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STA 2023 CH. 2 • Find the mean, median, and standard deviation of the data set { 5, 4, 7, 3, 1 } Mean = 4 Median= 4 Std. dev.= 2.24 CH. 2 • If the mean is greater than the median, the distribution is skewed right. • If the mean is less than the median, the distribution is skewed left. • If the mean is equal to the median, the distribution is not skewed. CH. 2 • A football team scores an average of 20 points a game with a standard deviation of 3 points. The distribution is approximately normal. (a) What is the probability that the team scores between 17 and 23 points? (b) Between 14 and 23 points? a.) About 68% b.) About 81.5% CH.2 • The mean of a data set is 8 with a std. dev. of 1.5. If we don’t know the shape of the distribution, what is the probability between 6.5 and 9.5? Between 5 and 11? • Chebychev: 1- 1/k2 Where k is the number of St. Devs away from the mean. Answer= At least 0% Answer= At least 75% CH. 3 • What is the difference between multiplicative, combination, and permutation? • Multiplicative: Just looking to find how many possible outcomes per trial “x” from “n” number of trials. (X)n • Combination: Number of ways we can select n items from N items without replacement. Use 𝑁! when order doesn’t matter. 𝑛! 𝑁−𝑛 ! CH. 3 • Permutation: Number of ways we can put n things out N in order. Use when order matters! • 𝑁! 𝑁−𝑛 ! CH. 3 • How many ways can we choose 5 flavors of ice cream out of 21? How many ways can we put 3 flavors in order from first to third? Answer: 𝑁! 𝑛! 𝑁−𝑛 ! Answer: 𝑁! 𝑁−𝑛 ! = = 21! 5! 21−5 ! 21! 21−3 ! = 20,349 = 7980 CH. 3 Currently Whipping Currently Nene Total Done Whipping 3 15 18 Done with Nene 7 8 15 Total 10 23 33 Find Probability of these dance combinations according to the table: (a) Done whipping or currently Nene; (b) Finished Nene and currently Nene CH. 3 • Answer (a): 0.79 A+𝐵− 𝐴∩𝐵 𝑇𝑜𝑡𝑎𝑙 = • Answer (b): 𝐴∩𝐵 𝑇𝑜𝑡𝑎𝑙 = 0.24 = 8 33 18+23−15 33 = 26 33 = CH. 3 • If P(A)= 0.3, P(B)= 0.5, and P(A U B)= 0.65: (a) what is P(A ∩ B)? (b) Are A and B independent? 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵) CH. 3 a.) 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∪ 𝐵 = 0.3 + 0.5 − 0.65 = 𝟎. 𝟏𝟓 b.) If independent, 𝑃 𝐴 𝑃 𝐵 = 𝑃 𝐴 ∩ 𝐵 0.3 0.5 = 0.15 Since 𝑃 𝐴 𝑃 𝐵 = 𝑃 𝐴 ∩ 𝐵 , A and B are independent CH. 4 X 2 3 5 7 P(x) 0.20 0.30 0.35 0.15 (a) Find μ and σ (b) Find P(x=5); P(x≤ 7); P(x > 2) 𝜇=𝐸 𝑥 =∑ 𝑥 𝑃 𝑥 𝜎= 𝜎2 = Σ 𝑥−𝜇 a.) μ= 4.1; σ= 1.67 b.) P(x=5)= 0.35; P(x≤ 7)= 1; P(x > 2)= 0.8 2 (𝑃 𝑥 ) CH. 4 You play roulette and place a $10 bet on red. There are 18 red spaces, 18 black spaces, and 2 green spaces. What is the expected result of your bet? Probability of red (success)= 18/38 Probability of black (failure) = 18/38 Probability of green (failure) = 2/38 Success= 18/38 Failure= 20/38 CH. 4 (gain) x P(success) – (lose) x P(failure) (10) X (18/38) – (10) x (20/38) 4.74 – 5.26 = -$0.52 We should expect to lose money. CH. 4 • How do you know when to use binomial or Poisson? Binomial: 2 possible outcomes (success and fail). 𝑛! 𝑝 𝑥 𝑞 𝑛−𝑥 𝑛−𝑥 ! 𝑥 ! Poisson: Use when dealing with time/rate. 𝛾 𝑥 𝑒 −𝛾 𝑥! where 𝛾 = 𝑚𝑒𝑎𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠, and x=number of successes we are interested in. CH. 5 The average semester grade in STA 2023 is 70 with a standard deviation of 3.5. What is the probability that the average grade this semester will be greater than 76? Less than 72? Equal to 68? Answer: 0.50 – 0.4564 = 0.0436 Answer: 0.50 + 0.2157 = 0.7157 Answer: Cannot be equal to a number because these are continuous random variables. P = 0. CH. 5 The average grade in STA 2023 is 70 with a standard deviation of 6. If you want to be in the 98th percentile, what is the minimum score you must obtain? Answer: 85.3 CH. 6 • We want to find out how many times Detroit Lions fans cry themselves to sleep per week. We randomly sample 40 Lions fans from a population with a mean of 5 and std. dev. of 1.5. What is the probability that the mean of our sample will be more than 4.5? Less than 4? P= 0.50 + 0.4826 = 0.9826 P= Our Z-score is -4.22. This is too negative to look up, so we assume P<-4.22 = 0 CH. 7 • At a set level of confidence, does our confidence interval increase or decrease as sample size increases? 𝐶. 𝐼. = 𝑥 ± 𝑆. 𝐸. 𝑆. 𝐸. = 𝑍 𝜎 𝑛 Answer: Decreases CH. 7 • At a set sample size, what happens to our confidence interval as our level of confidence increases? 𝐶. 𝐼. = 𝑥 ± 𝑆. 𝐸. 𝑆. 𝐸. = 𝑍 𝜎 𝑛 Answer: Increases CH. 7 • If constructing interval for 𝜇: When n> 30: 𝐶. 𝐼. = 𝑥 ± 𝑍𝛼/2 When n< 30: 𝐶. 𝐼. = 𝑥 ± 𝑡𝛼/2 • If finding n: 𝑍2𝜎 2 𝑛= 𝑆𝐸 2 𝜎 𝑛 𝜎 𝑛 CH. 7 • If constructing interval for P: When n> 30: 𝐶. 𝐼. = 𝑝 ± 𝑍𝛼/2 When n< 30: 𝐶. 𝐼 = 𝑝 ± 𝑡𝛼/2 𝑝𝑞 𝑛 𝑝𝑞 𝑛 • If finding n: 𝑛= 𝑍𝛼/2 2 𝑝𝑞 𝑆𝐸 2 Note: When finding n, if 𝒑 is unknown use 0.50. CH. 7 Construct a C.I. at 95% confidence with a sample mean of 70 and a std. dev. of 20 when n=49. When n=25 Answer = 70 ± 5.6 = (64.4, 75.6) Answer = 70 ±2.064 20 25 = 70 ± 8.26 CH. 7 30 pre-med students out of a sample of 40 say they have stress-induced acid reflux. Construct a 90% confidence interval to estimate the true proportion of pre-med students with stress-induced acid reflux. 𝑝 ± 𝑍𝛼/2 𝑝𝑞 𝑛 Answer = 0.75 ± 0.11 = (0.64, 0.86) CH. 7 • Determine the sample size needed to construct a 99% C.I. to estimate the true proportion to within 0.10 with 𝑝 = 0.60. What if we didn’t know 𝑝 ? • 𝑆. 𝐸. = 𝑍𝛼/2 • 𝑛= 𝑍𝛼/2 2 𝑆.𝐸.2 𝑝𝑞 𝑝𝑞 𝑛 = 𝟏𝟔𝟎 CH. 8 It is believed that the average grade on STA 2023 final exams is 70. A study of 36 students was run, and the results yielded a mean of 76 with a standard deviation of 18. Is this enough evidence to claim that the true mean score is greater than 70 at 𝛼 = 0.05? What is the level of significance? Answer: Yes, our test statistic lies in the RR; P=0.0228. CH. 8 It is estimated that 70 percent of college students enjoy going to Chipotle. A sample was conducted where 23 out of 29 students sampled said that they enjoy Chipotle. Is this enough evidence to say that more than 70 percent like Chipotle at 𝛼 = 0.1. Answer: No, our T.S. of 1.09 does not fall in the RR t>1.28. CH. 9 A study was run to see if there is a difference in mean test scores between students who play piano, and students who do not. 20 piano students and 18 non-piano students were studied. The mean of the piano group was 85 with s.d. of 8, and the mean of the non-piano group was 81 with s.d. equal to 7.5. Is this enough evidence to conclude that there is a difference at 𝛼 = 0.05? Construct a 95% C.I. CH. 9 𝑠𝑝 2 𝑠1 2 𝑛1 − 1 + 𝑠2 2 𝑛2 − 1 = 𝑛1 + 𝑛2 − 2 T.S.= 𝑥1 −𝑥2 𝑠𝑝 2 1 1 + 𝑛1 𝑛2 Answer: Test stat is not in RR, do not reject. Answer: 4 ± 5.23 = (−1.23,9.23) Hopefully you now feel less like this And more like Sheldon • https://www.youtube.com/watch?v=ay3dSzkf swE