If A and B are independent, then the probability of A occurring is not affected by event B occurring. Therefore P(A∣B)=P(A). Since P(A)=0.3, then P(A∣B)=0.3.

Question

asked 2021-05-05

A random sample of \( n_1 = 14 \) winter days in Denver gave a sample mean pollution index \( x_1 = 43 \).

Previous studies show that \( \sigma_1 = 19 \).

For Englewood (a suburb of Denver), a random sample of \( n_2 = 12 \) winter days gave a sample mean pollution index of \( x_2 = 37 \).

Previous studies show that \( \sigma_2 = 13 \).

Assume the pollution index is normally distributed in both Englewood and Denver.

(a) State the null and alternate hypotheses.

\( H_0:\mu_1=\mu_2.\mu_1>\mu_2 \)

\( H_0:\mu_1<\mu_2.\mu_1=\mu_2 \)

\( H_0:\mu_1=\mu_2.\mu_1<\mu_2 \)

\( H_0:\mu_1=\mu_2.\mu_1\neq\mu_2 \)

(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.

(Test the difference \( \mu_1 - \mu_2 \). Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)

(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?

At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are statistically significant.

At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are not statistically significant.

(f) Interpret your conclusion in the context of the application.

Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.

Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for

\( \mu_1 - \mu_2 \).

(Round your answers to two decimal places.)

lower limit

upper limit

(h) Explain the meaning of the confidence interval in the context of the problem.

Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.

Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.

Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.

Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.

asked 2021-03-07

This problem is about the equation

dP/dt = kP-H , P(0) = Po,

where k > 0 and H > 0 are constants.

If H = 0, you have dP/dt = kP , which models expontialgrowth. Think of H as a harvesting term, tending to reducethe rate of growth; then there ought to be a value of H big enoughto prevent exponential growth.

Problem: find acondition on H, involving \(\displaystyle{P}_{{0}}\) and k, that will prevent solutions from growing exponentially.

dP/dt = kP-H , P(0) = Po,

where k > 0 and H > 0 are constants.

If H = 0, you have dP/dt = kP , which models expontialgrowth. Think of H as a harvesting term, tending to reducethe rate of growth; then there ought to be a value of H big enoughto prevent exponential growth.

Problem: find acondition on H, involving \(\displaystyle{P}_{{0}}\) and k, that will prevent solutions from growing exponentially.

asked 2021-05-12

Suppose that \(X_{1}, X_{2} and X_{3}\) are three independent random variables with the same distribution as X.

What is the ecpected value of the sum \(X_{1}+X_{2}+X_{3}\)? The product \(X_{1}X_{2}X_{3}\)?

Suppose a discrete random variable X assumes the value \(\frac{3}{2}\) with probability 0.5 and assumes the value \(\frac{1}{2}\) with probability 0.5.

What is the ecpected value of the sum \(X_{1}+X_{2}+X_{3}\)? The product \(X_{1}X_{2}X_{3}\)?

Suppose a discrete random variable X assumes the value \(\frac{3}{2}\) with probability 0.5 and assumes the value \(\frac{1}{2}\) with probability 0.5.

asked 2021-01-24

If you used a random number generator for the numbers from 1 through 20 to play a game, what is the theoretical probability of getting each of these outcomes? a. A multiple of 3 or a multiple of 7, P(multiple of 3 or multiple of 7) b. P( even or odd) c. P(prime or 1) d. How did you find the probabilities of these events?

asked 2021-05-05

The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with \(\displaystyle\mu={1.5}\) and \(\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}\).

(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.

(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).

(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.

(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?

(e) What is the moment generating function for X?

(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.

(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).

(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.

(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?

(e) What is the moment generating function for X?

asked 2021-02-22

A truck engine transmits 28 kW (27.5 hp) to the driving wheels when the truck is traveling at a constant velocity of magnitude 60 km/h (37.3 m/h) on a level road. (a) What is the resisting force acting on the truck? (b) Assume that 65% of the resisting force is due to rolling friction and that the remainder is due to air resistance. If the force of rolling friction is independent of speed and the force of air resistance is proportional to the square of speed, what power will drive the truck at 30 km/h? at 120 km/h? give your answers in kilowatts and in horsepower.

asked 2020-11-08

According to Exercise 16, the probability that a U.S. resident has traveled to Canada is 0.18, to Mexico is 0.09, and to bith countries is 0.04.

What's the probability that someone who has traveled to Mexico has visited Canada too?

Are traveling to Mexico and to Canada disjoint events?

Are traveling to Mexico and to Canada independent events?

Explain.

What's the probability that someone who has traveled to Mexico has visited Canada too?

Are traveling to Mexico and to Canada disjoint events?

Are traveling to Mexico and to Canada independent events?

Explain.

asked 2021-01-28

Data is being processed from 35 data sources. Each dataset may or may or net generate dataset at the beginning of each timeslot, the probability that any individual source actually generates a dataset is 0.004,and the data sources are independent. In each time slot we can process up to two datasets. Now, the processing is real time with the processed results only being considered if they are done within the time slot. Determine the probability that all incoming datasets can be processed in any particular time slot?

asked 2021-05-23

Random variables \(X_{1},X_{2},...,X_{n}\) are independent and identically distributed. 0 is a parameter of their distribution.

If \(q(X,0)\sim N(0,1)\) is a pivotal function for 0, explain how you would use this result to obtain a symmetrical 95% confidence interval for 0.

If \(q(X,0)\sim N(0,1)\) is a pivotal function for 0, explain how you would use this result to obtain a symmetrical 95% confidence interval for 0.

asked 2021-05-01

If two random variables X and Y are independent with marginal pdfs \(fx(x)= 2x, 0\leq x \leq 1 and fy(y)=1,0\leq y \leq 1\)

Calculate P(Y|X>2)

Calculate P(Y|X>2)