t test Exercises (Dependent samples)

Transcription

1 t test Exercises (ependent samples) Steps to solving t test problems:. etermine the null and alternate hypotheses. Sketch out the problem 3. Calculate scores. Calculate the standard error a. Calculate the variance b. Calculate the standard error 5. Calculate t obt 6. etermine t crit and evaluate t obt Critical values: Formulas you ll need: SS = ( M ) s sm OR SS = n = s n SS = ( ) n df = n One tail t crit = + table value or table value Two tail t crit = +/ table value t obt = M μ s M A You just got out of your Psychology lecture where today s topic was Sleep. The teacher said that sleep was very important for memory, and thus staying up all night studying was not a good way to do well on exams. Since you and your friends always stay up all night studying before quizzes and exams, this news got you worried. You decide to test this claim. First, you select 5 friends that are all taking the same Calculus class. You know that this particular instructor gives a quiz every week, so on the first week of your experiment, you have your friends do their usual all night cramming session, and the next day after they take that week s quiz you ask them all what scores they got. The following week, you get your friends to NOT stay up all night studying, but instead go to bed early enough so they could get 8 hours of sleep. The following day after their quiz you got their quiz scores and compared them to the scores from the previous week using a dependent samples t test. Their scores are below: Friend No Sleep Good Sleep Beth 8 Sally Liam 5 Oren 5 8 Sarah 7 0 H : s = df = M = s M = t crit = YES / NO Quiz scores are significantly higher after getting sleep.

2 B You manage a juvenile detention unit at the local county jail, and since your budget is being cut you need to do something to reduce costs. Fights breaking out between the inmates have always been a problem, and besides leading to injuries, they are also expensive. amaged property needs to be repaired or replaced, injuries need to be treated, and often additional guards need to be on duty. If you could do something to reduce the frequency of fights, you could also reduce your costs and improve the situation for all involved. You think that something simple like putting a divider in the main recreation room to make two TV watching areas instead of just one like the room is set up now, might help with fights that break out over what to watch. You identify 6 inmates with the highest frequencies of starting fights, and you record how many fights they start during a week. You then have a divider installed in the recreation room, and you make the one room into two separate TV watching areas with their own TVs, chairs, etc. After a week to allow everybody to get used to the changes you again track these 6 inmates and record how many fights they start for a week. The numbers are below: Inmate Before nd TV After nd TV H : s = df = M = s M = t crit = YES / NO The inmates started significantly fewer fights after the second TV was installed. C You are the manager of a data entry shop with 6 full time data entry personnel that do the data entry. Basically, you receive things like surveys and registration cards that are hand written and you type them into electronic databases so they can be used. For example, you may get a subscription form that falls out of a magazine, decide you want to subscribe, and write in your name and address and mail it in. When the card is received it needs to be typed into some computer system so it can generate your bill, and create the labels so you can get your magazines. You are curious as to how playing music would affect the amount of work that got done, so you decided to measure the average number of entries each of your six employees made during a 5 day period. Then, you started playing music during work hours for weeks. uring the second of the weeks you again measured the average number of entries per employee. Their data are below: Employee No Music Music Sylvia arlene 6 67 Zach 6 5 ave 8 Nikita 7 00 olores 8 87 H : s = df = M = s M = t crit = YES / NO The number of entries is significantly different when music is played. It is (higher/lower) with music

3 You are concerned that a great social injustice is going unnoticed The bias against people whose name starts with a curvy letter. Yes, you believe that society has for too long preferred the Marks and Lindas over the Susans or Jims. To demonstrate this you select 6 people with names that start with straight line letters (e.g. L, M, N), and then you select 6 people whose name starts with a curvy letter (e.g. S, O, C) and are matched to one of the straight letter people based on level of education, age, gender, job, family size, and geographical location. For each of the people you recorded their yearly income for the previous year. If your claims are right, then straight letter people should make significantly more money than curvy letter people. Their data (in thousands of dollars) are below: Straight Letter Curvy Letter Person Income Person Income Lynda 7 Candace 76 Amy Jenny 0 3 Aida 3 Oprah 6 Edward 0 Sam 5 5 Lance 3 5 Quaid 00 6 Mark 35 6 Otis 3 H : s = df = M = s M = t crit = YES / NO Straight letter people make significantly more money than curvy letter people.

4 Answers A We are interested in seeing if quiz scores are HIGHER after gettingg 8 hours of sleep. Since we are only interested in one direction (we are not looking to see if quiz scores are LOWER, only HIGHER), this is a one tail test. Since we want to know if scores are HIGHER (a bigger value on the quiz scores for the quiz after Good Sleep than for the quiz after No Sleep), the way the numbers are currently set up (No Sleep Good Sleep =? This should give us a negative number if Good Sleep scoress are larger than No Sleep scores) this is a lower taill test. If we look at the table of critical values for t for one tail values for degrees of freedom at an α =.05 we get.3. Since the lower tail is critical, the critical value for t (t crit ) is.3 Friend No Sleep Beth 8 Sally Liam Oren 5 Sarah 7 Good Sleep Σ = 8 Σ = 0 H 0 : µ 0 H : µ < 0 M =.60 SS = 7.0 s =.80 s M = 0.60 t obt =.67 df = t crit =.3 YES Quiz scoress are significantly higher after getting sleep. t()=.67, p <.05

5 B We are interested in seeing if there are FEWER fights with a second TV. Since we are only interested in one direction (we are not looking to see if there are MORE fights, only LESS), this is a one tail test. Since we want to know if fights are LOWER (a smaller value for fights started with the second TV), the way the numbers are currently set up (Before nd TV After nd TV =? This should give us a positive number if Before fights are more than After fights) this is an upper tail test. If we look at the table of critical values for t for one tail values for 5 degrees of freedom at an α =.05 we get.05. Since the upper tail is critical, the critical value for t (t crit ) is Inmate Before nd TV After nd TV Σ = Σ = 3 H 0 0: µ 0 H : µ > 0 M =.83 SS = t ob bt = 3.05 s =.7 df = 5 s M = 0.60 t cr it =.05 YES The inmates started significantly fewer fights after the second TV was installed. t(5) = 3.06, p <.05

6 C In this question we are interested in seeing if there is a differencee in productivity between working with music and working without. Basically we want to know if one condition leads to more work getting done, and we don t have any reason to prefer one possibility over the other. We want to catch both possibilities, that employees are more productivee without music than with music, or that employees are more productive with music than without music. This will therefore be a two tail test. If we look at the table of critical values for t for two tail values for 5 degrees of freedom at an α =.05 we get.57. Since both tails are critical, the critical value for t (t crit ) is ±.57. Employee Sylvia arlene Zach ave Nikita olores No Music Music Σ = Σ = 3 H 0 : µ = 0 H : µ 0 M = 3.83 SS = 3.83 s = 6.7 s M =.08 t obt = 3.56 df = 5 t crit = ±.57 YES The number of entries is significantly different when music is played. It is higher with music t(5) = 3.56,, p <.0

7 In this question we are interested in seeing if peoplee with straight letter names make MORE money. Since we are only interested in one direction (we are not looking to see if they make LESS money, only MORE), this is a one tail test. Since we want to know if income is HIGHER (a larger value for income), the way the numbers are currently set up (Straight Letter Curvy Letter =? This should give us a positive number if Straight Letter incomes and more than Curvy Letter incomes) this is an upper tail test. If we look at the table of critical values for t for one tail values for 5 degrees of freedom at an α =.05 we get.05. Since the upper tail is critical, thee critical value for t (t crit ) is Straight Letter Person Income Lynda 7 Amy 3 Aida Edward 0 5 Lance 3 6 Mark 35 Curvy Letter Person Income Candace 76 Jenny 0 3 Oprah 6 Sam 5 5 Quaid 00 6 Otis Σ = 8 Σ = H 0 : µ 0 H : µ > 0 M = 3.00 SS = 7.00 s =.0 s M =.55 t obt =. df = 5 t crit = +.05 NO Straight letter people O NOT make significantly more money than curvy letter people. t(5) =.,, p = n.s.