USE THE FOLLOWING SCENARIO AND DATA SET FOR PROBLEMS 1 AND 2. As the Executive Vice President of Sales for ACME Nose Hair Trimmers Inc., your sales force is divided into four regions: NW, SW, NE, and SE. Your CFO and HR Director have determined that you can promote 400 of your sales force in the following year, and you allocate those promotions to your Regional VPs of Sales as follows: NW gets 80, SW gets 120, and NE and SE get 100 each. (This allocation is based on the relative sizes of the regional sales forces.) Your employees identify themselves by hair color, and when the promotions have been completed, your HR Director presents you with a demographic break-down of regional promotions by hair color as follows: REGION NW SW NE SE HAIR Brunette 30 30 40 25 COLOR Brown 20 20 20 15 Blond 10 40 20 25 Red 10 20 10 15 Gray 5 0 10 5 Bald 5 10 0 15 1. You receive a number of complaints from your employees that this years promotions were not assigned fairly (in that some VPs favored different hair colors), so you decide to determine if the distribution of promotions differed by region. You conduct a hypothesis test to this effect. a. State your null hypothesis for this test. b. State your alternative hypothesis for this test. c. Describe the Type I error for this test and what its implications are for ACME NHT Inc. d. Describe the Type II error for this test and what its implications are for ACME NHT Inc. e. What is the critical value of your test statistic if you are willing to reject your null hypothesis at the = .05 level of significance? (Ensure you identify what type of statistic it is.) f. What is the calculated value of your test statistic? (Show as much work as necessary to ensure partial credit if you are not confident of your answer.) g. What is the p-value for this test? What does it represent? h. What do you conclude about your hypothesis test, and why? i. What action will you take as the Executive Vice President of Sales? 2. Because of further employee complaints, the ACLU becomes involved in a discrimination lawsuit against ACME NHT Inc., accusing you of discriminatory promotion practices within your sales force. You immediately ask your HR Director for the distribution of your total sales force by hair color (expressed as percentages). She provides you with the following data: Brunette Brown Blond Red Gray Bald 30% 20% 25% 12% 5% 8% You decide to conduct a hypothesis test to determine if your total promotions follow the distribution of the demographics of your sales force. a. State your null hypothesis. b. State your alternative hypothesis. c. Describe the Type I error for this test and what its implications are for ACME NHT Inc. d. Describe the Type II error for this test and what its implications are for ACME NHT Inc. e. Fill in the following table with your expected number of promotions for each hair color if H0 is true. (Recall you had a total of 400 promotions.) Brunette Brown Blond Red Gray Bald f. What is the critical value of your test statistic if you are willing to reject your null hypothesis at the = .10 level of significance? (Ensure you identify what type of statistic it is.) g. What is the calculated value of your test statistic? (Show as much work as necessary to ensure partial credit if you are not confident of your answer.) h. What is the p-value for this test? What does it represent? i. What do you conclude about your hypothesis test, and why? j. What action will you take as the Executive Vice President of Sales? 3. The ACME NHT Inc. Executive Vice President of Marketing is considering a national advertising campaign, and asks you if the averages of your quarterly sales differ significantly by region. You already have available your quarterly sales totals for each region (in millions of dollars). They are as follows: REGION NW SW NE SE Q1 5 7 8 4 Q2 7 6 7 6 Q3 3 5 9 5 Q4 6 4 6 3 You decide to test if there is a statistical difference in your quarterly regional sales averages. a. State your null hypothesis for this test. b. State your alternative hypothesis for this test. c. Describe the Type I error for this test and what its implications are for ACME NHT Inc. d. Describe the Type II error for this test and what its implications are for ACME NHT Inc. e. What is the critical value of your test statistic if you are willing to reject your null hypothesis at the = .05 level of significance? (Ensure you identify what type of statistic it is.) f. What is the calculated value of your test statistic? (Show as much work as necessary to ensure partial credit if you are not confident of your answer.) g. What is the p-value for this test? What does it represent? h. What do you conclude about your hypothesis test, and why? i. What action will you take as the Executive Vice President of Sales? j. Which two regions are most likely different from each other in sales performance? 4. In your attempt to catch them all you hunt the prized Pokemon, Mew and Mewtwo. As you battle them with your top 25 Pokemon, you lose every battle. However, after each battle, you notice that the hit points (HP) of Mew and Mewtwo have been reduced. The remaining HPs of Mew and Mewtwo are shown below after battling your respective Pokemon. (A higher HP remaining indicates more success in battle.) You suspect that Mewtwo is the tougher opponent, and you want to test this hypothesis at the 95% level of confidence. Attacker Mew Mewtwo Bulbasaur 402 377 Ivysaur 305 354 Venusaur 348 343 Charmander 350 341 Charmeleon 320 347 Charizard 340 401 Squirtle 396 406 Wartortle 311 364 Blastoise 367 381 Caterpie 309 402 Metapod 391 387 Butterfree 385 347 Weedle 390 349 Kakuna 313 402 Beedrill 396 403 Pidgey 394 389 Pidgeotto 329 395 Pidgeot 375 356 Rattata 364 340 Raticate 340 400 Spearow 365 405 Fearow 395 402 Ekans 365 377 Arbok 396 341 Pikachu 381 399 Assume that before each battle, Mew and Mewtwo started with the same number of HPs. a. State the null hypotheses for this test. b. State the alternative hypotheses for this test. c. What type of test will you use to test this hypothesis? d. Describe the Type I Error for this test, and explain its implication for you as you continue your quest. e. Describe the Type II Error for this test, and explain its implication for you as you continue your quest. f. What is the rejection region for this hypothesis test? g. Calculate the test statistic for this hypothesis test. What is your conclusion? h. If you reject the null hypothesis for this test, what is the probability that you have committed a Type I Error? (i.e., find the p-value for this test.) i. What action will you take as you continue your quest based on your conclusion. j. Which do like better: Mew or Mewtwo? 5. You conduct an experiment where you collect the data in the table below. What population distribution does this sample come from? How confident are you of your conclusion? 14.79911 28.23858 22.7928 24.51667 22.50702 35.50089 21.75057 28.08752 23.47746 17.71201 26.95663 28.87379 30.59481 30.82701 24.77064 23.74223 25.52936 29.04185 36.87479 30.37362 27.93906 21.90754 26.42687 19.92638 22.93284 24.13246 20.7254 24.26107 18.39259 27.64909 34.6292 18.9683 13.42521 26.82254 31.90741 24.89723 27.22881 19.47299 29.76344 32.58799 16.85994 29.5746 27.5072 25.65619 15.6708 27.36716 27.79298 26.68962 22.06142 23.07119 31.60902 26.03893 32.98606 28.54943 20.34101 33.44006 26.55777 21.08564 23.87313 23.20798 29.21436 25.27635 28.70976 29.39177 22.21248 19.70519 23.34337 25.02365 16.32654 25.40277 22.36094 32.23156 22.65091 20.90823 33.97346 18.69098 19.22807 23.61038 30.16207 21.59024 28.39246 31.3317 26.16754 27.09202 20.13793 25.91088 20.53656 17.31394 21.42621 31.07193 24.64381 18.06844 26.29683 29.72477 24.00317 25.15 29.95899 21.25815 24.38912 25.78333 6. Going back to the home sales data for Delta County, CO (see your Project 1 work) you want to see when the market peaked and where it seemed to return to its previous levels. You consider the average sale price for the County for different years. a. Can you determine if the market peaked in 2007 or 2008? (Is there a difference between the averages prices in these two years?) b. Can we conclude that in 2010 and 2011 that the market had returned to 2006 levels? c. Did the market increase from 1995 to 2007? Are you sure? 7. For this problem, use the data file STAT4610Projecdt3ResidentialData2009. This file contains data on 75% of the homes sold in Colorado in 2009 (Sales2009 tab) and data on the listing real estate agents for those homes (Listing Broker Data tab). We are interested in investigating the differences in the Big 5 Realtor Associations in Colorado (identified under the Board field): Aurora Association of Realtors Boulder County Denver Metro Association of Realtors (Not Denver Metro Comm) Douglas/Elbert Realtor Association South Metro Denver Realtor Association The data we are interested in is Close Price, i.e., the actual price the homes sold for. Unfortunately, the Sales tab does not have the Realtor Association included, so the two files must be merged through a common key. (Fortunately, there is one!) The List Agent MLSID is a unique identifier for the listing agents, and it is included with the Sales data. By filtering out the close price data by Association and answer the following questions. a. What are the average closing prices for each Association in 2009. b. Are any of the averages different from the rest? (Justify your answer with an appropriate test and a p-value.) c. Is there a difference between Boulder and South Metro? (Justify your answer with an appropriate test and a p-value.) d. Is there a difference between Aurora and Douglas/Elbert? (Justify your answer with an appropriate test and a p-value.) e. Is there a difference between South Metro and Denver Metro? (Justify your answer with an appropriate test and a p-value.) f. Is there a difference between Boulder and Douglas/Elbert? (Justify your answer with an appropriate test and a p-value.)