Two events ?1 and ?2 are defined on the same probability space such that: ?(?1 ) = 0.7 , ?(?2 ) = 0.5 , and ?(?1 ??? ?2 ) = 0.3. a) Find ?(?1 ?? ?2 ). b) Find ?(?1 | ?2 ). c) Are ?1 and ?2 mutually exclusive (disjoint)? and why? d) Are ?1 and ?2 independent? and why?

he mayor of a town has proposed a plan for the construction of an adjoining bridge. A political study took a sample of 1000 voters in the town and found that 57% of the residents favored construction. Using the data, a political strategist wants to test the claim that the percentage of residents who favor construction is above 53%. Make the decision to reject or fail to reject the null hypothesis at the 0.20 level.

Chicago is a group dice game that requires no skill. The objective of the game is to accumulate points by rolling certain combinations (GamezBuff, 2017). How do you play Chicago? There are eleven rounds in the game, one for each combination that can be made by adding two dice, namely the numbers two through 12. Each round has a target combination starting with two and going up all the way to 12. Going clockwise, the players take turns to roll both dice one time. If players roll the target combination, then they score points equal to the target combination, otherwise they score zero. For example, if the round corresponds to target combination six, then the player scores six points if the two dice add up to six. Else, the player scores no points. The player with the highest score at the end of the eleventh round wins the game.

1. What is the probability that the player rolls the target combination?

One of the leading television providers has estimated the following demand equation after analyzing 36 regional markets:

  Q =   + 25,000 – 45P + 20A + 25Pc - 30Ac + 110 I

   (12000) (20.2) (14)   (9)      (48)       (50)                     

          R^{2   =} 0.86           F = 28.52

          The variables and their assumed values are

          Q = Quantity

          P = Price of the basic Model = 1200 (dollars)

          A = Advertising Expenditures = 90 (thousand dollars)

Pc = Average price of the competitor’s product = 1400 (dollars)

Ac = competitor’s advertising expenditures = 80 (thousand dollars)

  I = per capita income = 60 (thousand dollars)

1. Compute the elasticities for each variable. On this basis, discuss the relative impact that each variable has on the demand. What implications do these results have for the firm’s marketing, pricing, and production policies?
2. What would be the effect of a 6 unit increase in the competitor’s advertising expenditures?
3. What would be the change in your advertising expenditures to offset your competitor’s strategy?
4. Conduct a t-test for the statistical significance of each variable. Discuss the results of the t-tests in light of the policy implications mentioned.
5. What proportion of the variation in sales is explained by the independent variables in the equation? How confident are you about this answer? Explain conducting an F-test.

The following table was presented in an article summarizing a study to compare a new drug to a standard drug and to a placebo.

*Table entries and Mean (SD) or %

1. Are there any statistically significant differences in the characteristics shown among the

treatments?   Justify your answer.

2. Consider the test for differences in age among treatments. Write the hypotheses and the formula of the test statistic used (No computations required – formula only).

3. Consider the test for differences in insurance coverage among treatments. Write the hypotheses and the formula of the test statistic used (No computations required – formula only).

4. Consider the test for differences in disease stage among treatments. Write the hypotheses and the formula of the test statistic used (No computations required – formula only).

A fitness instructor wants to understand the relationship between her heart rate during a workout and the calories burned per minute during the workout. She recorded these two measurements over the course of several workouts. StatCrunch was used to create the output below:

For any numeric responses, round your answers to four decimal places.

1. Use the regression equation to provide a point estimate for a workout when the average heart rate is 150 beats per minute: y =
2. The fitness instructor would like to estimate the number of calories burned during a workout when her heart rate is 150 beats per minute. Which interval would be the most appropriate to provide?
   • prediction interval
   • confidence interval
3. Interpret the interval you selected in part b: With 95% confidence, we estimate the Select an answer number of calories burned during a workout mean number of calories burned during all workouts  is between  and  when the average heart rate is  beats per minute.

A professor of business at a local college wanted to determine which of his two methods of teaching statistics resulted in better grades. He randomly selected 30 students from his current classes and set up two special sub-sections of the course, one for each of his two methods, with 15 students in each sub-section. After the semester was over, he placed the final averages for each of the students in a spreadsheet and wanted to determine if the difference in the average scores of the samples indicated that there would be a difference in the scores of the population of students.

Use the correct Excel function found in Data Analysis to determine if the population average scores of the two methods are the same. At the bottom of the Excel worksheet, write a Decision Rule and Conclusion Statement

Method 1: 86, 69, 85, 68, 68, 68, 87, 85, 85, 72, 67, 73, 84, 66, 76

Method 2: 91, 94, 55, 96, 70, 91, 58, 87, 64, 87, 95, 65, 86, 75, 88

The professor would also like to know whether viewing the presidential debate would influence students’ intention to vote and randomly assigns students to one of two groups. Group one viewed the last presidential debate of the major candidates while the second group did not view the debate. After the debate had aired, both groups were asked to report their intention to vote. The question was formatted in a five-point scale as follows: 1= “I do not intend to vote in the next presidential election” to 5= “I absolutely intend to vote in the next presidential election.” Did exposure to the presidential debate relate to whether undergraduate students intend to vote in the presidential election?

1. State the following:H₀ :H_A:

A professional sports league claims that the average age of players in the league is 28 years old. A random sample of 20 players in the league was selected, and their ages are recorded in the following table.

Use Excel to test whether the mean player age is different from 28 years old, and then draw a conclusion in the context of the problem. Use α=0.05.

Age (years)
32
28
29
32
33
28
31
28
30
28
29
34
30
30
31
30
28
21
32
29

Select the correct answer below:

Reject the null hypothesis. There is sufficient evidence to conclude that the mean is not equal to 28.

Reject the null hypothesis. There is insufficient evidence to conclude that the mean is not equal to 28.

Fail to reject the null hypothesis. There is sufficient evidence to conclude that the mean is not equal to 28.

Fail to reject the null hypothesis. There is insufficient evidence to conclude that the mean is not equal to 28.

Find the following probability for a standard normal random variable, P(Z ≥ -2.16 )

Determine confidence intervals for each of the following:

1. In the results of a survey of 395 teaching university faculty and 266 research university faculty. Of the teaching university faculty, 224 said they were very satisfied with their jobs, whereas 126 of the research university faculty were very satisfied with their work. Estimate the difference between the proportion of all teaching university faculty who are very satisfied and research university faculty who are very satisfied by calculating and interpreting a 95% CI.

Broome county’s health department hires you to analyze data on COVID-19. They come to you with data from your county’s local hospitals, and tell you that they have randomly tested 147 people showing symptoms of coronavirus. Results showed that 68 of those were positive for the virus. They are worried about running out of room in their hospitals, so they want you to estimate the true proportion of people with the virus.

The county officials are saying that based on initial results they believe about 46.3% of their county could have COVID-19. Explain to them why this is incorrect by defining the point estimate in this scenario and the parameter it is estimating.

a. Give an estimate for the parameter in the form of a 90% confidence interval.

b. The county now wants a 90% confidence interval for the parameter with a margin of error of 0.05. How many total people will have to be tested before that can be achieved?

c. A neighboring county, Tioga, has 21 confirmed cased of the 55 that were tested in their hospitals similarly to how Broome county was sampled and tested. Estimate the true difference in proportions for these two counties using a 90% confidence interval.

A textile manufacturer wanted to know if the breaking load (kg/25 mm width) is greater for unabraded fabrics versus fabrics that were abraded. Abraded fabrics are produced by the use of friction to produce surface wear on the fabric. The manufacturer randomly selected equal-sized sections of 21 of their fabrics. Each section was then equally divided into two pieces, with the abrasion process being applied to one of them. The tensile strength measurements were taken for all of the fabric pieces with the results given in the following table. You will not need to use the information from all the rows. Please assume that the distribution is normal.

a) Should this situation be analyzed via a two-sample independent or paired method?

Please explain the correct answer. If this is a paired situation, please state the common characteristic that makes these data paired.

b) What is the alternative hypothesis for this situation?

Please explain the correct answer.

c) Is there any evidence to suggest that the mean tensile strength of the unabraded fabrics is greater than the mean strength of the abraded fabrics? Use α = 0.05.

Using R studio code, calculate the test statistic. Be sure that the information for the unabraided fabric is first.

Using R studio code, calculate the p-value.

Write the complete four steps of the hypothesis test for this situation.

d) Calculate the appropriate confidence interval or bound for the mean based on the question in part c). If you calculate a bound, type 10000 to indicate ∞.

Interpret the interval or bound calculated above.

e) In practical terms, does the data imply that the mean tensile strength of the unabraded fabrics is greater than that of the abraded fabrics? Please explain your reasoning. This part uses the information from parts c) and d)

f) Explain why the results in parts c) and d) are consistent with each other.

A teacher believes that the third homework assignment is a key predictor in how well students will do on the midterm. Let x represent the third homework score and y the midterm exam score. A random sample of last terms students were selected and their grades are shown below. Assume scores are normally distributed.

Based on your results, If your HW3 grades increases by 1 point, your Midterm grade, on average, increases by approximately how much?

Place your answer, rounded to 3 decimal places, in the blank. Do not use any stray punctuation marks or a dollar sign. For example, 34.567 would be a legitimate entry.