Evolutionary theories often emphasize that humans have adapted to their physical environment. One such theory hypothesizes that people should spontaneously follow a 24-hour cycle of sleeping and waking—even if they are not exposed to the usual pattern of sunlight. To test this notion, eight paid volunteers were placed (individually) in a room in which there was no light from the outside and no clocks or other indications of time. They could turn the lights on and off as they wished. After a month in the room, each individual tended to develop a steady cycle.Their cycles at the end of the study were as follows: 25, 27, 25, 23,24, 25, 26, and 25. Using the . 05 level of significance, what should we conclude about the theory that 24 hours is the natural cycle? (That is, does the average cycle length under these conditions differ significantly from 24 hours? ) (a) Use the steps of hypothesis testing. (b) Sketch the distributions involved. (c) Explain your answer to someone who has never taken a course in statistics. Part A. Step 1. Ho=M=24(null) H1=M24(alternative) Step 2. X| | | 25| 0| 0| 27| 2| 4| 25| 0| 0| 23| -2| 4| 24| -1| 1| 25| 0| 0| 26| 1| 1| 25| 0| 0| x= 200 S2=10/7= 1. 42857S2M=S2/N=10? 7/8=0. 17857 SM=Sm2= 0. 422577 Step 3. (=. 025, DF=7) =2. 365 =. 05 /2= . 025 (two tailed test) So if the sample > 1. 895 We reject H0 Step 4. Sample = 2. 366 Step 5. Since 2. 66>2. 365 we will reject the null hypothesis/HO Part B. . The results show that the average sleep cycle is not 24 hours. This allowed us to reject the null hypothesis. We used the data to determine if the average person’s sleep cycle was 24 hours. After examining the information we see that is not the case. 18. Twenty students randomly assigned to an experimental group receive an instructional program; 30 in a control group do not.After 6 months, both groups are tested on their knowledge. The experimental group has a mean of 38 on the test (with an estimated population standard deviation of 3); the control group has a mean of 35 (with an estimated population standard deviation of 5). Using the . 05 level, what should the experimenter conclude? (a) Use the steps of hypothesis testing, (b) sketch the distributions involved, and (c) explain your answer to someone who is familiar with the t test for a single sample but not with the t test for independent means. Exp Control a=. 05 N1=20 n2=30M1=38 m2= 35 S1=3 s2= 5 Step 1. Ho:M1=M2 H1:M1>M2 Step 2. S2 = 19(3)2 + 29(5)2 = 896 = 18. 667 30+20-2 48 (SM1) 2 = 18. 667/20 (SM2)2 = 18. 667/30 (SM1)2 = . 9333(SM2)2 = . 62222 S2 Difference = S2M1 + S2M2 = 1. 5556 DF= 20+30-2=48 Sdiff= 1. 5556 = 1. 247 Step 3 Cut off (=.. 05, DF= 48) = 1. 677 Step 4 Sample = 38-35 = 2. 405 1. 247 Step 5 Since 2. 405 > 1. 677 Reject H0 This exprerimanet shows that the on average the experiment group scored better than the controlled group. This is show that the treatment was successful. 17.Do students at various universities differ in how sociable they are? Twenty-five students were randomly selected from each of three universities in a region and were asked to report on the amount of time they spent socializing each day with other students. The result for University X was a mean of 5 hours and an estimated population variance of 2 hours; for University Y, ; and for University Z, . What should you conclude? Use the . 05 level. (a) Use the steps of hypothesis testing, (b) figure the effect size for the study; and (c) explain your answers to parts (a) and (b) to someone who has never had a course in statistics.College XCollegeYCollege Z M=5 S2=2M=4 S2=1. 5M=6 S2=2. 5 N=25N=25N=25 Step 1 Null H0: M1=M2=M3 (DON’T EXPECT DIFFERENCE) H1: At least one of them will differ. Step 2. Dfwi= 75-3=72 Dfbt= 3-2=1 F (2,72) Step 3. F( 2,72) =3. 125 Step 4. A) GM = 5+4+6 = 15 = 5 3 3 S2M = ( M-GM) 2 = (5-5)2 +(4-5)2 + (6-5)2 = 0+1+1 = 2 = 1 Dfbet3-1 2 2 S2wi= (2+1. 5+2. 5)/3 = 2 F= S2bet = 25 = 12. 5 S2wi 2 Step 5 12. 5>3. 125 which rejects H0 R2 = (S2 bet) (DF bet) = (25)(2) = 50 = . 577 (S2bet)(Dfbet) + (s2wi)(Dfwi) (25)(2) + (2)(72) 194 This study shows us that student do vary a great deal on how social they are from school to school. ******************Answers for 11, 12 ttached********************************* 11. Make up a scatter diagram with 10 dots for each of the following situations: (a) perfect positive linear correlation, (b) large but not perfect positive linear correlation, (c) small positive linear correlation, (d) large but not perfect negative linear correlation, (e) no correlation, (f) clear curvilinear correlation.For problems 12 to 14, do the following: (a) Make a scatter diagram of the scores; (b) describe in words the general pattern of correlation, if any; (c) figure the correlation coefficient; (d) figure whether the correlation is statistically significant (use the . 05 significance level, two-tailed); (e) explain the logic of what you have one, writing as if you are speaking to someone who has never heard of correlation (but who does understand the mean, deviation scores, and hypothesis testing); and (f) give three logically possible directions of causality, indicating for each direction whether it is a reasonable explanation for the correlation in light of the variables involved (and why). 12. Four research participants take a test of manual dexterity (high scores mean better dexterity)and an anxiety test (high scores mean more anxiety). The scores are as follows. (c) (e) (d) (f) Person Dexterity Anxiety 1 1 10 2 1 8 3 2 4 4 4 -2