The firm that manufactures patriot missiles purchases the guidance circuits from three different suppliers. Supplier A provides 30% of the guidance circuits and those circuits have a fault probability of pA =0.02, whereas the circuits from supplier B, which provides 25% of those purchased, have a fault probability of pB =0.025. The guidance circuits purchased from supplier C have pC =0.01. If a batch of 200 missiles is fired during a particular strategic offensive and three of the missiles fail to track to target, what is the probability that the batch of missiles contained guidance circuits obtained from supplier B?

Problem 6 (Inference via Bayes’ Rule)
Suppose we are given a coin with an unknown head probability θ ∈ {0.3,0.5,0.7}. In order to infer the value θ, we experiment with the coin and consider Bayesian inference as follows: Define events A1 = {θ = 0.3}, A2 = {θ = 0.5}, A3 = {θ = 0.7}. Since initially we have no further information about θ, we simply consider the prior probability assignment to be P(A1) = P(A2) = P(A3) = 1/3.
(a) Suppose we toss the coin once and observe a head (for ease of notation, we define the event B = {the first toss is a head}). What is the posterior probability P(A1|B)? How about P(A2|B) and P(A3|B)? (Hint: use the Bayes’ rule)
(b) Suppose we toss the coin for 10 times and observe HHTHHHTHHH (for ease of notation, we define the event C = {HHTHHHTHHH}). Moreover, all the tosses are known to be independent. What is the posterior probability P(A1|C), P(A2|C), and P(A3|C)? Given the experimental results, what is the most probable value for θ?
(c) Given the same setting as (b), suppose we instead choose to use a different prior probability assignment P(A1) = 2/5,P(A2) = 2/5,P(A3) = 1/5. What is the posterior probabilities P(A1|C), P(A2|C), and P(A3|C)? Given the experimental results, what is the most probable value for θ?

1. Stacy and Leslie are playing a very simple gambling game. They toss a coin and Stacy wins if it comes up “heads” while Leslie wins if it comes up “tails.” After 12 hours of gambling, Leslie begins to suspect that Stacy has been cheating because Stacy has won more games. Leslie accuses Stacy, but Stacy pleads innocent and proposes to test Leslie’s claim by doing an experiment in which the coin is tossed 14 times.

1. State the null and alternative hypotheses for this experiment.
2. If α = .01, what is the rejection region?
3. They do the experiment and “heads” occurs 8 times. What can they conclude? What if heads comes up 12 times?
4. Let’s assume that Stacy actually is cheating because she is using a coin that is biased to come up “heads” 55% of the time. What was the power of the experiment they did?
5. Given the power of the experiment, was Stacy clever or stupid to propose it as a test to Stacy?

6. A school system has a high rate of turn-over among new teachers. Specifically, 30% of the teachers that are hired leave within 2 years. The superintendent is concerned about the problem and institutes a program of teacher mentoring that he hopes will improve retention of the teachers. After the first 2 years of the program, he evaluates whether it is working by recording what happened with the 16 teachers who were hired at the start of the program. He finds that 3 of original 16 have left.

a. Complete the relevant hypothesis test, using α = .05.

b. Suppose that the mentoring program actually does improve retention to the point where the true probability of a teacher leaving is actually 10%. What was the power of the principal’s study? What does the number you compute mean in English? Explain the relevance (or lack of relevance) of your power calculation to your conclusion in part ‘a.’

1. The production manager of a company that produces an over-the-counter cold remedy wants to boost sales of the product. The product is considered effective by the people who have tried it, but many people decide not to buy it again because it tastes like day-old soapsuds. The manufacturer is trying to decide whether to add a lemon flavor to the product. Because the flavoring will increase production costs, the manager wants to be certain that people respond favorably to the flavoring before using it. Twenty people with colds are randomly sampled. On two difference occasions, each person uses the product and indicates which version (taste) is preferred. (They must pick one.)

1. State the competing hypotheses and the rejection region for α = .01.
2. What conclusion should be made if 16 people prefer the lemon-flavored product? What if 2 people prefer the lemon flavor?
3. Why would using α = .01 make sense in this case? If the results indicate that the null hypothesis should be rejected in favor of the alternative hypothesis, what additional problem(s) of interpretation might the production manager have to face?

Students will follow the hypothesis testing steps for each problem. They will compute the problem using the SPSS program. They will write the results in appropriate APA format and interpret the results. Steps of hypothesis testing will be typed out in a word document, as well as a copy and paste of the SPSS output.

For the following problems, you will:

• Do all steps of hypothesis testing
  • Populations and hypotheses
  • Write out the steps you would do to calculate t
  • Choose the t-cutoff score
  • Calculate the t-statistic using the computer program SPSS
  • Write the t-statistic using proper APA format
  • Decide whether or not you would reject the null hypothesis
• Interpret this result
• Be sure to include a copy of the SPSS output in the word document

1. Single Sample T-test

A researcher would like to study the effect of alcohol on reaction time. It is known that under regular circumstances the distribution of reaction times is normal with μ = 200. A sample of 10 subjects is obtained. Reaction time is measured for each individual after consumption of alcohol. Their reaction times were: 219, 221, 222, 222, 227, 228, 223, 230, 228, and 232. Use α = 0.05.

A social media survey found that 71%

of parents are​ "friends" with their children on a certain online networking site. A random sample of 140

parents was selected. Complete parts a through d below.

a. Calculate the standard error of the proportion.

sigma Subscript p

equals0.0383

​(Round to four decimal places as​ needed.)

b. What is the probability that

105

or more parents from this sample are​ "friends" with their children on this online networking​ site?

​P(105

or more parents from this sample are​ "friends" with their

​children)equals

nothing

​(Round to four decimal places as​ needed.)

1. According to the empirical rule, for a distribution that is symmetric and bell-shaped, approximately _______ of the data values will lie within 3 standard deviations on each side of the mean.

2. Assuming that the heights of boys in a high-school basket- ball tournament are normally distributed with mean 70 inches and standard deviation 2.5 inches, how many boys in a group of 40 are expected to be taller than 75 inches?

3. Let x be a random variable that represents the length of time it takes a student to complete Dr. Gill’s chemistry lab project. From long experience, it is known that x has a normal distribution with mean μ = 3.6 hours and standard deviation σ = 0.5.

Convert each of the following x intervals to standard z intervals.

(a) x ≥ 4.5 4. (a) __________________________

(b) 3 ≤ x ≤ 4 (b) __________________________

(c) x ≤ 2.5 (c) __________________________

Convert each of the following z intervals to raw-score x intervals.

(d) z ≤ −1 (d) __________________________

(e) 1 ≤ z ≤ 2 (e) __________________________

(f) z ≥ 1.5 (f)_ __________________________

Start StatCrunch and make the following sequence selection: Applets -> Distribution demos. Next select "Binomial" and click "Compute!". In the resulting popup window experiment by using the sliders to assign approximately 0.5 to p and successively assign the values 20, 30 and 40 to n. Discuss what you see in the subsequently drawn Binomial Distribution defined by your specified values for n and p. What value on the x axis (horizontal axis) does the top of the hump of the curve correspond to. Next set p and n to their extreme values? Discuss what you observed using the fact that the x-axis represents the number of successes and the height of the vertical lines represent the probability of getting x number of successes.

Problem 16-13 (Algorithmic)

The wedding date for a couple is quickly approaching, and the wedding planner must provide the caterer an estimate of how many people will attend the reception so that the appropriate quantity of food is prepared for the buffet. The following table contains information on the number of RSVP guests for the 145 invitations. Unfortunately, the number of guests does not always correspond to the number of RSVPed guests.

Based on her experience, the wedding planner knows it is extremely rare for guests to attend a wedding if they notified that they will not be attending. Therefore, the wedding planner will assume that no one from these 50 invitations will attend. The wedding planner estimates that the each of the 25 guests planning to come solo has a 75% chance of attending alone, a 20% chance of not attending, and a 5% chance of bringing a companion. For each of the 60 RSVPs who plan to bring a companion, there is a 90% chance that she or he will attend with a companion, a 5% chance of attending solo, and a 5% chance of not attending at all. For the 10 people who have not responded, the wedding planner assumes that there is an 80% chance that each will not attend, a 15% chance each will attend alone, and a 5% chance each will attend with a companion.

1. Assist the wedding planner by constructing a spreadsheet simulation model to determine the expected number of guests who will attend the reception. Round your answer to 2 decimal places.
   
   guests
   
2. To be accommodating hosts, the couple has instructed the wedding planner to use the Monte Carlo simulation model to determine X, the minimum number of guests for which the caterer should prepare the meal, so that there is at least a 90% chance that the actual attendance is less than or equal to X. What is the best estimate for the value of X? Round your answer to the neares whole number.
   
   guests

1. Grear Tire Company has produced a new tire with an estimated mean lifetime mileage of 36,500 miles. Management also believes that the standard deviation is 5000 miles and that tire mileage is normally distributed. To promote the new tire, Grear has offered to refund some money if the tire fails to reach 30,000 miles before the tire needs to be replaced. Specifically, for tires with a lifetime below 30,000 miles, Grear will refund a customer $1 per 100 miles short of 30,000.

   1. For each tire sold, what is the expected cost of the promotion? If required, round your answer to two decimal places.
      
      
      
   2. What is the probability that Grear will refund more than $50 for a tire? If required, round your answer to three decimal places.
      
      
      
   3. What mileage should Grear set the promotion claim if it wants the expected cost to be $2.00? If required, round your answer to the hundreds place.
      
      miles

1. Use the XLMiner Analysis ToolPak to find descriptive statistics for Sample 1 and Sample 2. Select "Descriptive Statistics" in the ToolPak, place your cursor in the "Input Range" box, and then select the cell range A1 to B16 in the sheet. Next, place your cursor in the Output Range box and then click cell D1 (or just type D1). Finally make sure "Grouped By Columns" is selected and all other check-boxes are selected. Click OK. Your descriptive statistics should now fill the shaded region of D1:G18. Use your output to fill in the blanks below.

   Sample 1 Mean:  (2 decimals)

   Sample 1 Standard Deviation:  (2 decimals)

   Sample 2 Mean:  (2 decimals)

   Sample 2 Standard Deviation:  (2 decimals)

2. Use a combination of native Excel functions, constructed formulas, and the XLMiner ToolPak to find covariance and correlation.

   In cell J3, find the covariance between Sample 1 and Sample 2 using the COVARIANCE.S function.

   (2 decimals)

   In cell J5, find the correlation between Sample 1 and Sample 2 using the CORREL function.
   (2 decimals)

   In cell J7, find the correlation between Sample 1 and Sample 2 algebraically, cov/(sx*sy), by constructing a formula using other cells that are necessary for the calculation.

   (2 decimals)

   Use the XLMiner Analysis ToolPak to find the correlation between Sample 1 and Sample 2. Place your output in cell I10.

   (2 decimals)

3. Calculate z-scores using a mix of relative and absolute cell references. In cell A22, insert the formula =ROUND((A2-$E$3)/$E$7,2). Next grab the lower-right corner of A22 and drag down to fill in the remaining green cells of A23 to A36. Note how the formula changes by looking in Column D. Changing a cell from a relative reference such as E3 to an absolute reference such as $E$3 means that cell remains "fixed" as you drag. Therefore the formula you entered into A22 takes each data observation such as A2, A3, A4..., subtracts $E$3 and then divides by $E$7. Since the last two cells have absolute references they will not change as you drag. The ROUND function simply rounds the z-score to two digits.

   Now find the z-scores for Sample 2 using the same method you learned above by editing the formula to refer to the correct cells for Sample 2. Make sure each z-score is rounded to 2 places.

   Sample 2 z-scores

Decisions about alpha level may be different, especially as it relates from hard sciences to social sciences. For example, medical trials for cancer treatments are conducted at an alpha of 0.0001. For "hard" and social sciences, alpha of 0.05 is used. Do you agree with these alpha levels? Why or why not? Provide a specific example and interpretation of "significance" in your answer.

Tommy has recently graduated from SUSS and has joined a well-known retailer that operates 3 department stores in Singapore. His job function is that of a business analyst. It has been well-reported that retail business in Singapore is on the decline and his employer would like to determine if the forecast for the next few years will be equally bad. Tommy has been tasked to perform the analysis and his output will provide insights in the company's hiring and expansion plan in Singapore.

The first thing Tommy did was to download the Retail Sales Index data from the Department of Statistics Singapore website. He specifically extracted the data for "Department Stores" from Q1 2008 to Q4 2018. (2017 is set as the base year with Index = 100). Refer to data below:

Year/Quarter Retail Sales Index (Department Stores)

2008 1Q 92.4

2008 2Q 93

2008 3Q 84.9

2008 4Q 105.3

2009 1Q 87.1

2009 2Q 88.6

2009 3Q 83.2

2009 4Q 103.6

2010 1Q 94.3

2010 2Q 93.9

2010 3Q 90.7

2010 4Q 109.3

2011 1Q 99.6

2011 2Q 99

2011 3Q 95.8

2011 4Q 119.3

2012 1Q 104.9

2012 2Q 99.9

2012 3Q 95

2012 4Q 117

2013 1Q 106.1

2013 2Q 100.9

2013 3Q 97.5

2013 4Q 118.6

2014 1Q 107.1

2014 2Q 98.7

2014 3Q 95.7

2014 4Q 117.6

2015 1Q 108.2

2015 2Q 101.1

2015 3Q 98.8

2015 4Q 114.4

2016 1Q 103.8

2016 2Q 93.9

2016 3Q 93.2

2016 4Q 112.9

2017 1Q 100.3

2017 2Q 93

2017 3Q 94.6

2017 4Q 112.1

2018 1Q 102.7

2018 2Q 92.3

2018 3Q 93.6

2018 4Q 112.8

(a) Use the techniques of time series modelling and with justifications, discuss the considerations on how Tommy would forecast the quarterly Retail Sales Index in 2019 and 2020.

(b) Upon confirmation of the forecast technique to undertake in Part 3(a), determine the quarterly RSI forecast in 2019 and 2020. You are required to show the essential steps in deriving the forecast and comment on any limitations of your technique.

• Explain the difference between two independent samples and two dependent samples.
• Provide examples of each type of pairs of samples.
• Compare and contrast the hypothesis testing process of two independent and two dependent samples. (Do not conduct the hypothesis testing.)