Suppose you have a bag with 18 small colored stones. There are 6 red stones, 7 green stones, and the rest of the stones  are blue.

a. Would reaching into the bag, pulling out a stone, and recording the color be considered a random experiment? Explain your answer.

b. If you reach into the bag and pull out a stone, what is the probability that the stone is blue?

c. What is the probability that you will not pull out a blue stone?

d. Suppose you pull out a red stone. If you do not replace the first stone, what is the probability that you will pull out a blue stone this time?

Consider a new hotel deciding on cleaning staff hiring for the upcoming season. Cleaning times depend on whether it is a stay-over room or a check-out. Suppose that a guest will check-out on a given day with probability 40%. From your experience in similar hotels you estimate that a stay-over room cleaning time is well-described with normal distribution with average 15 minutes and standard deviation 1 minute. Check-out room cleaning time is also normal but with average 30 minutes and standard deviation 10 minutes.

i. Consider an occupied room (stay-over or check-out), what is the average cleaning time for such a room?

ii. Find the variance for the cleaning time for an occupied room.

iii. Suppose that the hotel has 200 rooms, and you estimate that on a given day a room will be occupied with probability 90%. Only occupied rooms need cleaning. Find the average total cleaning time for the hotel. iv. Find the variance of the total cleaning time for the hotel.

Hints: remember var(X) = EX^2 − (EX)^2 .

betting theory on tennis

A bookmaker has quoted odds on a tennis match between players I and II. The match consists of the best two out of three sets, i.e., if a player wins the first two sets, the third set is not played and the bet on it is canceled. The bookmaker is giving odds of 5 to 2 that player I will win the match and odds of 3 to 2 that player I will win each set. A bettor has 100 dollars which he can distribute by betting on either player I or II to win the match and any of the sets. All bets are made before the match starts (if there are only two sets, all bets on the third set are returned to the bettor).
(a) Find a way of placing bets so that no matter what happens the bettor is assured of winning an amount z where z is as large as possible. Formulate this problem as a linear programming and solve it using AMPL.
(b) What if now we have best three out of five sets, i.e., once a player wins three sets, no more sets are played and their corresponding bets are canceled, and everything else keeps the same? Re-solve the problem and compare the answer with part (a).

Researcher conducts a study to decide whether support groups improve academic performance for at-risk high school students. Ten such students are randomly selected to take part in the support group for a semester, while the other 10 at-risk students serve as a control group. At the end of the semester, the improvement in GPA versus the previous semester is recorded for each student.
Support Group: 0.5, 0.8, 0.7, 0.7, -0.1, 0.2, 0.4, 0.4, 0.5, 0.4
Control Group: -0.3, 0.0, -0.1, 0.2, -0.1, -0.2, -0.2, 0.0, -0.1, 0.1

At the 10% level, use R to compare the two groups using a permutation test (with 100,000 randomly generated permutations). You need to write your hypotheses, the test statistic, the pvalue, and the decision/conclusion in the context of the problem.

R code for reference:

SupportGroup <- c(0.5, 0.8, 0.7, 0.7, -0.1, 0.2, 0.4, 0.4, 0.5, 0.4)
ControlGroup <- c(-0.3, 0.0, -0.1, 0.2, -0.1, -0.2, -0.2, 0.0, -0.1, 0.1)

mean(SupportGroup);sd(SupportGroup)
mean(ControlGroup);sd(ControlGroup)

#permutation test on difference of means
choose(20,10)#number of possible permutations
new.dat <- c(SupportGroup,ControlGroup)
obs.mean.diff <- mean(SupportGroup) - mean(ControlGroup)
nsim <- 100000
sim.mean.diff <- rep(NA,length=nsim)
for (i in 1:nsim){
grps <- sample(c(rep(1,10),rep(2,10)),replace=FALSE)
sim.mean.diff[i] <- mean(new.dat[grps==1]) - mean(new.dat[grps==2])
}

hist(sim.mean.diff);abline(v=obs.mean.diff,col="red",lty=2)
length(sim.mean.diff[sim.mean.diff<=obs.mean.diff])/nsim #estimated p-value

A random sample of 49 measurements from one population had a sample mean of 16, with sample standard deviation 3. An independent random sample of 64 measurements from a second population had a sample mean of 18, with sample standard deviation 4. Test the claim that the population means are different. Use level of significance 0.01. (a) What distribution does the sample test statistic follow? Explain.

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.     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.

(b) State the hypotheses.

H₀: μ₁ ≠ μ₂; H₁: μ₁ = μ₂H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂     H₀: μ₁ = μ₂; H₁: μ₁ < μ₂H₀: μ₁ = μ₂; H₁: μ₁ > μ₂

(c) Compute

x₁ − x₂.

x₁ − x₂ =

Compute the corresponding sample distribution value. (Test the difference μ₁ − μ₂. Round your answer to three decimal places.)

An eating disorders clinic would like to assess the efficacy of their 10-week mindfulness training program with clients who have Binge Eating Disorder (BED). The clinic researchers first measured the number of binges in the previous week for 16 clients through self-report. One month after the mindfulness training sessions were conducted, the clients were again asked to report the number of binges in the last week. The data are listed in the table below. The clinic researchers have set the significance level at α = .05. # of Binges per week Subject Before Training After Training

Part I. (25 points total) a) Identify the outcome (dependent) variable and the independent variable (that differentiates the two populations being compared). What are the “samples” in this paired-samples t test? (Or, what are the “means” in this dependent-means t test?) (1 point) b) The clinic researchers predict the number of binges per week will decrease after the mindfulness training. In other words, the researchers believe the mindfulness training will be helpful in reducing binge eating. What would be the null and alternative hypotheses in both words and symbol notations? c) Calculate the difference scores by subtracting the “before” scores from the “after” scores. (In other words, set up the columns to calculate after minus before.) Create a table below for “difference score.” d) Calculate the mean from the sample of the difference scores. e) Estimate the standard deviation of the comparison population (that represents the null hypothesis). f) Calculate the standard error (standard deviation of the sampling distribution). g) Calculate the t statistic for the sample. h) Because the hypotheses are directional, a one-tailed test can be performed. Determine the critical t value based on the degrees of freedom and the preset alpha level. Compare the t statistic with the critical t value. Is the calculated t statistic more extreme or less extreme than the critical t value? Then make a decision about the hypothesis test, stating explicitly “reject” or “fail to reject” accordingly. i) Interpret the result in 1-2 sentences (you may restate the hypothesis accepted or explain it in your own words). ( a) Calculate the raw and standardized effect size of this hypothesis test. The clinic researchers could also set up the hypothesis to see if there are any differences (increases or decreases) in binge eating behavior after mindfulness training. a) What would be the null and alternative hypotheses for this alternative analysis? Compose them in symbol notations only. b) Since a non-directional hypothesis is examined with a two-tailed test, determine the critical t values for the two-tailed test using the same alpha level and degree of freedom. c) Compare the t statistic with the critical t values. Is the calculated t statistic more extreme or less extreme than the critical t value? What is the decision of the hypothesis test now? d) Was the two-tailed test result (Part II) different from the one-tailed test result (from Part I)? Why or why not?

This is the first question and I know how to solve this one, but I am confused by the second one (The admissions office of a small, selective liberal-arts college will only offer admission to applicants who have a certain mix of accomplishments, including a combined SAT score of 1,300 or more. Based on past records, the head of admissions feels that the probability is 0.58 that an admitted applicant will come to the college. If 500 applicants are admitted, what is the probability that 310 or more will come? Note that “310 or more” means the set of values {310, 311, 312, …, 500}. )

The following is the second question

Consider the admissions office in the previous problem. Based on financial considerations, the college would like a class size of 310 or more. Find the smallest n, number of people to admit, for which the probability of getting 310 or more to come to the college is at least 0.95.

The data provided give the gasoline mileage​ (in miles per​ gallon) based on the horsepower of a​ car's engine and the weight of the car​ (in pounds). Using the data​ provided, determine the VIF for each independent variable in the model. Is there reason to suspect the existence of​ collinearity?

Determine the VIF for each independent variable in the model.

Round to three decimal places as​ needed.

An experiment on memory was performed, in which 16 subjects were randomly assigned to one of two groups, called "Sentences" or "Intentional". Each subject was given a list of 50 words. Subjects in the "Sentences" group were told to form multiple sentences, each using at least two words from the list, and to keep forming sentences until all the words were used at least once. Subjects in the "Intentional" group were told to spend five minutes memorizing as many of the 50 words as possible. Subjects from both groups were then asked to write down as many words from their lists as they could recall. The data are in the table below. Number of words recalled "Sentences" group 29 30 35 33 32 29 33 34 "Intentional" group 31 36 36 32 34 33 30 33 Enter this data into JMP in "long form" (e.g. each column should be a variable and each row should be an observation). We are interested in determining if there is a significant difference in the average number of words recalled for subjects in the "sentences" group vs. subjects in the "intentional" group, using α = 0.05. Use JMP to answer the questions below, and round all answers to three decimal places.

standard error of (x_sentences - x_intentional) =
test statistic: t =  
p-value =  

Report the 95% confidence interval JMP gives for μ_sentences - μ_intentional

Lower bound =  
Upper bound =

From these results, our statistical conclusion should be:
(You have two attempts at this question.)

a.The means for "sentences" and "intentional" differ significantly, because the p-value is less than α and zero is inside the confidence interval

b. The means for "sentences" and "intentional" differ significantly, because the p-value is less than α and zero is outside the confidence interval  

c. The means for "sentences" and "intentional" differ significantly, because the p-value is less than α and -1.25 is inside the confidence interval

d. The means for "sentences" and "intentional" differ significantly, because the p-value is less than α and -1.25 is outside the confidence interval

e.The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and zero is inside the confidence interval

f.The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and zero is outside the confidence interval

g.The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and -1.25 is inside the confidence interval

h. The means for "sentences" and "intentional" do not differ significantly, because the p-value is greater than α and -1.25 is outside the confidence interval

Suppose as part of a national study of economic competitiveness a marketing research firm randomly sampled 200 adults between the ages of 27 and 35 living in metropolitan Seattle and 180 adults between the ages of 27 and 35 living in metropolitan Minneapolis. Each adult selected in the sample was asked, among other things, whether they had a college degree. From the Seattle sample 66 adults answered yes and from the Minneapolis sample 63 adults answered yes when asked if they had a college degree. Based on the sample data, can we conclude that there is a difference between the population proportions of adults between the ages of 27 and 35 in the two cities with college degrees? Use a level of significance of 0.10 to conduct the appropriate hypothesis test.

Group of answer choices

A Since the test statistic, 1.8214, is greater than the critical value of 1.645, reject the null hypothesis and conclude that there is a higher proportion of Seattle adults that have a college degree

B Since the test statistic, 2.0112, is greater than the critical value of 1.645, reject the null hypothesis and conclude that there is a higher proportion of Seattle adults that have a college degree.

C Since the test statistic, 0.8921, is not greater than the critical value of 1.645, do not reject the null hypothesis and conclude that there is not a higher proportion of Seattle adults that have a college degree.

D Since the test statistic, -0.411, is not greater than the critical value of 1.645, do not reject the null hypothesis and conclude that there is not a higher proportion of Seattle adults that have a college degree.

Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data.

(a) Compute Σx, Σx², Σy, Σy².

(b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to two decimal places.)

(c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)

Use the intervals to compare the two funds.

75% of the returns for the balanced fund fall within a narrower range than those of the stock fund.75% of the returns for the stock fund fall within a narrower range than those of the balanced fund.    25% of the returns for the balanced fund fall within a narrower range than those of the stock fund.25% of the returns for the stock fund fall within a wider range than those of the balanced fund.

You have been asked to conduct a study related to consumer loyalty toward three different retail formats (i.e., department stores, specialty stores, and off-price retailers). After gathering background information, you decide to focus your study on three research streams; retail service quality, consumer satisfaction, and consumer loyalty. The literature suggests that there are five dimensions of retail service quality; physical aspect, reliability, personal interaction, problem solving, and policy. According to the literature, consumer satisfaction is a unidimensional construct and consumer loyalty consists of two dimensions; word-of-mouth and behavioral intention. You want to learn whether or not influencing factors of consumer loyalty are different between these three types of retailers.

State the null hypothesis and the alternative hypothesis

What are the dependent and independent variables

What statistical test would you run in SPSS

Dr. Mack Lemore, an expert in consumer behavior, wants to estimate the average amount of money that people spend in thrift shops. He takes a small sample of 8 individuals and asks them to report how much money they had in their pockets the last time they went shopping at a thrift store. Here are the data: 13.66, 41.35, 21.43, 10.49, 25.57, 37.04, 17.5, 27.07. Find the lower bound of a 98% confidence interval for the true mean amount of money individuals carry with them to thrift stores, to two decimal places. Take all calculations toward the final answer to three decimal places.

In a small town, there are 5 high school districts. Each district includes 100 high school

students. In total, there are 500 high school students in the town including 240 male

students and 260 female students. Researchers would like to select a sample of 200

students.

(a) Explain how the sample can be obtained using random cluster sampling.

(b) Explain how the sample can be obtained using stratified random sampling.

(c) Explain how the sample can be obtained using systematic random sampling.