1. One sample has SS=100 and a second sample has SS=124.
a. If n=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 the 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.
2. Two separate samples receive different treatments. After treatment, the first sample has n=10 with SS=500, and the second sample has n=6 with SS=340.
a. Compute pooled variance for the two samples.
b. Calculate 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 using a two-tailed test with α=.05?
3. In a recent study, Piff, Kraus, Cote, Cheng and Keitner (2010) found that people from lower social economic 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 α=.01.
b. Construct an 90% confidence interval to estimate the size of the population mean difference.
4. Recent research has demonstrated that music-based physical training for elderly peopld can improve balance and walking efficiency and reduce the risk of falls (Trombetti et al., 2011). As part of the training, participants walked in time to music and responded to changes in the music's rhythm during a 1-hour per week exercise program. After 6 months, participants in the training group increased their walking speed and their stride length compared to individuals in the control group. The following data are similar to the results obtained in the study.
Exercise Group Control Group
Stride length Stride length
24, 25, 22, 24 26, 23, 20, 23
26, 17, 21, 22 20, 16, 21, 17
22, 19, 24, 23 18, 23, 16, 20
23, 28, 25, 23 25, 19, 17, 16
a. Do the results indicate a significant difference in the stride length for the two groups? Use a two-tailed test with α=.05.
5. 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 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, 8, 10, 6, 8 9, 11, 13, 10, 11
5, 7, 12, 5 9, 15, 14, 10
a. Is there a significant difference in reported performance between the two conditions? Use a two-tailed test with α=.05.
b. Compute Cohen's d to estimate the size of the treatment effect.
6. A sample of difference scores from a repeated-measures experiment has a mean of MD=3 with a standard deviation of s=4.
a. If n=16, is this sample sufficient to reject the null hypothesis using a two-tailed test with α=.05?
b. If s=12, is this sample sufficient to reject the null hypothesis using a two-tailed test with α=.05?
7. A researcher would like to examine how the chemical tryptophan, contained in foods such as turkey, can affect mental alertness. A sample of n=9 college students is obtained, and each student's performance on a familiar video game is measured before and after eating a traditional Thanksgiving dinner including roast turkey. The average score dropped by MD=14 points after the meal with SS=1152 for the difference scores.
a. Is there significant difference in performance before eating versus after eating?
Use a two-tailed test with α=.05.
b. Compute Cohen's d to measure the size of the treatment effect.
8. A teacher gives a reading skills test to a third-grade class of n=25 at the beginning of the school year. To evaluate the changes that occur during the year, students are tested again at the end of the year. Their test scores showed an average improvement of MD=12.7 points with s2=100.
a. Are the results sufficient to conclude that there is a significant improvement? Use a one-tailed test with α=.01.
b. Construct a 90% confidence interval for the mean difference to describe the size
of the improvement.
9. Researchers have noted a decline in cognitive functioning as people age (Bartus, 1990). However, the results from other research suggest that the antioxidants in foods such as blueberries may reduce and even reverse these age-related declines (Joseph et al., 1999). To examine this phenomenon, suppose that a researcher obtains a sample of n=16 adults who are between the ages of 65 and 75. The researcher uses a standardized test to measure cognitive performance for each individual. The participants then begin a 2- month program in which they receive daily doses of a blueberry supplement. At the end of the 2-month period, the researcher again measures cognitive performance for each participant. The results show an average increase in performance of MD=7.4 with SS=1215.
a. Does this result support the conclusion that the antioxidant supplement has a significant effect on cognitive performance? Use a two-tailed test with α=.01.
b. Construct a 80% confidence interval to estimate the average cognitive
performance improvement for the population of older adults.
10. One of the primary advantages of a repeated-measures design, compared to independent-measures, is that it reduces the overall variability by removing variance caused by individual differences. The following data are from a research study comparing two treatment conditions.
a. Assume that the data are from an independent-measures study using two separate samples of n=6. Compute the pooled variance and the estimated standard error for the mean difference.
b. Now assume that the data are from a repeated-measures study using the same sample of n = 6 participants in both treatment conditions. Compute the variance for the sample of difference scores and the estimated standard error for the mean difference. (You should find that the repeated-measures design substantially reduces the variance and the standard error.)
Treatment I Treatment II ___Difference__
10 13 3
12 12 0
8 10 2
6 10 4
5 6 1
7 9 2
M = 8 M = 10 MD = 2
SS = 34 SS = 30 SS = 10
11. A researcher uses a matched-samples design to investigate whether single people with pets are happier than singles without pets. A mood inventory survey is given to a group of 20- to 29- year-old non-pet owners and a similar age group of pet owners. The pet owners are matched one to one with the non-pet owners for income, number of close friendships, and general health. The data follow:
Matched Pair Non-Pet Owner Pet Owner
A 12 14
B 8 7
C 10 13
D 9 9
E 7 13
F 10 12
a. Is there a significant difference in the mood scores for non-pet owners versus pet owners? Test with σ = .05 for two tails.
b. Construct the 90% confidence interval to estimate the size of the mean difference in mood between the population of pet owners and the population of non-pet owners. (You should find that a mean difference of µD = 0 is an acceptable value, which is consistent with the conclusion from the hypothesis test.)