Question 1

Calculate the mean for the discrete probability distribution shown here. $$\begin{array} { l | c c c c } 
X & 3 & 4 & 8 & 11 \
\hline P ( X ) & 0.26 & 0.1 & 0.06 & 0.58
\end{array}$$

Accepted Answer

The mean of a discrete probability distribution is calculated by multiplying each value of X by its corresponding probability and summing the results: (3 * 0.26) + (4 * 0.1) + (8 * 0.06) + (11 * 0.58) = 0.78 + 0.4 + 0.48 + 6.38 = 8.04.

Question 2

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 10$ and $d = 50$. Find $P ( x > 50 )$.

Accepted Answer

In a uniform distribution, the probability of any single point is 0. Since $x$ cannot be greater than 50 in the range [10, 50], $P(x > 50) = 0$.

Question 3

Use the standard normal distribution to find P $$\mathrm { P } ( \mathrm { z } < - 2.33 \text { or } z > 2.33 )$$

Accepted Answer

The standard normal distribution is a bell-shaped curve with a mean of 0 and a standard deviation of 1. The probability of a z-score being less than -2.33 or greater than 2.33 is the area under the curve that lies outside of these two z-scores. This area is equal to 0.0198, which is choice A.

Question 4

Suppose that x has an exponential distribution with $$\theta = 5 . \text { Find } P ( x \leq 10 )$$

Accepted Answer

The cumulative distribution function (CDF) for an exponential distribution with parameter $\theta$ is given by $P(x \leq t) = 1 - e^{-t/\theta}$. Substituting $\theta = 5$ and $t = 10$, we get $P(x \leq 10) = 1 - e^{-10/5} = 1 - e^{-2} \approx 0.864665$.

Question 5

Given that $x$ is a hypergeometric random variable, compute $p ( x ) \text { for } N = 6 , n = 3 , r = 3 \text {, and } x = 1 \text {. }$

Accepted Answer

The answer of Given that $x$ is a hypergeometric random...

Question 6

According to a recent study, 1 in every 10 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that more than 22 of the women sampled have not been the victim of domestic abuse.

Accepted Answer

The answer of According to a recent study, 1 in...

Question 7

Suppose a random variable $x$ is best described by a normal distribution with $\mu = 60$ and $\sigma = 5$. Find the $z$-score that corresponds to the value $x = 60$.

Accepted Answer

The answer of Suppose a random variable $x$ is best...

Question 8

A machine is set to pump cleanser into a process at the rate of 7 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 6.5 to 9.5 gallons per minute. Find the probability that between 7.0 gallons and 8.0 gallons are pumped during a randomly selected minute.

Accepted Answer

The probability density function of the uniform distribution is constant over the interval, so the probability of any subinterval is the length of the subinterval divided by the length of the entire interval. In this case, the probability of pumping between 7.0 and 8.0 gallons is (8.0 - 7.0) / (9.5 - 6.5) = 0.33.

Question 9

Transportation officials tell us that 90% of drivers wear seat belts while driving. What is the probability that between 525 and 530 drivers in a sample of 600 drivers wear seat belts?

Accepted Answer

The problem can be solved using the normal approximation to the binomial distribution. The mean (μ) is np = 600 * 0.9 = 540, and the standard deviation (σ) is sqrt(np(1-p)) = sqrt(600 * 0.9 * 0.1) = 7.348. We find the z-scores for 524.5 and 530.5 (using continuity correction) and calculate the probability between these z-scores, which is approximately 0.0811.

Question 10

The probability that an individual is left-handed is 0.13. In a class of 70 students, what is the mean and standard deviation of the number of left-handed students? Round to the nearest hundredth When necessary.

Accepted Answer

The mean number of left-handed students is 0.13 * 70 = 9.1. The standard deviation is sqrt(0.13 * 0.87 * 70) = 2.81.

Question 11

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$. Find the probability that $x$ will assume a value within the interval $\mu \pm 2 \sigma$.

Accepted Answer

The answer of Suppose that the random variable $x$ has...

Question 12

Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $700 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the point in the distribution
Below which 2.5% of the PCE's fell.

Accepted Answer

To find the point below which 2.5% of the PCE's fell, we need to find the z-score that corresponds to the 2.5th percentile, which is approximately -1.96. Using the formula: X = μ + zσ, where μ = $700 and σ = $50, we get X = 700 + (-1.96)(50) = $602.00.

Question 13

The binomial distribution can be used to model the number of rare events that occur over a given time period.

Accepted Answer

The answer of The binomial distribution can be used to...

Question 14

Suppose that x has an exponential distribution with $\theta = 1.5 . \text { Find } P ( x > 1 )$

Accepted Answer

The answer of Suppose that x has an exponential distribution...

Question 15

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the approximate probability that more than 17 and less than 26 of the sampled students had at least one tattoo.

Accepted Answer

Using the normal approximation to the binomial distribution, the mean is 20 and the standard deviation is approximately 3.87. The probability of having more than 17 and less than 26 students with tattoos is approximately 0.6644.

Question 16

A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 40 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 358 seconds.

Accepted Answer

To find the probability, we need to calculate the z-score for 358 seconds using the formula: z = (X - μ) / σ, where X is 358, μ is 450, and σ is 40. The z-score is (358 - 450) / 40 = -2.3. Using the standard normal distribution table, a z-score of -2.3 corresponds to a probability of approximately 0.0107.

Question 17

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the approximate probability that more than 30 of the sampled students had at least one tattoo.

Accepted Answer

The answer of A study of college students stated that...

Question 18

The Fresh Oven Bakery knows that the number of pies it can sell varies from day to day. The owner believes that on 50% of the days she sells 100 pies. On another 25% of the days she sells 150 pies, and she sells 200 pies on the remaining 25% of the days. To make sure she has enough product, the owner bakes 200 pies each day at a cost of $2 each. Assume any pies that go unsold are thrown out at the end of the day. If she sells the pies for $5 each, find the probability distribution for her dailyProfit.

Accepted Answer

The answer of The Fresh Oven Bakery knows that the...

Question 19

Suppose a random variable $x$ is best described by a normal distribution with $\mu = 60$ and $\sigma = 16$. Find the $z$-score that corresponds to the value $x = 0$.

Accepted Answer

The z-score is calculated as (x - mu) / sigma, where x is the value of the random variable, mu is the mean, and sigma is the standard deviation. In this case, x = 0, mu = 60, and sigma = 16. Plugging these values into the formula, we get (0 - 60) / 16 = -3.75. Therefore, the z-score that corresponds to the value x = 0 is -3.75.

Question 20

Assume that $x$ is a binomial random variable with $\mathrm { n } = 400$ and $\mathrm { p } = 0.30$. Use a normal approximation to find $P ( x \geq 100 )$.

Accepted Answer

The correct answer is A.The mean of the binomial distribution is $\mu = np = 400 \times 0.30 = 120$. The standard deviation is $\sigma = \sqrt{np(1-p)} = \sqrt{400 \times 0.30 \times 0.70} = 10.95$.Using the normal approximation to the binomial distribution, we have:$$P(X \geq 100) \approx P(Z \geq \frac{100 - 120}{10.95}) = P(Z \geq -1.82) = 0.9875$$where Z is a standard normal random variable.

Quiz 4: Random Variables and Probability Distributions

Calculate the mean for the discrete probability distribution shown here. $\begin{array} { l | c c c c } X & 3 & 4 & 8 & 11 \\\hline P ( X ) & 0.26 & 0.1 & 0.06 & 0.58\end{array}$

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 10$ and $d = 50$ . Find $P ( x > 50 )$ .

Use the standard normal distribution to find P $\mathrm { P } ( \mathrm { z } < - 2.33 \text { or } z > 2.33 )$

Suppose that x has an exponential distribution with $\theta = 5 . \text { Find } P ( x \leq 10 )$

Given that $x$ is a hypergeometric random variable, compute $p ( x ) \text { for } N = 6 , n = 3 , r = 3 \text {, and } x = 1 \text {. }$

Suppose a random variable $x$ is best described by a normal distribution with $\mu = 60$ and $\sigma = 5$ . Find the $z$ -score that corresponds to the value $x = 60$ .

Transportation officials tell us that 90% of drivers wear seat belts while driving. What is the probability that between 525 and 530 drivers in a sample of 600 drivers wear seat belts?

The probability that an individual is left-handed is 0.13. In a class of 70 students, what is the mean and standard deviation of the number of left-handed students? Round to the nearest hundredth When necessary.

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the probability that $x$ will assume a value within the interval $\mu \pm 2 \sigma$ .

The binomial distribution can be used to model the number of rare events that occur over a given time period.

Suppose that x has an exponential distribution with $\theta = 1.5 . \text { Find } P ( x > 1 )$

Suppose a random variable $x$ is best described by a normal distribution with $\mu = 60$ and $\sigma = 16$ . Find the $z$ -score that corresponds to the value $x = 0$ .

Assume that $x$ is a binomial random variable with $\mathrm { n } = 400$ and $\mathrm { p } = 0.30$ . Use a normal approximation to find $P ( x \geq 100 )$ .

Quiz 4: Random Variables and Probability Distributions

Calculate the mean for the discrete probability distribution shown here. X34811P(X) 0.260.10.060.58\begin{array} { l | c c c c } X & 3 & 4 & 8 & 11 \\\hline P ( X ) & 0.26 & 0.1 & 0.06 & 0.58\end{array}XP(X) ​30.26​40.1​80.06​110.58​​

Suppose xxx is a random variable best described by a uniform probability distribution with c=10c = 10c=10 and d=50d = 50d=50 . Find P(x>50) P ( x > 50 ) P(x>50) .

Use the standard normal distribution to find P P(z<−2.33 or z>2.33) \mathrm { P } ( \mathrm { z } < - 2.33 \text { or } z > 2.33 ) P(z<−2.33 or z>2.33)

Suppose that x has an exponential distribution with θ=5. Find P(x≤10) \theta = 5 . \text { Find } P ( x \leq 10 ) θ=5. Find P(x≤10)

Given that xxx is a hypergeometric random variable, compute p(x) for N=6,n=3,r=3, and x=1. p ( x ) \text { for } N = 6 , n = 3 , r = 3 \text {, and } x = 1 \text {. }p(x) for N=6,n=3,r=3, and x=1.

Suppose a random variable xxx is best described by a normal distribution with μ=60\mu = 60μ=60 and σ=5\sigma = 5σ=5 . Find the zzz -score that corresponds to the value x=60x = 60x=60 .

Transportation officials tell us that 90% of drivers wear seat belts while driving. What is the probability that between 525 and 530 drivers in a sample of 600 drivers wear seat belts?

The probability that an individual is left-handed is 0.13. In a class of 70 students, what is the mean and standard deviation of the number of left-handed students? Round to the nearest hundredth When necessary.

Suppose that the random variable xxx has an exponential distribution with θ=1.5\theta = 1.5θ=1.5 . Find the probability that xxx will assume a value within the interval μ±2σ\mu \pm 2 \sigmaμ±2σ .

The binomial distribution can be used to model the number of rare events that occur over a given time period.

Suppose that x has an exponential distribution with θ=1.5. Find P(x>1) \theta = 1.5 . \text { Find } P ( x > 1 ) θ=1.5. Find P(x>1)

Suppose a random variable xxx is best described by a normal distribution with μ=60\mu = 60μ=60 and σ=16\sigma = 16σ=16 . Find the zzz -score that corresponds to the value x=0x = 0x=0 .

Assume that xxx is a binomial random variable with n=400\mathrm { n } = 400n=400 and p=0.30\mathrm { p } = 0.30p=0.30 . Use a normal approximation to find P(x≥100) P ( x \geq 100 ) P(x≥100) .

Calculate the mean for the discrete probability distribution shown here. $\begin{array} { l | c c c c } X & 3 & 4 & 8 & 11 \\\hline P ( X ) & 0.26 & 0.1 & 0.06 & 0.58\end{array}$

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 10$ and $d = 50$ . Find $P ( x > 50 )$ .

Use the standard normal distribution to find P $\mathrm { P } ( \mathrm { z } < - 2.33 \text { or } z > 2.33 )$

Suppose that x has an exponential distribution with $\theta = 5 . \text { Find } P ( x \leq 10 )$

Given that $x$ is a hypergeometric random variable, compute $p ( x ) \text { for } N = 6 , n = 3 , r = 3 \text {, and } x = 1 \text {. }$

Suppose a random variable $x$ is best described by a normal distribution with $\mu = 60$ and $\sigma = 5$ . Find the $z$ -score that corresponds to the value $x = 60$ .

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the probability that $x$ will assume a value within the interval $\mu \pm 2 \sigma$ .

Suppose that x has an exponential distribution with $\theta = 1.5 . \text { Find } P ( x > 1 )$

Suppose a random variable $x$ is best described by a normal distribution with $\mu = 60$ and $\sigma = 16$ . Find the $z$ -score that corresponds to the value $x = 0$ .

Assume that $x$ is a binomial random variable with $\mathrm { n } = 400$ and $\mathrm { p } = 0.30$ . Use a normal approximation to find $P ( x \geq 100 )$ .