X 2. s 1 n 1 s 2. n 2. s 2. 2 r 12

Transcription

1 Homework for t-tests -- one sample, two inepenent samples, an correlate samples Formulas X One sample t-test: t s/ n Two inepenent samples t-test: t X SE X s 1 s n 1 n Correlate samples t-test: t X SE s 1 s n 1 X n r 1 where SE is the stanar error of the ifference between means. s 1 n 1 s n For correlate samples t-test, the formula above can be teious, so a simpler formula can also be use: t s N s N SE In this formula, is the mean of the ifference between pairs of scores N an SE is the stanar error of the ifference SE s N where s is the variance of the ifference scores an is calculate just like a regular variance, except in this case it is the variance of the ifference scores s N 1 In short, the correlate t-test is simply the mean of the ifference between two sets of scores,, ivie by the stanar error of that ifference, SE, i.e., t /SE. These formulas will be illustrate in the exercises below.

2 Instructions For each of the problems below, provie the following: (a) Ientification of which statistical test shoul be use (b) Written null hypothesis (c) Symbolic null hypothesis () Written non-irectional hypothesis (e) Symbolic non-irectional hypothesis (f) Obtaine (calculate) statistic (g) f, critical t statistics (taken from table in back of text), an ecision (h) Written results in APA style. See Reporting statistical outcomes on the course web page for examples. I recommen that the first four exercises be calculate by han so you become more familiar with these test proceures. 1. A principal is concerne that isciplinary referrals are on the rise. Last year at the principal's mile school, the average number of iscipline referrals across all teachers uring the first month was 5.5. This year, the total number of referrals uring the first month from each teacher is recore below. Is there any evience that iscipline referrals have change from last year? (Calculate this one by han.) Teacher Number of Discipline Referrals Mae Betty 10 LaTonya Sue 8 Mary 7 Fre 6 John Wilma 9 Jim 3 Saunra 8 Wenel 10 Marie 1

3 . Male an Female stuents from a grauate class were aske their opinion about a particular textbook. The scale on which they replie range from 1strongly unfavorable to 5strongly favorable. Which group appears to have more favorable opinions regaring the textbook, males or females? (Calculate this one by han.) Male Opinion s Female Opinion s Below are incoming freshmen mathematics SAT scores for both males an females. Is there any evience of a ifference between them? (Calculate this one by han.) Males Females M 585 M 610 s 86.7 S 99.8 n 56 n Laies at Weight Watchers wish to know whether participating in Weight Watchers seems to have any effect on weight change. Below are ata collecte from several participants before an after attening Weight Watchers for a given perio of time. Does it appear that weight change results from attening Weight Watchers? (Calculate this one by han.) Weight Watcher Participant Weight before Weight after Betty LaTonya Sue Mary Frea Johnnie Wilma Jimmie Saunra Wenella Marie

4 5. A school counselor is intereste in learning whether small-group counseling helps freshmen stuents become more ientifie with school uring their first year of high school. That is, oes small-group counseling help stuents feel more attache to school; o they perceive school to be more important to them. At the beginning of the school year 15 freshmen stuents are ranomly selecte an require to atten small-group counseling sessions for six weeks. Prior to attening, an instrument esigne to measure their level of acaemic ientification with school is aministere to each stuent (higher scores on this instrument inicate greater levels of acaemic ientification). At the en of counseling sessions six week later, the same instrument is re-aministere to each stuent. Is there any evience stuents in small-group counseling show levels of acaemic ientification that iffer from levels taken at the beginning of the school year? Data below show levels of acaemic ientification for stuents before (start) an after (en) small-group counseling. Stuent Start En Stuent Start En Stuent Start En John 5 Brena Gaylor 5 19 Jerry 6 33 Bonnie 8 5 Ginger 9 33 Jeff 5 5 Briget 9 3 Gigi 8 35 James 10 1 Buffy 0 7 George 1 Jimmy Bertha 11 9 Georj One-group before-after type experimental esigns such as that use in #5 above are notoriously problematic an poor. Such esigns shoul never be use for serious ecision making. It is extremely important to have comparison groups so changes within groups can be compare. Given this, counselors intereste in performing a small-group intervention ientical to the type use in #5 above revise that stuy to inclue two groups of stuents. One group receive the small-group counseling, an the other i not. Measurement of school ientification was taken from both groups after six weeks of school (six weeks of counseling for the treatment group). Is there any evience that those in counseling have levels of acaemic ientification that iffer from those not receiving counseling? Stuents who receive small-group counseling Stuent Ientification Stuent Ientification Stuent Ientification John 5 Brena 33 Gaylor 19 Jerry 33 Bonnie 5 Ginger 33 Jeff 5 Briget 3 Gigi 35 James 1 Buffy 7 George Jimmy 35 Bertha 9 Georj 19

5 Stuents who i not receive small-group counseling Stuent Ientification Stuent Ientification Stuent Ientification Denver 6 Frank 19 Zoila 6 Doug 19 Frances 8 Zela 1 Deira 33 Freie 33 Zoraia Debbie 35 Florence 9 Zola 8 Dave 5 Felicia 30 Zora In the 1990s I use a test in EDUR 7130 (Introuction to Eucational Research) in which stuents, across many class sections, average about 89. That class was taught face-to-face. Since 000 all sections of that class are taught on-line rather than face-to-face. Recently I began using, in my on-line classes, the same test that I use in the 1990s for face-to-face classes. Below are actual scores from one on-line class from that test. Is there any evience that stuent achievement, as measure by this one test, iffers from typical stuent achievement in face-to-face classes? Test s from On-line Stuents

6 Answers 1. A principal is concerne that isciplinary referrals are on the rise. Last year at the principal's mile school, the average number of iscipline referrals across all teachers uring the first month was 5.5. This year, the total number of referrals uring the first month from each teacher is recore below. Is there any evience that iscipline referrals have change from last year? (Calculate this one by han.) (a) Ientification of which statistical test shoul be use one sample t-test (b) Written null hypothesis The number of isciplinary referrals from the first month of school this year will equal the average from the first month of school last year (M 5.5). (c) Symbolic null hypothesis H0: μ 5.5 () Written non-irectional hypothesis The number of isciplinary referrals from the first month of school this year will iffer from the average from the first month of school last year (M 5.5). (e) Symbolic non-irectional hypothesis H0: μ 5.5 (f) Obtaine (calculate) statistic One sample t-test: t X s/ n Mean for these ata is M 7.00 Stanar eviation for these ata is s SD Sample size is n 11 Calculate t is: t X s/ n / / (g) f, critical t statistics (taken from table in back of text), an ecision f n 1 10 critical t at α.05 is ±.8 Since 1.46 is less than.8 (oes not fall in the rejection region), fail to reject. Note also that the p- value, if statistical software is use to perform this test, is greater than.05 (p >.05, p.175), so fail to reject null.

7 (h) Written results in APA style. See Reporting statistical outcomes on the course web page for examples. The number of isciplinary referrals for the first month of this year oes not appear to iffer statistically, at the.05 level, from the average number of referrals uring the first month of last school year (M 7.00, SD 3.41, 95% CI 0.79 to 3.79, n 11, t 1.46, f 10, p.175). Although this year's mean is slightly higher, this fining oes not iffer from what one might expect from chance an therefore these ata o not support the principal's concern that number of isciplinary referrals is on the rise.. Male an Female stuents from a grauate class were aske their opinion about a particular textbook. The scale on which they replie range from 1strongly unfavorable to 5strongly favorable. Which group appears to have more favorable opinions regaring the textbook, males or females? (Calculate this one by han.) (a) Ientification of which statistical test shoul be use two inepenent samples t-test (b) Written null hypothesis There will be no ifference in opinion between males an females regaring the textbook. (c) Symbolic null hypothesis H0: μm μf () Written non-irectional hypothesis There will be a ifference in opinion between males an females regaring the textbook. (e) Symbolic non-irectional hypothesis H1: μm μf (f) Obtaine (calculate) statistic Two inepenent samples t-test: t Males M 4.333, s 0.817, n 6 Females M 1.833, s 0.983, n 6 X SE X s 1 s n 1 n X M F M F t s 1 s n 1 n M F 6 M 6 F M 6 M F 6 F M F

8 4.333 M F (g) f, critical t statistics (taken from table in back of text), an ecision f n 10 critical t at α.05 is ±.8 Since 4.79 is greater than.8 (falls in the rejection region), reject the null. Note also that the p-value, if statistical software is use to perform this test, is less than.05 (p <.05, p.001), so reject null. (h) Written results in APA style. See Reporting statistical outcomes on the course web page for examples. Table 1: Results of t-test an Descriptive Statistics of Text Opinion by Sex Outcome Group Male Female M SD n M SD n 95% CI for Mean Difference Text Opinion , * 10 * p <.05. Results of this stuy show that there is a statistically significant ifference in opinions about the text between males an females. As the means in Table 1 emonstrate, males hel more favorable opinions about the text than i females. t f 3. Below are incoming freshmen mathematics SAT scores for both males an females. Is there any evience of a ifference between them? (Calculate this one by han.) (a) Ientification of which statistical test shoul be use two inepenent samples t-test (b) Written null hypothesis There will be no ifference in mathematics SAT scores between males an females. (c) Symbolic null hypothesis H0: μm μf () Written non-irectional hypothesis There will be a ifference in mathematics SAT scores between males an females. (e) Symbolic non-irectional hypothesis H1: μm μf

9 (f) Obtaine (calculate) statistic Two inepenent samples t-test: t X SE X s 1 s n 1 n X 585 M 610 F 585 M 610 F t s 1 s n 1 n 86.7 M 99.8 F 56 M 531 F M 56 M F 531 F 585 M 610 F M 610 F (g) f, critical t statistics (taken from table in back of text), an ecision f n 1055 critical t at α.05 is ± 1.96 Since is less than a negative (falls in the rejection region), reject the null. Note also that the p- value, if statistical software is use to perform this test, is less than.05 (p <.05, p.001), so reject null. (h) Written results in APA style. See Reporting statistical outcomes on the course web page for examples. Table 1: Results of t-test an Descriptive Statistics for SAT by Sex Outcome SAT Math. scores * p <.05. Group Male Female M SD n M SD n 95% CI for Mean Difference , * 1055 t f Results of this stuy show that there is a statistically significant ifference in SAT scores between male an female stuents. Females, on average, ha the higher SAT mathematics scores.

10 4. Laies at Weight Watchers wish to know whether participating in Weight Watchers seems to have any effect on weight change. Below are ata collecte from several participants before an after attening Weight Watchers for a given perio of time. Does it appear that weight change results from attening Weight Watchers? (Calculate this one by han.) Weight Watcher Participant Weight before Weight after Betty LaTonya Sue Mary Frea Johnnie Wilma Jimmie Saunra Wenella Marie (a) Ientification of which statistical test shoul be use Correlate samples t-test (b) Written null hypothesis There will be no ifference in weight measurement before an after joining weight watchers. (c) Symbolic null hypothesis H0: μb μa () Written non-irectional hypothesis There will be a ifference in weight measurement before an after joining weight watchers. (e) Symbolic non-irectional hypothesis H1: μb μa (f) Obtaine (calculate) statistic t s N s N SE

11 Before After Difference, Mbefore ; SDbefore ; Mafter ; SDafter ; SS ; s SS/n / t s N (g) f, critical t statistics (taken from table in back of text), an ecision f n 1 10 critical t, at.05, is ±.8 Since is not larger than.8, fail to reject 95%CI ± (tcrit)(se) 95%CI ± (.8)(3.68) 95%CI ± , 7.83

12 (h) Written results in APA style. See Reporting statistical outcomes on the course web page for examples. Table 1: Descriptive Statistics an t-test Results for Weight Outcome Before Weight Watchers Time After Weight Watchers M SD M SD n 95% CI for Mean Difference Weight , * p <.05 t f Note --- correlation not calculate with irect ifference approach, so remove from table. There is not a statistically significant ifference in women's weight before or after joining Weight Watchers. Results show that weight loss oes not occur after joining Weight Watchers, at least for the 11 women sample. 5. A school counselor is intereste in learning whether small-group counseling helps freshmen stuents become more ientifie with school uring their first year of high school. That is, oes small-group counseling help stuents feel more attache to school; o they perceive school to be more important to them. At the beginning of the school year 15 freshmen stuents are ranomly selecte an require to atten small-group counseling sessions for six weeks. Prior to attening, an instrument esigne to measure their level of acaemic ientification with school is aministere to each stuent (higher scores on this instrument inicate greater levels of acaemic ientification). At the en of counseling sessions six week later, the same instrument is re-aministere to each stuent. Is there any evience stuents in small-group counseling show levels of acaemic ientification that iffer from levels taken at the beginning of the school year? Data below show levels of acaemic ientification for stuents before (start) an after (en) small-group counseling. Stuent Start En Stuent Start En Stuent Start En John 5 Brena Gaylor 5 19 Jerry 6 33 Bonnie 8 5 Ginger 9 33 Jeff 5 5 Briget 9 3 Gigi 8 35 James 10 1 Buffy 0 7 George 1 Jimmy Bertha 11 9 Georj 6 19 (a) Ientification of which statistical test shoul be use correlate samples t-test (b) Written null hypothesis There will be no ifference in level of acaemic ientification before an after small group counseling.

13 (c) Symbolic null hypothesis H0: μb μa () Written non-irectional hypothesis There will be a ifference in level of acaemic ientification before an after small group counseling. (e) Symbolic non-irectional hypothesis H1: μb μa (f) Obtaine (calculate) statistic t (g) f, critical t statistics (taken from table in back of text), an ecision f tcrit ±.145 Reject null since is less than (h) Written results in APA style. See Reporting statistical outcomes on the course web page for examples. Table 1: Descriptive Statistics an t-test Results for Acaemic Ientification Outcome Time 95% CI for Before After Mean Counseling Counseling Difference M SD M SD n Acaemic Ientification * p <.05 r t f , * 14 There is a statistically significant ifference in level of acaemic ientification isplaye by stuents before an after receiving small group counseling. After small group counseling, stuents isplaye a much higher level of acaemic ientification with school than before small group counseling. 6. One-group before-after type experimental esigns such as that use in #5 above are notoriously problematic an poor. Such esigns shoul never be use for serious ecision making. It is extremely important to have comparison groups so changes within groups can be compare. Given this, counselors intereste in performing a small-group intervention ientical to the type use in #5 above revise that stuy to inclue two groups of stuents. One group receive the small-group counseling, an the other i not. Measurement of school ientification was taken from both groups after six weeks of school (six weeks of counseling for the treatment group). Is there any evience that those in counseling have levels of acaemic ientification that iffer from those not receiving counseling?

14 Stuents who receive small-group counseling Stuent Ientification Stuent Ientification Stuent Ientification John 5 Brena 33 Gaylor 19 Jerry 33 Bonnie 5 Ginger 33 Jeff 5 Briget 3 Gigi 35 James 1 Buffy 7 George Jimmy 35 Bertha 9 Georj 19 Stuents who i not receive small-group counseling Stuent Ientification Stuent Ientification Stuent Ientification Denver 6 Frank 19 Zoila 6 Doug 19 Frances 8 Zela 1 Deira 33 Freie 33 Zoraia Debbie 35 Florence 9 Zola 8 Dave 5 Felicia 30 Zora 31 (a) Ientification of which statistical test shoul be use two inepenent samples t-test (b) Written null hypothesis There will be no ifference in level of school ientification between those stuents who receive small group counseling an those stuents who i not receive small group counseling. (c) Symbolic null hypothesis H0: μyes μno () Written non-irectional hypothesis There will be a ifference in level of school ientification between those stuents who receive small group counseling an those stuents who i not receive small group counseling. (e) Symbolic non-irectional hypothesis H1: μyes μno (f) Obtaine (calculate) statistic t

15 (g) f, critical t statistics (taken from table in back of text), an ecision f n 30 8 tcrit ±.048 Fail to reject null since is not less than (h) Written results in APA style. See Reporting statistical outcomes on the course web page for examples. Table 1: Results of t-test an Descriptive Statistics for School Ientification by Counseling Outcome School Ientification * p <.05. Small Group Counseling Yes No M SD n M SD n 95% CI for Mean Difference , t f Results of this stuy show that there is no statistically significant ifference in level of school ientification between those stuents who receive small group counseling an those who i not. Contrary to the previous stuy's finings, results of this stuy show that small group counseling oes not appear to show ifferences in school ientification for stuents. 7. In the 1990s I use a test in EDUR 7130 (Introuction to Eucational Research) in which stuents, across many class sections, average about 89. That class was taught face-to-face. Since 000 all sections of that class are taught on-line rather than face-to-face. Recently I began using, in my on-line classes, the same test that I use in the 1990s for face-to-face classes. Below are actual scores from one on-line class from that test. Is there any evience that stuent achievement, as measure by this one test, iffers from typical stuent achievement in face-to-face classes? Test s from On-line Stuents (a) Ientification of which statistical test shoul be use One sample t-test

16 (b) Written null hypothesis Average achievement on EDUR 7130 test scores for on-line stuents will not iffer from the average of test scores for face-to-face classes (M 89). (c) Symbolic null hypothesis H0: μ 89 () Written non-irectional hypothesis Average achievement on EDUR 7130 test scores for on-line stuents will iffer from the average of test scores for face-to-face classes (μ 89). (e) Symbolic non-irectional hypothesis H0: μ 89 (f) Obtaine (calculate) statistic t (g) f, critical t statistics (taken from table in back of text), an ecision f n critical t at α.05 is ±.056 Since is not less than (oes not fall in the rejection region), so fail to reject. Note also that the p-value, if statistical software is use to perform this test, is greater than.05 (p >.05, p.175), so fail to reject null. (h) Written results in APA style. See Reporting statistical outcomes on the course web page for examples. The average test scores for stuents on-line (M 86.69, SD 8.59, n 7) oes not iffer, statistically, from the average of face-to-face stuents (M 89; t -1.40, f 6, p.18, 95%CI to 1.09). These finings suggest that test performance seems to be similar between the two instructional formats.