Midterm Exam 1Changing your grade • – – given that grading is on a curve, the result would be Answers to the 10 (or so) most frequently missed questions #1 • (a) 0.8% (b) 12.5% (c) 1.25% (d) 4% (e) 50% #1 • If you got this question right, • The TA’s will not give you credit for a question that has an answer different from that in our answer key. •Why? Our answers are correct. If you were confused by some aspect of a question, chances are a bunch of other people were as well. We’d have to regrade everyone’s exams, to be fair, and little or no change to anyone’s grade. On planet gargu there are five different genders. A child can have only one of these genders. Picking a person at random, the probability of each of these genders is .2. Let’s say parents want to have three children in a row. What are the odds (before they have any children) that all three children will have the same gender? congratulations – you were one of the few in the class to get it right! Nonetheless, this is a simple probability problem, I expect you to be able to solve problems like this, and you should expected to be tested more on this on the final. 2---------------------------------------------------------------------The catch • • E.G. • One way of solving the problem • – (0.2)(0.2)(0.2) = 0.008 • • Most people said the answer was (a) 0.8% However, this is the (correct) probability of having 3 children who all have a particular gender. it’s the probability of having 3 children who all have gender A The question doesn’t ask for that. It asks for the probability of having the 3 children have the same gender – this could be any of the 5 genders What is the probability of getting three kids all with a particular gender, e.g. A? Many people gave this answer, 0.8%. If you gave this answer, you didn’t take into account that this is just one possible gender, and you need to count all 5. (They are mutually exclusive events, so you can just add the probabilities.) Answer = (5)(0.008) = 0.04 = 4% Another way of solving the problem • – AAA, BBB, CCC, DDD, EEE = 5 • – 5·5·5 = 53 • –5/53 = 1/52 = 4/100 = 4% #2 • book: Degrees of 25% 10% 5% 2.5% 1% 0.5% 9 0.70 1.38 1.83 2.26 2.82 3.25 10 0.70 1.37 1.81 2.23 2.76 3.17 11 0.70 1.36 1.80 2.20 2.72 3.11 obt How many possible ways can someone have 3 children of the same gender? Call the genders A, B, C, D, and E How many possible combinations of 3 children are there? What’s the probability of getting 3 kids all the same gender? The following lines appear in the t-table at the back of your freedom I calculate an observed value tof -2.24, from a sample of size 10. What do I conclude? 3------------------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------Your choices 0: µ = µ0a: µ ≠ µ0, and α=0.05. 0 : µ = µ0, with p<0.05, if a: µ ≠ µ0. only applies to the upper tail. (d) I accept the null hypothesis that µ < µ0. (e) I reject the null hypothesis, with p<0.025, if my a: µ ≠µ0. Degrees of 25% 10% 5% 2.5% 1% 0.5% 9 0.70 1.38 1.83 2.26 2.82 3.25 10 0.70 1.37 1.81 2.23 2.76 3.17 11 0.70 1.36 1.80 2.20 2.72 3.11 tobt = -2.24, N=10 • dealing with? ((a) I accept the null hypothesis H , if my alternative hypothesis is H(b) I reject the null hypothesis, Hmy alternative hypothesis is H(c) I reject the null hypothesis if my alternative hypothesis alternative hypothesis is Hfreedom First of all, which line of the table are we – Degrees of freedom = 9 Most got this right.) Degrees of 25% 10% 5% 2.5% 1% 0.5% 9 0.70 1.38 1.83 2.26 2.82 3.25 10 0.70 1.37 1.81 2.23 2.76 3.17 11 0.70 1.36 1.80 2.20 2.72 3.11 tobt = -2.24, N=10 • obt to indicate |tobta : µ < µ0 Because tobt < 0 Degrees of 25% 10% 5% 2.5% 1% 0.5% 9 0.70 1.38 1.83 2.26 2.82 3.25 10 0.70 1.37 1.81 2.23 2.76 3.17 11 0.70 1.36 1.80 2.20 2.72 3.11 tobt = -2.24, N=10 • – obt| singletail – a : µ < µ0 – a : µ ≠ µ0 • freedom What needs to be true for our ta significant result? –Well, | > 1.83, which corresponds to 5% in a single tail of the t-distribution – So, significant at the 0.05 level if HWhy < (lower tail)? freedom What needs to be true for our tobt to indicate a significant result? Well, |t > 1.83, which corresponds to 5% in a of the t-distribution So, significant at the 0.05 level if HOr, significant at the 0.10 level if HThese are the cases in which you reject the null hypothesis 4If Ha : µ < µ0, reject H0 at the 0.05 level If Ha : µ ≠ µ0j0 at the 0.10 level 0: µ = µ0a: µ ≠ µ0, and α=0.05. 0 : µ = µ0, with p<0.05, if a: µ ≠ µ0. only applies to the upper tail. (d) I accept the null hypothesis that µ < µ0. (e) I reject the null hypothesis, with p<0.025, if my a: µ ≠µ0. Needs p<0.10 If Ha : µ < µ0, reject H0 at the 0.05 level If Ha : µ ≠ µ0j0 at the 0.10 level 0: µ = µ0a: µ ≠ µ0, and α=0.05. 0 : µ = µ0, with p<0.05, if a: µ ≠ µ0. only applies to the upper tail. (d) I accept the null hypothesis that µ < µ0. (e) I reject the null hypothesis, with p<0.025, if my a: µ ≠µ0. ,reect H(a) I accept the null hypothesis H , if my alternative hypothesis is H(b) I reject the null hypothesis, Hmy alternative hypothesis is H(c) I reject the null hypothesis if my alternative hypothesis alternative hypothesis is H,reect H(a) I accept the null hypothesis H , if my alternative hypothesis is H(b) I reject the null hypothesis, Hmy alternative hypothesis is H(c) I reject the null hypothesis if my alternative hypothesis alternative hypothesis is H#3 • closed. Someone tells you that there is a prize behind one (a) 33.33% (b) 67.67% (c) 50% (d) 0% (e) 100% • worth spending much time on. • questions like this; (2) the question sounds ambiguous, and might be in casual conversation, but isn’t in this context. #3 • closed. Someone tells you that there is a prize behind one • – ) answer 100% – ( If ) You come upon three doors, one already open and two and only one of the two closed doors. What is the probability that the prize is behind one of these doors? This was intended as a bit of a trick question, and it’s not It seems tricky because (1) in real life people don’t ask You come upon three doors, one already open and two