STAT4610Project3

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4610

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Statistics

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Jun 12, 2024

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STAT4610 Project 3 Statistical Tests 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 year’s 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: Brunett e 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 H 0 is true. (Recall you had a total of 400 promotions.) Brunett e Brown Blon d 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 Charmande r 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 sWeedle 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? 14.7991 1 28.2385 8 22.7928 24.5166 7 22.5070 2 35.5008 9 21.7505 7 28.0875 2 23.4774 6 17.7120 1 26.9566 3 28.8737 9 30.5948 1 30.8270 1 24.7706 4 23.7422 3 25.5293 6 29.0418 5 36.8747 9 30.3736 2 27.9390 6 21.9075 4 26.4268 7 19.9263 8 22.9328 4 24.1324 6 20.7254 24.2610 7 18.3925 9 27.6490 9 34.6292 18.9683 13.4252 1 26.8225 4 31.9074 1 24.8972 3 27.2288 1 19.4729 9 29.7634 4 32.5879 9 16.8599 4 29.5746 27.5072 25.6561 9 15.6708 27.3671 6 27.7929 8 26.6896 2 22.0614 2 23.0711 9 31.6090 2 26.0389 3 32.9860 6 28.5494 3 20.3410 1 33.4400 26.5577 21.0856 23.8731 23.2079

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