A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 40% of the time she travels on airline #1, 20% of the time on airline #2, and the remaining 40% of the time on airline #3. For airline #1, flights are late into D.C. 40% of the time and late into L.A. 15% of the time. For airline #2, these percentages are 30% and 15%, whereas for airline #3 the percentages are 35% and 20%. If we learn that on a particular trip she arrived late at exactly one of the two destinations, what are the posterior probabilities of having flown on airlines #1, #2, and #3? Assume that the chance of a late arrival in L.A. is unaffected by what happens on the flight to D.C. [Hint: From the tip of each first-generation branch on a tree diagram, draw three second-generation branches labeled, respectively, 0 late, 1 late, and 2 late.] (Round your answers to four decimal places.)

Appropriate sampling is a critical component in developing a good research project. Using your approved research questions and research topic, explain your anticipated sampling method and why this is appropriate for your research proposal. What is your sample size? Next, read and review two of your classmates’ posts and analyze their sampling approach. Are their sampling approaches appropriate? Why or why not?

The number of floods that occur in a certain region over a given year is a random variable having a Poisson distribution with mean 2, independently from one year to the other. Moreover, the time period (in days) during which the ground is flooded, at the time of an arbitrary flood, is an exponential random variable with mean 5. We assume that the durations of the floods are independent. Using the central limit theorem, calculate (approximately)
(a) the probability that over the course of the next 50 years, there will be at least 80 floods in this region. Assume that we do not need to apply half-unit correction for this question.

(b) the probability that the total time during which the ground will be flooded over the course of the next 50 floods will be smaller than 200 days.

A Theater has n numbered seats, and n tickets are distributed among n persons. Compute the probability that

(a) exactly two persons will be seated at seats corresponding to their ticket numbers if all the seats are occupied at random.

(b) at least two persons will be seated at seats corresponding to their ticket numbers if all the seats are occupied at random.

A turfgrass scientist is looking for an optimal approach to control a certain plant disease in Kentucky blue grass. He compares three different management strategies designed so that they would reduce spread of the disease. He has set up a field study with a total of 20 experimental plots and 4 treatments (3 disease prevention treatments and a control treatment. Each treatment has been assigned to 5 randomly selected experimental plots. Plant biomass is then measured from each pot at the end of the experiment. ANOVA table and the treatment means are shown below.

ANOVA:

a) Do all pairwise comparisons between the treatment means using LSD, (a=0.05). Present the results using letters assigned to treatment means (Put the letters in the column Letters for part a) in the above table)

b) Do all pairwise comparisons between the treatment means using Tukey’s HSD (a=0.05). Present the results using letters assigned to treatment means. (Put the letters in the column Letters for part b) in the above table)

c) Comment on differences in conclusions obtained using the two methods. Which method would you use for this analysis? For full credit, provide an explanation of your choice.

Let X be a random variable with CDF F(x) = e^-e(µ-x)/β, where β > 0 and -∞ < µ, x < ∞.

1. What is the median of X?

2. Obtain the PDF of X. Use R to plot, in the range -10<x<30, the pdf for µ = 2, β = 5.

3. Draw a random sample of size 1000 from f(x) for µ = 2, β = 5 and draw a histogram of the values in the random sample drawn. Compare this histogram with the answer in 2 above.

You may need to use the appropriate appendix table or technology to answer this question.

Individuals filing federal income tax returns prior to March 31 received an average refund of $1,053. Consider the population of "last-minute" filers who mail their tax return during the last five days of the income tax period (typically April 10 to April 15).

(a)

A researcher suggests that a reason individuals wait until the last five days is that on average these individuals receive lower refunds than do early filers. Develop appropriate hypotheses such that rejection of

H0

will support the researcher's contention.

H0: μ > $1,053
Ha: μ ≤ $1,053H0: μ = $1,053
Ha: μ ≠ $1,053    H0: μ ≥ $1,053
Ha: μ < $1,053H0: μ < $1,053
Ha: μ ≥ $1,053H0: μ ≤ $1,053
Ha: μ > $1,053

(b)

For a sample of 400 individuals who filed a tax return between April 10 and 15, the sample mean refund was $910. Based on prior experience, a population standard deviation of

σ = $1,600

may be assumed.

What is the test statistic? (Round your answer to two decimal places.)

What is the p-value? (Round your answer to four decimal places.)

p-value =

(c)

At

α = 0.05,

what is your conclusion?

Do not reject H0. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.Reject H0. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.    Reject H0. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less than or equal $1,053.Do not reject H0. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less or equal than $1,053.

(d)

Repeat the preceding hypothesis test using the critical value approach.

State the null and alternative hypotheses.

H0: μ > $1,053
Ha: μ ≤ $1,053H0: μ = $1,053
Ha: μ ≠ $1,053    H0: μ ≥ $1,053
Ha: μ < $1,053H0: μ < $1,053
Ha: μ ≥ $1,053H0: μ ≤ $1,053
Ha: μ > $1,053

Find the value of the test statistic. (Round your answer to two decimal places.)

State the critical values for the rejection rule. (Use α = 0.05. Round your answer to two decimal places. If the test is one-tailed, enter NONE for the unused tail.)

test statistic≤test statistic≥

State your conclusion.

Do not reject H0. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.Reject H0. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.    Reject H0. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less than or equal $1,053.Do not reject H0. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less or equal than $1,053.

Which variables measure level of happiness? using Descriptive statistics and bivariate statistics.

List the characteristics of a multinomial experiment. (Select all that apply.)

The number of successes is evenly distributed over all k categories.

We are interested in x, the number of events that occur in a period of time or space.

The trials are independent.

The experiment consists of n identical trials.

Its mean is 0 and its standard deviation is 1.

The outcome of each trial falls into one of k categories.

The experiment contains M successes and N − M failures.

The probability that the outcome of a single trial falls into a particular category remains constant from trial to trial.

The probability that the outcome of a single trial falls between two categories is equal to the area under the curve between those categories.

We are interested in x, the number of successes observed during the n trials.

The experimenter counts the observed number of outcomes in each category.

Each trial results in one of only two possible outcomes.

A diagnostic test has a 95% probability of giving a positive result when given to a person who has a certain disease. It has a 10% probability of giving a (false) positive result when given to a person who doesn’t have the disease. It is estimated that 15% of the population suffers from this disease.

(a) What is the probability that a test result is positive?

(b) A person recieves a positive test result. What is the probability that this person actually has the disease? (probability of a true positive)

(c) A person recieves a positive test result. What is the probability that this person doesn’t actually have the disease? (probability of a false negative)

Two cards are drawn one after the other from a standard deck of 52 cards.

(a) In how many ways can one draw first a spade and then a heart?
(b) In how many ways can one draw first a spade and then a heart or a diamond?
(c) In how many ways can one draw first a spade and then another spade?

(d) Do the previous answers change if the first card is put back in the deck before the second card is drawn?

Note that the book states to use a value of 25% if you don’t know what a good value is for the population estimate. Would we want to use this value in planning election polling? Why or why not? What would be the sample sizes needed to get a 95% confidence interval of plus or minus 3% given that the initial estimate of the population proportion is either 1%, 25%, 50%, 75% or 99% (calculate the five intervals). What do you notice that is interesting?

A biologist looked at the relationship between number of seeds a plant produces and the percent of those seeds that sprout. The results of the survey are shown below.

1. Find the correlation coefficient: r=r=   Round to 2 decimal places.
2. The null and alternative hypotheses for correlation are:
   H0:H0: ?ρμr == 0
   H1:H1: ?rρμ ≠≠ 0
   The p-value is: (Round to four decimal places)
   
3. Use a level of significance of α=0.05α=0.05 to state the conclusion of the hypothesis test in the context of the study.
   • There is statistically significant evidence to conclude that a plant that produces more seeds will have seeds with a lower sprout rate than a plant that produces fewer seeds.
   • There is statistically insignificant evidence to conclude that a plant that produces more seeds will have seeds with a lower sprout rate than a plant that produces fewer seeds.
   • There is statistically insignificant evidence to conclude that there is a correlation between the number of seeds that a plant produces and the percent of the seeds that sprout. Thus, the use of the regression line is not appropriate.
   • There is statistically significant evidence to conclude that there is a correlation between the number of seeds that a plant produces and the percent of the seeds that sprout. Thus, the regression line is useful.
4. r2r2 = (Round to two decimal places)
5. The equation of the linear regression line is:   
   ˆyy^ = + xx   (Please show your answers to two decimal places)
   
6. Use the model to predict the percent of seeds that sprout if the plant produces 57 seeds.
   Percent sprouting = (Please round your answer to the nearest whole number.)

Please answer the following:

1. United Dairies, Inc., supplies milk to several independent grocers throughout Dade County, Florida. Managers at United Dairies want to develop a forecast of the number of half gallons of milk sold per week. Sales data for the past 12 weeks are:

1. Compute four-week and five-week moving averages for the time series.

1. 1. Compute the MSE for the four-week and five-week moving average forecasts.
   2. What appears to be the best number of weeks of past data (three, four, or five) to use in the moving average computation? The MSE for three weeks is 23527.78
2. Show the exponential smoothing forecasts using α = 0.1.
   1. Applying the MSE measure of forecast accuracy, would you prefer a smoothing constant of α=0.1 or α= 0.2 for the United Dairies sales time series?
   2. Are the results the same if you apply MAE as the measure of accuracy?
   3. What are the results if MAPE is used?

3.    Use exponential smoothing with a α = 0.4 to develop a forecast of demand for week 13. What is the resulting MSE?