ (using a calculator or table) is approximately 0.2358. Then $P(X > 4)=1 - 0.2358 = 0.7642$. Converting to a percentage, we get $0.7642\times100=76.4\%$ (rounded to the nearest tenth of a percent).

c) Probability of spending between 4 days and 8 days in recovery

We already know from part (a) that $P(X < 8)\approx0.9332$ and from part (b) that $P(X < 4)\approx0.2358$. Then $P(4 < X < 8)=P(X < 8)-P(X < 4)$.

So, $P(4 < X < 8)=0.9332 - 0.2358 = 0.6974$. Converting to a percentage, we get $0.6974\times100 = 69.7\%$ (rounded to the nearest tenth of a percent).

Part 2: Fly Balls (Normal Distribution, $\mu = 242$, $\sigma = 60$)

a) Probability of traveling fewer than 240 feet

Calculate the z - score for $x = 240$:
$z=\frac{240 - 242}{60}=\frac{-2}{60}\approx - 0.0333$

We want $P(X < 240)$, which is the area to the left of $z=-0.0333$. Using a calculator or table, the area to the left of $z = - 0.0333$ is approximately 0.4867. Converting to a percentage, we get $0.4867\times100 = 48.7\%$ (rounded to the nearest tenth of a percent).

b) Probability of traveling more than 252 feet

First, calculate the z - score for $x = 252$:
$z=\frac{252 - 242}{60}=\frac{10}{60}\approx0.1667$

We want $P(X > 252)$, which is equal to $1 - P(X\leq252)$. The area to the left of $z = 0.1667$ (using a calculator or table) is approximately 0.5662. Then $P(X > 252)=1 - 0.5662 = 0.4338$. Converting to a percentage, we get $0.4338\times100 = 43.4\%$ (rounded to the nearest tenth of a percent).

Final Answers

Patient Recovery Time

a) $\boldsymbol{93.3\%}$
b) $\boldsymbol{76.4\%}$
c) $\boldsymbol{69.7\%}$

Fly Balls

a) $\boldsymbol{48.7\%}$
b) $\boldsymbol{43.4\%}$

Answer:

Part 1: Patient Recovery Time (Normal Distribution, $\mu = 5.3$, $\sigma = 1.8$)

a) Probability of spending less than 8 days in recovery

To find $P(X < 8)$ for a normal distribution, we first calculate the z - score:
The formula for the z - score is $z=\frac{x-\mu}{\sigma}$.
For $x = 8$, $\mu = 5.3$, and $\sigma = 1.8$, we have:
$z=\frac{8 - 5.3}{1.8}=\frac{2.7}{1.8}=1.5$

Using a graphing calculator (or standard normal table) to find the area to the left of $z = 1.5$. The area to the left of $z = 1.5$ is approximately 0.9332. Converting this to a percentage, we get $0.9332\times100 = 93.3\%$ (rounded to the nearest tenth of a percent).

b) Probability of spending more than 4 days in recovery

First, calculate the z - score for $x = 4$:
$z=\frac{4 - 5.3}{1.8}=\frac{- 1.3}{1.8}\approx - 0.7222$

We want $P(X>4)$, which is equal to $1 - P(X\leq4)$. Using the z - score, we find the area to the left of $z=-0.7222$ (using a calculator or table) is approximately 0.2358. Then $P(X > 4)=1 - 0.2358 = 0.7642$. Converting to a percentage, we get $0.7642\times100=76.4\%$ (rounded to the nearest tenth of a percent).

c) Probability of spending between 4 days and 8 days in recovery

We already know from part (a) that $P(X < 8)\approx0.9332$ and from part (b) that $P(X < 4)\approx0.2358$. Then $P(4 < X < 8)=P(X < 8)-P(X < 4)$.

So, $P(4 < X < 8)=0.9332 - 0.2358 = 0.6974$. Converting to a percentage, we get $0.6974\times100 = 69.7\%$ (rounded to the nearest tenth of a percent).

Part 2: Fly Balls (Normal Distribution, $\mu = 242$, $\sigma = 60$)

a) Probability of traveling fewer than 240 feet

Calculate the z - score for $x = 240$:
$z=\frac{240 - 242}{60}=\frac{-2}{60}\approx - 0.0333$

We want $P(X < 240)$, which is the area to the left of $z=-0.0333$. Using a calculator or table, the area to the left of $z = - 0.0333$ is approximately 0.4867. Converting to a percentage, we get $0.4867\times100 = 48.7\%$ (rounded to the nearest tenth of a percent).

b) Probability of traveling more than 252 feet

First, calculate the z - score for $x = 252$:
$z=\frac{252 - 242}{60}=\frac{10}{60}\approx0.1667$

We want $P(X > 252)$, which is equal to $1 - P(X\leq252)$. The area to the left of $z = 0.1667$ (using a calculator or table) is approximately 0.5662. Then $P(X > 252)=1 - 0.5662 = 0.4338$. Converting to a percentage, we get $0.4338\times100 = 43.4\%$ (rounded to the nearest tenth of a percent).

Final Answers

Patient Recovery Time

a) $\boldsymbol{93.3\%}$
b) $\boldsymbol{76.4\%}$
c) $\boldsymbol{69.7\%}$

Fly Balls

a) $\boldsymbol{48.7\%}$
b) $\boldsymbol{43.4\%}$