?For each of the following, assume that the two samples are obtained from populations

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

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

One sample has SS = 72 and a second sample has SS = 24. a. If \(n=7\) 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 \(n=7\) for the first sample and \(n=11\) 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. Equation Transcription: Text Transcription: n=7 n=11

One sample has SS = 80 and a second sample has SS = 48. a. If n = 17 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 = 17 for the first sample and n = 5 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.

Two separate samples, each with n = 9 individuals, receive different treatments. After treatment, the first sample has SS = 546 and the second has SS = 606. 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?

Two separate samples receive different treatments. After treatment, the first sample has n = 5 with SS = 60, and the second has n = 9 with SS = 84. 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 7 points, is this enough to reject the null hypothesis using a two-tailed test with \(\alpha=.05\)?

Research results suggest a relationship between the TV viewing habits of 5 -year-old children and their future performance in high school. For example, Anderson, Huston, Wright, and Collins (1998) report that high school students who regularly watched Sesame Street as children had better grades in high school than their peers who did not watch Sesame Street. Suppose that a researcher intends to examine this phenomenon using a sample of 20 high school students. The researcher first surveys the students' parents to obtain information on the family's TV viewing habits during the time that the students were 5 years old. Based on the survey results, the researcher selects a sample of \(n=10\) students with a history of watching Sesame Street and a sample of \(n=10\) students who did not watch the program. The average high school grade is recorded for each student and the data are as follows: Use an independent-measures \(t\) test with \(\alpha=.01\), two-tailed, to determine whether there is a significant difference between the two types of high school student. Equation Transcription: Text Transcription: n=10 t alpha=.01

Does posting calorie content for menu items affect people's choices in fast-food restaurants? According to results obtained by Elbel, Gyamfi, and Kersh (2011), the answer is no. The researchers monitored the calorie content of food purchases for children and adolescents in four large fast-food chains before and after mandatory labeling began in New York City. Although most of the adolescents reported noticing the calorie labels, apparently the labels had no effect on their choices. Data similar to the results obtained show an average of \(M=786\) calories per meal with \(s=85\) for \(n=100\) children and adolescents before the labeling, compared to an average of \(M=772\) calories with \(s=91\) for a similar sample of \(n=100\) children after the mandatory posting. a. Use a two-tailed test with \(\alpha=.05\) to determine whether the mean number of calories after the posting is significantly different than before calorie content was posted. b. Calculate \(r^{2}\) to measure effect size for the mean difference. Equation Transcription: Text Transcription: M=786 s=85 n=100 M=772 s=91 alpha=.05 r^2

A long history of psychology research has demonstrated that memory is usually improved by studying material on multiple occasions rather than one time only. This effect is commonly known as distributed practice, or spacing effects. In a recent paper examining this effect, Cepeda et al. (2008) looked at the influence of different delays or gaps between study sessions. The results suggest that optimal long-term memory occurs when the study periods are spaced one to three weeks apart. In one part of the study, a group of participants studied a set of obscure trivia facts one day, returned the next day for a second study period, and then was tested five weeks later. A second group went through the same procedure but had a one-week gap between the two study sessions. The following data are similar to the results obtained in the study. Do the data indicate a significant difference between the two study conditions? Test with \(\alpha = .05\), two-tailed.

Recent research has shown that creative people are more likely to cheat than their less-creative counterparts (Gino & Ariely, 2012). Participants in the study first completed creativity assessment questionnaires and then returned to the lab several days later for a series of tasks. One task was a multiple-choice general knowledge test for which the participants circled their answers on the test sheet. Afterward, they were asked to transfer their answers to a bubble sheet for computer scoring. However, the experimenter admitted that the wrong bubble sheet had been copied so that the correct answers were still faintly visible. Thus, the participants had an opportunity to cheat and inflate their test scores. Higher scores were valuable because participants were paid based on the number of correct answers. However, the researchers had secretly coded the original tests and the bubble sheets so that they could measure the degree of cheating for each participant. Assuming that the participants were divided into two groups based on their creativity scores, the following data are similar to the cheating scores obtained in the study. _______________________________________________________ High-Creativity Low-Creativity Participants Participants _______________________________________________________ 27 27 7.41 4.78 749.5 830 ________________________________________________________ a. Use a one-tailed test with \(\alpha=.05\) to determine whether these data are sufficient to conclude that high-creativity people are more likely to cheat than people with lower levels of creativity. b. Compute Cohen’s d to measure the size of the effect. c. Write a sentence demonstrating how the results from the hypothesis test and the measure of effect size would appear in a research report. Equation Transcription: Text Transcription: alpha=.05

Anxiety affects our ability to make decisions. Remmers and Zander (2018) demonstrated that anxiety also prevents us from intuiting about our environments. In their experiment, 111 participants were randomly assigned to receive either an anxiety-inducing statement (e.g., “Safety is guaranteed neither in our neighborhoods nor in our own homes”) accompanied by a photograph of a dangerous situation, or an emotionally neutral statement (e.g., “A rolling pin is a kitchen tool that helps to extend dough”) accompanied by an innocuous image. Researchers then measured participants’ intuition index, which assesses ability to identify a word (e.g., “sea”) that was semantically related to a list of three words (e.g., “foam,” “deep,” and “salt”). Researchers observed reduced intuition among subjects who received anxiety inducing statements and imagery than among those who received innocuous stimuli. Data like those observed by the researchers are listed below: a. Are the test scores significantly lower for the participants who received anxiety-inducing statements? Use a two-tailed test with \(\alpha\) = .05. b. Compute the value of \(r^2\) (percentage of variance accounted for) for these data.

Positive events are great, but recent research suggests that unexpected positive outcomes (e.g., an unseasonably sunny day) predict greater-than-normal amounts of risk-taking and gambling (Otto, Fleming, & Glimcher, 2016). Researchers demonstrated this by comparing lottery sales—indicative of risk-taking—on normal days with lottery sales on days when some unexpected positive event occurred in the city. They observed increased sales after unexpected positive outcomes. Suppose that a researcher extends this observation to the laboratory and randomly assigns participants to two groups. Group 1 receives an unexpectedly large payment for participating and Group 2 receives the expected amount of compensation. The researcher then measures how much money the participants are willing to gamble in a game of chance. Test the one-tailed hypothesis that an unexpected positive outcome increased the amount of money that participants were willing to gamble. Use \(\alpha\) = .01.

Binge-watching a television show might not be the best way to enjoy a television series (Horvath, Horton, Lodge, & Hattie, 2017). Participants in an experiment watched an entire television series in the laboratory during either daily one-hour sessions or a single binge session. Participants were asked to rate their enjoyment of the television series on a scale of 0–100. Data like those observed by the authors are listed below. a. Test the hypothesis that binge-watching the television series resulted in less enjoyment of the show. Use \(\alpha = .05\), two-tailed. b. Compute Cohen’s d to measure the size of the effect. c. Write the results as they would appear in a scientific journal article.

What causes us to overeat? One surprising factor might be the material of the plate on which our food is served. Williamson, Block, and Keller (2016) gave n = 68 participants two donuts each and measured the amount of food that was wasted by each participant. In an independent samples design, participants received their donuts either on a disposable paper plate or on a reusable plastic plate. Data like those observed by the authors are listed below. a. Test the hypothesis that participants who received donuts on a paper plate wasted more food than participants who were served donuts on a plastic, reusable plate. Use \(\alpha\) = .05, two-tailed. b. Construct a 95% confidence interval to estimate the size of the mean difference. c. Write the results as they would appear in a scientific journal article.

In a classic study in the area of problem solving, Katona (1940) compared the effectiveness of two methods of instruction. One group of participants was shown the exact, step-by-step procedure for solving a problem and was required to memorize the solution. Participants in a second group were encouraged to study the problem and find the solution on their own. They were given helpful hints and clues, but the exact solution was never explained. The study included the problem in the following figure showing a pattern of five squares made of matchsticks. The problem is to change the pattern into exactly four squares by moving only three matches. (All matches must be used, none can be removed, and all the squares must be the same size.) After three weeks, both groups returned to be tested again. The two groups did equally well on the matchstick problem they had learned earlier. But when they were given new problems (similar to the matchstick problem), the memorization group had much lower scores than the group who explored and found the solution on their own. The following data demonstrate this result. a. Is there a significant difference in performance on new problems for these two groups? Use a two tailed test with \(\alpha\) = .05. b. Construct a 90% confidence interval to estimate the size of the mean difference. Incidentally, if you still have not discovered the solution to the matchstick problem, keep trying. According to Katona’s results, it would be very poor teaching strategy for us to give you the answer. If you still have not discovered the solution, however, check Appendix C, page 627.

A researcher conducts an independent-measures study comparing two treatments and reports the t statistic as \(t(20)=2.09\) a. How many individuals participated in the entire study? b. Using a two-tailed test with \(\alpha=.05\), is there a significant difference between the two treatments? c. Using a two-tailed test with \(\alpha=.01\), is there a significant difference between the two treatments? d. Compute \(r^{2}\) to measure the percentage of variance accounted for by the treatment effect. Equation Transcription: Text Transcription: t(20)=2.09 alpha=.05 alpha=.01 r^2

In a recent study, Piff, Kraus, Côté, Cheng, and Keltner (2010) found that people from lower socioeconomic classes tend to display greater prosocial behavior than their higher-class counterparts. In one part of the study, participants played a game with an anonymous partner. Part of the game involved sharing points with the partner. The lower economic class participants were significantly more generous with their points compared with the upper-class individuals. Results similar to those found in the study show that n = 12 lower-class participants shared an average of M = 5.2 points with SS = 11.91, compared to an average of M = 4.3 with SS = 9.21 for the n = 12 upper-class participants. a. Are the data sufficient to conclude that there is a significant mean difference between the two economic populations? Use a two-tailed test with \(\alpha\) = .05. b. Construct a 90% confidence interval to estimate the size of the population mean difference.

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

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

As noted on page 332, 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 \(\mu_{1}=\mu_{2}\) and calculate how much difference should be expected between the two sample means. a. One sample has \(n=6\) scores with \(S S=500\) and the second sample has \(n=12\) scores with \(S S=524\). b. One sample has \(n=6\) scores with \(S S=600\) and the second sample has \(n=12\) scores with \(S S=696\). 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 magnitude of the standard error for the sample mean difference? Equation Transcription: Text Transcription: mu_1 = mu_2 n=6 SS=500 n=12 SS=524 SS=600 SS=696

Two samples are selected from the same population. For each of the following, calculate how much difference is expected, on average, between the two sample means. a. One sample has n = 6, the second has n = 10, and the pooled variance is 135. b. One sample has n = 12, the second has n = 15, and the pooled variance is 135. c. In Part b, the sample sizes are larger but the pooled variance is unchanged. How does larger sample size affect the magnitude of the standard error for the sample mean difference?

For each of the following, assume that the two samples are obtained from populations with the same mean, and calculate how much difference should be expected, on average, between the two sample means. a. Each sample has n = 7 scores with \(s^2\) = 142 for the first sample and \(s^2\) = 110 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 = 28 scores with \(s^2\) = 142 for the first sample and \(s^2\) = 110 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?

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 \(s^2 = 17\), and the second sample has n = 8 scores and a variance of \(s^2 = 27\). b. Now the sample variances are increased so that the first sample has n = 4 scores and a variance of \(s^2 = 68\), and the second sample has n = 8 scores and a variance of \(s^2 = 108\). c. Comparing your answers for Parts a and b, how does increased variance influence the size of the estimated standard error?

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: a. Use an independent-measures t test with \(\alpha=.05\) to determine whether there is a significant difference between the two conditions and compute \(r^{2}\) to measure effect size. b. Now, increase the variability by doubling the two SS values to \(S S_{1}=1,020 \text { and } S S_{2}=828\). Repeat the hypothesis test and the measure of effect size. c. Comparing your answers for Parts a and b, describe how sample variability influences the outcome of the hypothesis test and the measure of effect size. Equation Transcription: Text Transcription: alpha =.05 r^2 SS_1 =1,020, and SS_2 =828