Let Y = (Y₁, Y₂,..., Y_n) and let θ > 0. Let Y|θ ∼ Pois(θ). Derive the posterior density of θ given Y assuming the prior distribution of θ is Gamma(a,b) where a > 1. Then find the prior and posterior means and prior and posterior modes of θ.

Of 180 students in a massive group, 135 are scholarship holders, 146 dedicate part of their time to work; and 114 are scholarship holders and dedicate part of their time to work. If a student is selected at random, what is the probability that: a) The chosen student is awarded a scholarship or dedicates part of his time to work b) The selected student is not awarded a scholarship and does not spend part of his time working c) A sample of 6 students was chosen, which is the probability that the sample will have at least 5 scholars. (Here only consider two groups, the scholars and the non-scholars)

How can I resolve this case study by using R studio from chapter 10 , book business statistics written by Jaggia/Kelly third edition.can you please resolve for me

CASE STUDY 10.3

Paige Thomsen is about to graduate from college at a local university in San Francisco. Her options are to look for a job in San Francisco or go home to Denver and search for work there. Recent data report that average starting salaries for college graduates is $48,900 in San Francisco and $40,900 in Denver (Forbes, June 26, 2008). Suppose these data were based on 100 recent graduates in each city where the population standard deviation is $16,000 in San Francisco and $14,500 in Denver. For social reasons, Paige is also interested in the percent of the population who are in their twenties. The same report states that 20% of the population are in their twenties in San Francisco; the corresponding percentage in Denver is 22%.

In a report, use the sample information to

1. Determine whether the average starting salary in San Francisco is greater than Denver’s average starting salary at the 5% significance level.
2. Determine whether the proportion of the population in their twenties differs in these two cities at the 5% significance level.

The amounts of nicotine in a certain brand of cigarette are normally distributed with a mean of 0.908 g and a standard deviation of 0.319 g. The company that produces these cigarettes claims that it has now reduced the amount of nicotine. The supporting evidence consists of a sample of 46 cigarettes with a mean nicotine amount of 0.856 g.

Assuming that the given mean and standard deviation have NOT changed, find the probability of randomly seleting 46 cigarettes with a mean of 0.856 g or less.
P(x-bar < 0.856 g) =
Enter your answer as a number accurate to 4 decimal places.

Based on the result above, is it valid to claim that the amount of nicotine is lower?

• No. The probability of obtaining this data is high enough to have been a chance occurrence.
• Yes. The probability of this data is unlikely to have occurred by chance alone.

A particular report included the following table classifying 716 fatal bicycle accidents according to time of day the accident occurred.

(a) Assume it is reasonable to regard the 716 bicycle accidents summarized in the table as a random sample of fatal bicycle accidents in that year. Do these data support the hypothesis that fatal bicycle accidents are not equally likely to occur in each of the 3-hour time periods used to construct the table? Test the relevant hypotheses using a significance level of .05. (Round your χ² value to two decimal places, and round your P-value to three decimal places.)

(b) Suppose a safety office proposes that bicycle fatalities are twice as likely to occur between noon and midnight as during midnight to noon and suggests the following hypothesis: H₀: p₁ = 1/3, p₂ = 2/3, where p₁ is the proportion of accidents occurring between midnight and noon and p₂ is the proportion occurring between noon and midnight. Do the given data provide evidence against this hypothesis, or are the data consistent with it? Justify your answer with an appropriate test. (Hint: Use the data to construct a one-way table with just two time categories. Use α = 0.05. Round your χ² value to two decimal places, and round your P-value to three decimal places.)

10. The director of the project wants to test if the weight of wood needed for cooking with the improved stove is significantly less than the weight of wood needed for cooking with the old stove.

A. What are the appropriate hypotheses for this test? Note: Consider the intended test in context of the reduction variable.

B. Based only on the previous parts, which of the following options is true for the value of the p-value for this test? a. The p-value is less than 0.05. b. The p-value is less than 0.10. c. The p-value is greater than 0.05. d. The p-value is greater than 0.10.

C. Based only on the previous parts, is there sufficient evidence to reject the null hypothesis at the 10% level of significance? Explain.

D. Based only on the previous parts, state the appropriate conclusion of the test in context.

E. Interpret the level of significance, α = 0.1, in context.

F. What is the power of this test using the pilot study design to detect an improvement of 0.3 kg? G. Based on the result from part F., what is the probability of a Type II error? please help!

You may assume that the conditions needed for inference to be reliable are satisfied. You may assume (based on many similar studies) that the population standard deviation of reduction of firewood used is 0.7 kg

How can I resolve this case study by using R studio from chapter 10 , book business statistics written by Jaggia/Kelly third edition.can you please resolve for me

CASE STUDY 10.2

The Speedo LZR Racer Suit is a high-end, body-length swimsuit that was launched on February 13,2008. When 17 world records fell at the December 2008 European Short Course Championships in Croatia, many believed a modification in the rules surrounding swimsuits was necessary. The FINA Congress, the international governing board for swimming, banned the LZR Racer and all other body-length swimsuits from competition, effective January 2010. In a statement to the public, FINA defended its position with the following statement: “FINA wishes to recall the main and core principle that swimming is a sport essentially based on the physical performance of the athlete” (BBC Sport, March 14, 2009).

Luke Johnson, a freelance journalist, wonders if the decision made by FINA has statistical backing. He conducts an experiment with the local university’s Division I swim team. He times 10 of the swimmers swimming the 50-meter breaststroke in his/her bathing suit and then retests them while wearing the LZR Racer. A portion of the results is shown in the accompanying table.

Data for Case Study 10.2 50-Meter Breaststroke Times (in seconds)

In a report, use the sample information to

1. Determine whether the LZR Racer improves swimmers’ times at the 5% significance level. Assume that the time difference is normally distributed.
2. Comment on whether the data appear to support FINA’s decision.

The Bahamas is a tropical paradise made up of 700 islands sprinkled over 100,000 square miles of the Atlantic Ocean. According to the figures released by the government of the Bahamas, the mean household income in the Bahamas is $34,803 and the median income is $31,729. A demographer decides to use the lognormal random variable to model this nonsymmetric income distribution. Let Y represent household income, where for a normally distributed X, Y = e^X. In addition, suppose the standard deviation of household income is $13,000. Use this information to answer the following questions. [You may find it useful to reference the z table.]

a. Compute the mean and the standard deviation of X. (Round your intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)

b. What proportion of the people in the Bahamas have household income above the mean? (Round your intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

c. What proportion of the people in the Bahamas have household income below $21,000? (Round your intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

d. Compute the 65th percentile of the income distribution in the Bahamas. (Round your intermediate calculations to at least 4 decimal places, “z” value to 3 decimal places, and final answer to the nearest whole number.)

1. Using a short paragraph explain each part in 10 lines:

a) The meaning of an “interaction” term in a general linear model.

b) The importance of ensuring data points are independent.

c) The value of an orthogonal design.

d) The difference between a random and a fixed effect.

e) The difference between a general linear model and a generalized linear model

f) The difference between a Type I and Type II error.

7) From a population of No. 900 elements, we would like to extract a sample of size n x 15. Using the random number table, designate which 15 individuals make up the sample.

Police records show the following numbers of daily crime reports for a sample of days during the winter months and a sample of days during the summer months.

Use a 0.05 level of significance to determine whether there is a significant difference between the winter and summer months in terms of the number of crime reports.

State the null and alternative hypotheses.

H₀: Median number of daily crime reports for winter − Median number of daily crime reports for summer < 0
H_a: Median number of daily crime reports for winter − Median number of daily crime reports for summer = 0

H₀: The two populations of daily crime reports are identical.
H_a: The two populations of daily crime reports are not identical.    

H₀: Median number of daily crime reports for winter − Median number of daily crime reports for summer ≤ 0
H_a: Median number of daily crime reports for winter − Median number of daily crime reports for summer > 0

H₀: The two populations of daily crime reports are not identical.
H_a: The two populations of daily crime reports are identical.

H₀: Median number of daily crime reports for winter − Median number of daily crime reports for summer ≥ 0
H_a: Median number of daily crime reports for winter − Median number of daily crime reports for summer < 0

Find the value of the test statistic.

W =

Find the p-value. (Round your answer to four decimal places.)

p-value =

What is your conclusion?

Reject H₀. There is sufficient evidence to conclude that there is a significant difference between the winter and summer months in terms of the number of crime reports.

Reject H₀. There is not sufficient evidence to conclude that there is a significant difference between the winter and summer months in terms of the number of crime reports.   

Do not reject H₀. There is sufficient evidence to conclude that there is a significant difference between the winter and summer months in terms of the number of crime reports.

Do not reject H₀. There is not sufficient evidence to conclude that there is a significant difference between the winter and summer months in terms of the number of crime reports.

Samples of starting annual salaries for individuals entering the public accounting and financial planning professions follow. Annual salaries are shown in thousands of dollars.

(a)

Use a 0.05 level of significance and test the hypothesis that there is no difference between the starting annual salaries of public accountants and financial planners.

State the null and alternative hypotheses.

H₀: Median salary for public accountants − Median salary for financial planners ≤ 0
H_a: Median salary for public accountants − Median salary for financial planners > 0

H₀: The two populations of salaries are not identical.
H_a: The two populations of salaries are identical.    

H₀: Median salary for public accountants − Median salary for financial planners > 0
H_a: Median salary for public accountants − Median salary for financial planners = 0

H₀: The two populations of salaries are identical.
H_a: The two populations of salaries are not identical.

H₀: Median salary for public accountants − Median salary for financial planners ≥ 0
H_a: Median salary for public accountants − Median salary for financial planners < 0

Find the value of the test statistic.

W =

Find the p-value. (Round your answer to four decimal places.)

p-value =

What is your conclusion?

Reject H₀. There is sufficient evidence to conclude that there is a significant difference between the starting annual salaries of public accountants and financial planners.

Do not reject H₀. There is sufficient evidence to conclude that there is a significant difference between the starting annual salaries of public accountants and financial planners.   

Do not reject H₀. There is not sufficient evidence to conclude that there is a significant difference between the starting annual salaries of public accountants and financial planners.

Reject H₀. There is not sufficient evidence to conclude that there is a significant difference between the starting annual salaries of public accountants and financial planners.

(b)

What are the sample median annual salaries (in $) for the two professions?

Public Accountants sample median=$

Financial Planners sample median=$

what is a grant and what does grant seeker want to see in an application?

l. Find the standard deviation.

            m. Find the coefficient of variation.

            n. What is the 80^th percentile of this data set?

            o. Does this data set have any outliers? Use statistics in answering this question (not just your opinion).

            p. Would you use the Empirical rule or Chebyshev’s Theorem here and why?

            q. Based on the mean and median calculated above, would you expect this data to be symmetric, skewed to the left or skewed to the right and why?

Requirements:

Moving Averages. Use the below actual sales to calculate a one-year average which will be used as the forecast for next periods (chapter 14, text). Choose a moving average period that best supports this calculation.

Exponential Smoothing. Use the same data to forecast sales for the next periods with α=.40 (chapter 14, text).

Regression Analysis on Excel. Draw a scatter graph from Insert/Graph/Scatter graph selections in Excel (chapter 15, text).

Month Actual Sales

1 3050

2 2980

3 3670

4 2910

5 3340

6 4060

7 4750

8 5510

9 5280

10 5504

11 5810

12 6100