Psy201 Stats Ch 10 - Independent measures test

3 Assumptions of an independent t-test

(1) Observation of each sample must be independent (2) The two populations from which the sample is selected must be normal (3) The population from which the samples are selected must have equal variances "homogenize of variance""

If H0 is U1-U2=0 and the confidence interval is 3.798 to 12.202 and 0 is concluded as accepted in the confidence interval then it is the same saying __________

failure to reject H0

2. The homogeneity of variance assumption requires that the two sample variances be equal. (True or false?)

2. False. The assumption is that the two population variances are equal.

11. For each of the following, calculate the pooled variance and the estimated standard error for the sample mean difference a. The first sample has n = 4 scores and a variance of s2 = 55, and the second sample has n = 6 scores and a variance of s2 = 63. b. Now the sample variances are increased so that the first sample has n = 4 scores and a variance of s2 = 220, and the second sample has n = 6 scores and a variance of s2 = 252. c. Comparing your answers for parts a and b, how does increased variance influence the size of the estimated standard error?

11. a. The pooled variance is 60 and the estimated standard error is 5. b. The pooled variance is 240 and the estimated standard error is 10. c. Increasing the sample variance produces an increase in the standard error.

2. A researcher report states that there is a significant difference between treatments for an independent-measures design with t(28) = 2.27. a. How many individuals participated in the research study? (Hint: Start with the dfvalue.) b. Should the report state that p > .05 or p < .05?

2. a. The df = 28, so the total number of participants is 30. b. A significant result is indicated by p < .05.

4. Describe the homogeneity of variance assumption and explain why it is important for the independentmeasures t test

4. The homogeneity of variance assumption specifies that the variances are equal for the two populations from which the samples are obtained. If this assumption is violated, the t statistic can cause misleading conclusions for a hypothesis test.

standard error and it's relationship to sample varience and sample size (a) (Increase) Sample varience

(a) (Increase) standard error (increase) Sample error (b) (Increase) sample size (decrease) Sample errorq

1. Describe the basic characteristics of an independent measures, or a between-subjects, research study.

1. An independent-measures study uses a separate sample for each of the treatments or populations being compared.

2. Describe what is measured by the estimated standard error in the bottom of the independent-measures t statistic

2. The standard error for the independent measures t provides an estimate of the standard distance between a sample mean difference (M1 - M2) and the population mean difference (μ1 - μ2). When the two samples come from populations with the same mean (when H0 is true), the standard error indicates the standard amount of error (distance) between two sample means.

3. When you are using an F-max test to evaluate the homogeneity of variance assumption, you usually do not want to find a significant difference between the variances. (True or false?)

3. True. If there is a significant difference between the two variances, you cannot do the t test with pooled variance

6. One sample has SS = 70 and a second sample has SS = 42. a. Ifn = 8 for both samples, find each of the sample variances, and calculate the pooled variance. Because the samples are the same size, you should find that the pooled variance is exactly halfway between the two sample variances. b. Now assume that n = 8 for the first sample and n = 4 for the second. Again, calculate the two sample variances and the pooled variance. You should find that the pooled variance is closer to the variance for the larger sample

6. a. The first sample has a variance of 10, the second sample variance is 6, and the pooled variance is 8 (halfway between). b. The first sample has a variance of 10, the second sample variance is 14, and the pooled variance is 112/10 = 11.2 (closer to the variance for the larger sample).

9. Two separate samples receive two different treatments. The first sample has n = 9 with SS = 710, and the second has n = 6 with SS = 460. a. Compute the pooled variance for the two samples. b. Calculate the estimated standard error for the sample mean difference. c. If the sample mean difference is 10 points, is this enough to reject the null hypothesis using a twotailed test with a = .05? d. If the sample mean difference is 13 points, is this enough to reject the null hypothesis using a two tailed test with a = .05?

9. a. The pooled variance is 90. b. The estimated standard error is 5. c. A mean difference of 10 points produces t = 2.00. With critical boundaries of ±2.160, fail to reject H0 d. A mean difference of 13 points produces t = 2.60. With critical boundaries of ±2.160, reject H0

What's the difference between between subject design and repeated measures design?

Between subject design is the study of two different groups while repeated measures research/within subject design is two sets of data which is taken from the same group of participants

homogenize of variance

The population from which the samples are selected must have equal variances

What does Hartey's F-Max test do?

This test tries to get an understanding of how different the sample variances are. If it fails to reject H0 then it is good news because the differences are not extreme enough to be a problem

Three assumptions that should be satisfied before you use the independentmeasures t formula for hypothesis testing:

1. The observations within each sample must be independent (see p. 254). 2. The two populations from which the samples are selected must be normal. 3. The two populations from which the samples are selected must have equal variances.

1. A researcher is using an independent-measures design to evaluate the difference between two treatment conditions with n = 8 in each treatment. The first treatment produces M = 63 with a variance of s2 = 18, and the second treatment has M = 58 with s2 = 14. a. Use a one-tailed test with a = .05 to determine whether the scores in the first treatment are significantly greater than the scores in the second. (Note: Because the two samples are the same size, the pooled variance is simply the average of the two sample variances.) b. Predict how the value for the t statistic would be affected if the two sample variances were increased to s2 = 68 and s2 — 60. Compute the new t to confirm your answer. c. Predict how the value for the t statistic for the original samples would be affected if each sample had n = 32 scores (instead of n = 8). Compute the new t to confirm your answer.

1. a. The pooled variance is 16, the estimated standard error is 2, and t(14) = 2.50. With a one-tailed critical value of 1.761, reject the null hypothesis. Scores in the first treatment are significantly higher than scores in the second. b. Increasing the variance should lower the value of t. The new pooled variance is 64, the estimated standard error is 4, and t(14) = 1.25. c. Increasing the sample sizes should increase the value of t. The pooled variance is still 16, but the new standard error is 1, and t(62) = 5.00.

. An educational psychologist would like to determine whether access to computers has an effect on grades for high school students. One group of n = 16 students has home room each day in a computer classroom in which each student has a computer. A comparison group of n = 16 students has home room in a traditional classroom. At the end of the school year, the average grade is recorded for each student. The data are as follows: Computer Traditional M = 86 M = 82.5 SS = 1005 SS = 1155 a. Is there a significant difference between the two groups? Use a two-tailed test with a = .05. b. Compute Cohen's d to measure the size of the difference. c. Write a sentence that demonstrates how the outcome of the hypothesis test and the measure of effect size would appear in a research report. d. Compute the 90% confidence interval for the population mean difference between a computer classroom and a regular classroom.

1. a. The pooled variance is 72, the standard error is 3, and t = 1.17. With a critical value of t = 2.042, fail to reject the null hypothesis. b. Cohen's d = 3.5A172 = 0.412 c. The results show no significant difference in grades for students with computers compared to students without computers, t(30) = 1.17, p > .05, d = 0.412. d. With df = 30 and 90% confidence, the t values for the confidence interval are ± 1.697. The interval is p., — p,2 = 3.5 ± 1.697(3). Thus, the population mean difference is estimated to be between —1.591 and 8.591. The fact that zero is an acceptable value (inside the interval) is consistent with the decision that there is no significant difference between the two population means

10. For each of the following, assume that the two samples are selected from populations with equal means and calculate how much difference should be expected, on average, between the two sample means. a. Each sample has n = 5 scores with s2 = 38 for the first sample and s2 = 42 for the second. (Note: Because the two samples are the same size, the pooled variance is equal to the average of the two sample variances.) b. Each sample has n = 20 scores with s2 = 38 for the first sample and s2 = 42 for the second. c. In part b, the two samples are bigger than in part a, but the variances are unchanged. How does sample size affect the size of the standard error for the sample mean difference?

10. a. The estimated standard error for the sample mean difference is 4 points. b. The estimated standard error for the sample mean difference is 2 points. c. Larger samples produce a smaller standard error.

12. A researcher conducts an independent-measures study comparing two treatments and reports the t statistic as t(30) = 2.085. a. How many individuals participated in the entire study b. Using a two-tailed test with a = .05, is there a significant difference between the two treatments? c. Compute r2 to measure the percentage of variance accounted for by the treatment effect.

12. a. The two samples combined have a total of 32 participants. b. With df = 30 and α = .05, the critical region consists of t values beyond 2.042. The t statistic is in the critical region. Reject H0 and conclude that there is a significant difference. c. r2 =4.35/34.35 = 0.127 or 12.7%

13. Hallam, Price, and Katsarou (2002) investigated the influence of background noise on classroom performance for children aged 10 to 12. In one part of the study, calming music led to better performance on an arithmetic task compared to a no-music condition. Suppose that a researcher selects one class of n = 18 students who listen to calming music each day while working on arithmetic problems. A second class of n = 18 serves as a control group with no music. Accuracy scores are measured for each child and the average for students in the music condition is M = 86.4 with SS = 1550 compared to an average of M = 78.8 with SS = 1204 for students in the nomusic condition. a. Is there a significant difference between the two music conditions? Use a two-tailed test with a = .05. b. Compute the 90% confidence interval for the population mean difference. c. Write a sentence demonstrating how the results from the hypothesis test and the confidence interval would appear in a research report

13. a. Using df = 30, , because 34 is not listed in the table, and α = .05, the critical region consists of t values beyond 2.042. The pooled variance is 81, the estimated standard error is 3, and t(34) = 7.6/3 = 2.53. The t statistic is in the critical region. Reject H0 and conclude that there is a significant difference. b. For 90% confidence, the t values are 1.697 (using df = 30), and the interval extends from 2.509 to 12.691 points higher with the calming music. c. Classroom performance was significantly better with background music, t(34) = 2.53, p < .05, 95% CI [2.509, 12.691].

14. Do you view a chocolate bar as delicious or as fattening? Your attitude may depend on your gender. In a study of American college students, Rozin, Bauer, and Catanese (2003) examined the importance of food as a source of pleasure versus concerns about food associated with weight gain and health. The following results are similar to those obtained in the study. The scores are a measure of concern about the negative aspects of eating. Males Females n = 9 n = 15 M = 33 M = 42 SS = 740 SS = 1240 a. Based on these results, is there a significant difference between the attitudes for males and for females? Use a two-tailed test with a = .05. b. Compute ?, the percentage of variance accounted for by the gender difference, to measure effect size for this study. c. Write a sentence demonstrating how the result of the hypothesis test and the measure of effect size would appear in a research report.

14. a. The pooled variance is 90, the estimated standard error is 4, and t = 9/4 = 2.25. With df = 22 the critical value is 2.074. Reject the null hypothesis and conclude that there is a significant difference in attitude between males and females. b. r2 = 5.06/27.06 = 0.187 or 18.7% c. The results show a significant difference between males and females in their attitude toward food, t(22) = 2.25, p < .05, r2 = 0.187.

15. In a study examining overweight and obese college football players, Mathews and Wagner (2008) found that on average both offensive and defensive linemen exceeded the at-risk criterion for body mass index (BMI). BMI is a ratio of body weight to height squared and is commonly used to classify people as overweight or obese. Any value greater than 30 kg/m2 is considered to be at risk. In the study, a sample of n = 17 offensive linemen averaged M = 34.4 with a 348 CHAPTER 10 THE t TEST FOR TWO INDEPENDENT SAMPLES standard deviation ofs = 4.0. A sample ofn = 19 defensive linemen averaged M = 31.9 with s = 3.5. a. Use a single-sample t test to determine whether the offensive linemen are significantly above the at-risk criterion for BMI. Use a one-tailed test with a = .01. b. Use a single-sample t test to determine whether the defensive linemen are significantly above the at-risk criterion for BMI. Use a one-tailed test with a = .01. c. Use an independent-measures t test to determine whether there is a significant difference between the offensive linemen and the defensive linemen. Use a two-tailed test with a = .01.

15. a. For the offensive linemen, the standard error is 0.97 and t = 4.54. For a one-tailed test with df = 16, the critical value is 2.583. Reject the null hypothesis. The offensive linemen are significantly above the criterion for BMI. b. For the defensive linemen, the standard error is 0.80 and t = 2.375. For a one-tailed test with df = 18, the critical value is 2.552. Fail to reject the null hypothesis. The defensive linemen are not significantly above the criterion for BMI. c. For the independent-measures t, the pooled variance is 14.01, the estimated standard error is 1.25, and t(34) = 2.00. For a two-tailed test using df = 30 (because 34 is not listed), the critical value is 2.750. Fail to reject the null hypothesis. There is no significant difference between the two groups.

16. Functional foods are those containing nutritional supplements in addition to natural nutrients. Examples include orange juice with calcium and eggs with omega-3. Kolodinsky, et al. (2008) examined attitudes toward functional foods for college students. For American students, the results indicated that females had a more positive attitude toward functional foods and were more likely to purchase them compared to males. In a similar study, a researcher asked students to rate their general attitude toward functional foods on a 7-point scale (higher score is more positive). The results are as follows: Females Male --= 8 n = 12 M = 4.69 M = 4.43 SS = 1.60 SS = 2.72 a. Do the data indicate a significant difference in attitude for males and females? Use a two-tailed test with a = .05. b. Compute r2, the amount of variance accounted for by the gender difference, to measure effect size. c. Write a sentence demonstrating how the results of the hypothesis test and the measure of effect size would appear in a research report

16. a. The pooled variance is 0.24, the estimated standard error is 0.22, and t = 1.18. For a two-tailed test with df = 18 the critical value is 2.101. Fail to reject the null hypothesis. There is no significant difference between the two groups. b. For these data, r2 = 1.39/19.39 = 0.072 or 7.2%. c. The data showed no significant difference in attitude toward functional foods for males compared with females, t(18) = 1.18, p > .05, r2 = 0.072.

17. In 1974, Loftus and Palmer conducted a classic study demonstrating how the language used to ask a question can influence eyewitness memory. In the study, college students watched a film of an automobile accident and then were asked questions about what they saw. One group was asked, "About how fast were the cars going when they smashed into each other?" Another group was asked the same question except the verb was changed to "hit" instead of "smashed into." The "smashed into" group reported significantly higher estimates of speed than the "hit" group. Suppose a researcher repeats this study with a sample of today's college students and obtains the following results. Estimated Speed Smashed into n = 15 M = 40.8 SS = 510 a. Do the results indicate a significantly higher estimated speed for the "smashed into" group? Use a one-tailed test with a = .01. b. Compute the estimated value for Cohen's d to measure the size of the effect. c. Write a sentence demonstrating how the results of the hypothesis test and the measure of effect size ld appear in a research report Hit = IS M = 34.0 SS = 414

17. a. The research prediction is that participants who hear the verb "smashed into" will estimate higher speeds than those who hear the verb "hit." For these data, the pooled variance is 33, the estimated standard error is 2.10, and t(28) = 3.24. With df = 28 and α = .01, the critical value is t = 2.467. The sample mean difference is in the right direction and is large enough to be significant. Reject H0. b. The estimated Cohen's d = 6.8/√33 = 1.18. c. The results show that participants who heard the verb "smashed into" estimated significantly higher speeds than those who heard the verb "hit," t(28) = 3.24, p < .01, d = 1.18.

18. Numerous studies have found that males report higher self-esteem than females, especially for adolescents (Kling, Hyde, Showers, & Buswell, 1999). Typical results show a mean self-esteem score ofM = 39.0 with SS = 60.2 for a sample ofn = 10 male adolescents and a mean ofM = 35.4 withSS = 69.4 for a sample of n = 10 female adolescents. a. Do the results indicate that self-esteem is significantly higher for males? Use a one-tailed test with a = .01. b. Use the data to make a 95% confidence interval estimate of the mean difference in self-esteem between male and female adolescents. c. Write a sentence demonstrating how the results from the hypothesis test and the confidence interval would appear in a research report.

18. a. The pooled variance is 7.2, the estimated standard error is 1.2, and t(18) = 3.00. For a one-tailed test with df = 18 the critical value is 2.552. Reject the null hypothesis. There is a significant difference between the two groups. b. For 95% confidence, the t values are 2.101, and the interval extends from 1.079 to 6.121 points higher for boys. c. The results indicate that adolescent males have significantly higher self-esteem than girls, t(18) = 3.00, p < .01, one tailed, 95% CI [1.079, 6.121].

19. A researcher is comparing the effectiveness of two sets of instructions for assembling a child's bike. A sample of eight fathers is obtained. Half of the fathers are given one set of instructions and the other half receives the second set. The researcher measures how much time is needed for each father to assemble the bike. The scores are the number of minutes needed by each participant. Instruction Set I Instruction Set II 8 14 4 10 8 6 4 10 a. Is there a significant difference in time for the two sets of instructions? Use a two-tailed test at the .05 level of significance. b. Calculate the estimated Cohen's d and r2 to measure effect size for this study.

19. a. The null hypothesis states that there is no difference between the two sets of instructions, H0: μ1 - μ2 = 0. With df = 6 and α = .05, the critical region consists of t values beyond ±2.447. For the first set, M = 6 and SS = 16. For the second set, M = 10 with SS = 32. For these data, the pooled variance is 8, the estimated standard error is 2, and t(6) = 2.00. Fail to reject H0. The data are not sufficient to conclude that there is a significant difference between the two sets of instructions. b. For these data, the estimated d = 4/√8 = 1.41 (a very large effect) and r2 = 4/10 = 0.40 (40%).

20. When people learn a new task, their performance usually improves when they are tested the next day, but only if they get at least 6 hours of sleep (Stickgold, Whidbee, Schirmer, Patel, & Hobson, 2000). The following data demonstrate this phenomenon. The participants learned a visual discrimination task on one day, and then were tested on the task the following day. Half of the participants were allowed to have at least 6 hours of sleep and the other half were kept awake all night. Is there a significant difference between the two conditions? Use a two-tailed test with a = .05. Performance Scores 6 Hours of Sleep No Sleep n = 14 n = 14 M = 72 M = 65 SS = 932 SS = 706

20. The pooled variance is 63, the estimated standard error is 3.00, and t = 7/3 = 2.33. With df = 26 the critical value is 2.056. Reject the null hypothesis and conclude that there is a significant difference between the two sleep conditions.

21. Steven Schmidt (1994) conducted a series of experiments examining the effects of humor on memory. In one study, participants were given a mix of humorous and nonhumorous sentences and significantly more humorous sentences were recalled. However, Schmidt argued that the humorous sentences were not necessarily easier to remember, they were simply preferred when participants had a choice between the two types of sentence. To test this argument, he switched to an independent-measures design in which one group got a set of exclusively humorous sentences and another group got a set of exclusively nonhumorous sentences. The following data are similar to the results from the independentmeasures study. Humorous Sentences Non humorous Sentence 4 5 2 4 6 3 5 3 6 7 6 6 3 4 2 6 2 5 4 3 4 3 4 4 3 3 5 3 5 2 6 4 Do the results indicate a significant difference in the recall of humorous versus nonhumorous sentences? Use a two-tailed test with a = .05.

21. The humorous sentences produced a mean of M = 4.25 with SS = 35, and the non- humorous sentences had M = 4.00 with SS = 26. The pooled variance is 2.03, the estimated standard error is 0.504, and t = 0.496. With df = 30, the critical value is 2.042. Fail to reject the null hypothesis and conclude that there is no significant difference in memory for the two types of sentences

22. Downs and Abwender (2002) evaluated soccer players and swimmers to determine whether the routine blows to the head experienced by soccer players produced long-term neurological deficits. In the study, neurological tests were administered to mature soccer players and swimmers and the results indicated significant differences. In a similar study, a researcher obtained the following data. Swimmers Soccer players 10 7 8 4 7 9 9 3 13 7 7 6 12 a. Are the neurological test scores significantly lower for the soccer players than for the swimmers in the control group? Use a one-tailed test with a = .05. b. Compute the value of r2 (percentage of variance accounted for) for these data.

22. a. The null hypothesis states that the type of sport does not affect neurological performance. For a one-tailed test, the critical boundary is t = 1.796. For the swimmers, M = 9 and SS = 44. For the soccer players, M = 6 and SS = 24. The pooled variance is 6.18 and t(11) = 2.11. Reject H0. The data show that the soccer players have significantly lower scores. b. For these data, r2 = 0.288 (28.8%).

23. Research has shown that people are more likely to show dishonest and self-interested behaviors in darkness than in a well-lit environment (Zhong, Bohns, & Gino, 2010). In one experiment, participants were given a set of 20 puzzles and were paid $0.50 for each one solved in a 5-minute period. However, the participants reported their own performance and there was no obvious method for checking their honesty. Thus, the task provided a clear opportunity to cheat and receive undeserved money. One group of participants was tested in a room with dimmed lighting and a second group was tested in a well-lit room. The reported number of solved puzzles was recorded for each individual. The following data represent results similar to those obtained in the study. Well-Lit Room Dimly Lit Room 7 9 8 II 10 13 6 10 8 II 5 7 IS 12 14 5 10 a. Is there a significant difference in reported performance between the two conditions? Use a two-tailed test with a = .01. b. Compute Cohen's d to estimate the size of the treatment effect.

23. a. The null hypothesis states that the lighting in the room does not affect behavior. For the well-lit room the mean is M = 7.55 with SS = 42.22. For the dimly-lit room, M = 11.33 with SS = 38. The pooled variance is 5.01, the standard error is 1.06, and t(16) = 3.57. With df = 16 the critical values are 2.921. Reject the null hypothesis and conclude that the lighting did have an effect on behavior. b. d = 3.78/2.24 = 1.69

3. If other factors are held constant, explain how each of the following influences the value of the independent-measures t statistic and the likelihood of rejecting the null hypothesis: a. Increasing the number of scores in each sample. b. Increasing the variance for each sample.

3. a. The size of the two samples influences the magnitude of the estimated standard error in the denominator of the t statistic. As sample size increases, the value of t also increases (moves farther from zero), and the likelihood of rejecting H0 also increases. b. The variability of the scores influences the estimated standard error in the denominator. As the variability of the scores increases, the value of t decreases (becomes closer to zero), and the likelihood of rejecting H0 decreases.

5. One sample has SS = 48 and a second sample has SS = 32. a. If n = 5 for both samples, find each of the sample variances and compute the pooled variance. Because the samples are the same size, you should find that the pooled variance is exactly halfway between the two sample variances. b. Now assume that n = 5 for the first sample and n = 9 for the second. Again, calculate the two sample variances and the pooled variance. You should find that the pooled variance is closer to the variance for the larger sample

5. a. The first sample has s2 = 12 and the second has s2 = 8. The pooled variance is 80/8 = 10 (halfway between). b. The first sample has s2 = 12 and the second has s2 = 4. The pooled variance is 80/12 = 6.67 (closer to the variance for the larger sample).

7. As noted on page 320, when the two population means are equal, the estimated standard error for the independent-measures t test provides a measure of how much difference to expect between two sample means. For each of the following situations, assume that p = i.1.2 and calculate how much difference should be expected between the two sample means. a. One sample has n = 8 scores with SS = 45 and the second sample has n = 4 scores with SS = 15. b. One sample has n = 8 scores with SS = 150 and the second sample has n = 4 scores with SS = 90. c. In part b, the samples have larger variability (bigger SS values) than in part a, but the sample sizes are unchanged. How does larger variability affect the size of the standard error for the sample mean difference?

7. a. The pooled variance is 6 and the estimated standard error is 1.50. b. The pooled variance is 24 and the estimated standard error is 3. c. Larger variability produces a larger standard error.

8. Two separate samples, each with n = 12 individuals, receive two different treatments. After treatment, the first sample has SS = 1740 and the second has SS = 1560. a. Find the pooled variance for the two samples. b. Compute the estimated standard error for the sample mean difference. c. If the sample mean difference is 8 points, is this enough to reject the null hypothesis and conclude that there is a significant difference for a two-tailed test at the .05 level? PROBLEMS 347 d. If the sample mean difference is 12 points, is this enough to indicate a significant difference for a two-tailed test at the .05 level? e. Calculate the percentage of variance accounted for (r2) to measure the effect size for an 8-point mean difference and for a 12-point mean difference.

8. a. The pooled variance is 150. b. The estimated standard error is 5.00. c. A mean difference of 8 would produce t = 8/5 = 1.60. With df = 22 the critical values are ±2.074. Fail to reject H0. d. A mean difference of 12 would produce t = 12/5 = 2.40. With df = 22 the critical values are ±2.074. Reject H0. e. With a mean difference of 8 points, r2 = 0.104. With a difference of 12 points, r2 = 0.207.