# Give a complete and correct answer to the question asked what Posterior Probabilities?

Question
Modeling data distributions
Give a complete and correct answer to the question asked what Posterior Probabilities?

2021-01-24
A posterior probability, in Bayesian statistics, is the revised or updated probability of an event occurring after taking into consideration new information. The posterior probability is calculated by updating the prior probability using Bayes' Theorem. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred.
A posterior probability, in Bayesian statistics, is the revised or updated probability of an event occurring after taking into consideration new information.
The posterior probability is calculated by updating the prior probability using Bayes' theorem.
In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred.
Bayes' theorem can be used in many applications, such as medicine, finance, and economics. In finance, Bayes' theorem can be used to update a previous belief once new information is obtained. Prior probability represents what is originally believed before new evidence is introduced, and posterior probability takes this new information into account. Posterior probability distributions should be a better reflection of the underlying truth of a data generating process than the prior probability since the posterior included more information. A posterior probability can subsequently become a prior for a new updated posterior probability as new information arises and is incorporated into the analysis.

### Relevant Questions

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
$$\mu_1 - \mu_2$$.
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.
The following table represents the Frequency Distribution and Cumulative Distributions for this data set: 12, 13, 17, 18, 18, 24, 26, 27, 27, 30, 30, 35, 37, 41, 42, 43, 44, 46, 53, 58 Class Frequency Relative Cumulative Frequency Frequency 10 but less than 20 5 20 but less than 30 4 30 but less than 4 4 40 but less than 50 5 50 but less than 60 2 TOTAL What is the Relative Frequency for the class: 20 but less than 30? State you answer as a value with exactly two digits after the decimal. for example 0.30 or 0.35

Give a full correct answer for given question 1- Let W be the set of all polynomials $$\displaystyle{a}+{b}{t}+{c}{t}^{{2}}\in{P}_{{{2}}}$$ such that $$\displaystyle{a}+{b}+{c}={0}$$ Show that W is a subspace of $$\displaystyle{P}_{{{2}}},$$ find a basis for W, and then find dim(W) 2 - Find two different bases of $$\displaystyle{R}^{{{2}}}$$ so that the coordinates of $$b= \begin{bmatrix} 5\\ 3 \end{bmatrix}$$ are both (2,1) in the coordinate system defined by these two bases

Give a complete answer and describe Bayes' theorem
Give full and correct answer for this questions 1) A t-test is a ? 2) Which of the following statement is true? a)The less likely one is to commit a type I error, the more likely one is to commit a type II error, b) A type I error has occurred when a false null hypothesis has been wrongly accepted. c) A type I error has occurred when a two-tailed test has been performed instead of a one-tailed test, d) None of the above statements is true. 3)Regarding the Central Limit Theorem, which of the following statement is NOT true? a.The mean of the population of sample means taken from a population is equal to the mean of the original population. b. The frequency distribution of the population of sample means taken from a population that is not normally distributed will approach normality as the sample size increases. c. The standard deviation of the population of sample means is equal to the standard deviation of the, original population. d. The frequency distribution of the population of sample means taken from a population that is not normally distributed will show less dispersion as the sample size increases.
Solve and give the correct answer with using second derivative of the function as follows $$\displaystyle f{{\left({x}\right)}}={\left({x}+{9}\right)}^{2}$$
1. The standard error of the estimate is the same at all points along the regression line because we assumed that A. The observed values of y are normally distributed around each estimated value of y-hat. B. The variance of the distributions around each possible value of y-hat is the same. C. All available data were taken into account when the regression line was calculated. D. The regression line minimized the sum of the squared errors. E. None of the above.
Census data are often used to obtain probability distributions for various random variables. Census data for families in a particular state with a combined income of \$50,000 or more show that 22% of these families have no children, 30% have one child, 27% have two children, and 21% have three children. From this information, construct the probability distribution for x, where x represents the number of children per family for this income group. (Give your answers correct to two decimal places.) $$P(0 \text{children}) =$$
$$P(1 \text{child}) =$$
$$P(2 \text{children}) =$$
$$P(3 \text{children}) =$$
Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task. $$\displaystyle{\left({1},{3}\right)},{\left({2},{6}\right)},{\left({3},{12}\right)},{\left({4},{24}\right)}$$
a. Test for a difference in the means in the two populations using an $$[\alpha={.05}{t}-{t}{e}{s}{t}.]$$