1.Use the concepts of sampling error and z-scores to explain the concept of distribution of sample means. (this is a paragraph answer needed)
2. Describe the distribution of sample means shapefor samples of n=36 selected from a population with a mean of μ=100 and a standard deviation of o=12. , expected value, and standard error)
3. The distribution of sample means is not always a normal distribution. Under what circumstances is the distribution of sample means not normal?
4. For a population with a mean of μ=70 and a standard deviation of o=20, how much error, on average, would you expect between the sample mean (M) and the population mean for each of the following sample sizes?
a. n=4 scores
b. n=16 scores
c. n=25 scores
5. If the population standard deviation is o=8, how large a sample is necessary to have a standard error that is:
a. less than 4 points?
b. less than 2 points?
c. less than 1 point?
6. For a population with a mean of μ=80 and a standard deviation of o=12, find the z-score corresponding to each of the following samples.
a. M=83 for a sample of n=4 scores
b. M=83 for a sample of n=16 scores
c. M=83 for a sample of n=36 scores
7. A population forms a normal distribution with a mean of μ=80 and a standard deviation of o=15. For each of the following samples, compute the z-score for the sample mean and determine whether the sample mean is a typical, representative value or an extreme value for a sample of this size.
a. M=84 for n=9 scores
b. M=84 for n=100 scores
7. The population of IQ scores forms a normal distribution with a mean of μ=100 and a standard deviation of o=15. What is the probability of obtaining a sample mean greater than M =97,
a. for a random sample of =9 people?
b. for a random sample of n=25 people?
8. A population of scores forms a normal distribution with a mean of μ=40 and a standard deviation of o=12.
a. What is the probability of randomly selecting a score less than X=34?
b. What is the probability of selecting a sample of n=9 scores with a mean less than M=34?
c. What is the probability of selecting a sample of n=36 scores with a mean less than M=34?
9. At the end of the spring semester, the Dean of Students sent a survey to the entire freshman class. One question asked the students how much weight they had gained or lost since the beginning of the school year. The average was a gain of μ=9 pounds with a standard deviation of o=6. The distribution of scores was approximately normal. A sample of n=4 students is selected and the average weight change is computed for the sample.
a. What is the probability that the sample mean willbe greater than M=10 pounds? In symbols, what is p(M>10)?
b. Of all of the possible samples, what proportion will show an average weight loss? In symbols, what is p(M<0)?
c. What is the probability that the sample mean will be a gain of between M=9 and M=12. In symbols, what is p(9
10. The average age for licensed drivers in the county is μ =40.3 years with a standard deviation of o=13.2 years.
a. A researcher obtained a random sample of n=16 parking tickets and computed an average age of M=38.9 years for the drivers. Compute the z-score for the sample mean and find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. Is it reasonable to conclude that this set of n=16 people is a representative sample of licensed drivers?
b. The same researcher obtained a random sample of n=36 speeding tickets and computed an average age of M=36.2 years for the drivers. Compute the z-score for the sample mean and find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. Is it reasonable to conclude that this set of n _ 36 people is a representative sample of licensed drivers?
11. Welsh, Davis, Burke, and Williams (2002) conducted a study to evaluate the effectiveness of a carbohydrate-electrolyte drink on sports performance and endurance. Experienced athletes were given either a carbohydrate-electrolyte drink or a placebo while they were tested on a series of high-intensity exercises. One measure was how much time it took for the athletes to run to fatigue. Data similar to the results obtained in the study are shown in the following table.
Time to Run to Fatigue (in minutes)
Mean SE
Placebo 21.7 2.2
Carbohydrate-electrolyte 28.6 2.7
a. Construct a bar graph that incorporates all of the information in the table.
b. Looking at your graph, do you think that the carbohydrate-electrolyte drink helps performance?
12. The value of the z-score in a hypothesis test is influenced by a variety of factors. Assuming that all other variables are held constant, explain how the value of z is influenced by each of the following:
a. Increasing the difference between the sample mean and the original population mean.
b. Increasing the population standard deviation.
c. Increasing the number of scores in the sample.
13. If the alpha level is changed from α=.05 to α=.01,
a. What happens to the boundaries for the critical region?
b. What happens to the probability of a Type I error?
14. Childhood participation in sports, cultural groups, and youth groups appears to be related to improved self-esteem for adolescents (McGee, Williams, Howden- Chapman, Martin, &Kawachi, 2006). In a representative study, a sample of n=100 adolescents with a history of group participation is given a standardized self-esteem questionnaire. For the general population of adolescents, scores on this questionnaire form a normal distribution with a mean of µ=40 and a standard deviation of o=12. The sample of group-participation adolescents had an average of M=43.84.
a. Does this sample provide enough evidence to conclude that self-esteem scores for these adolescents are significantly different from those of the general population? Use a two-tailed test with α=.01.
b. Compute Cohen's d to measure the size of the difference.
c. Write a sentence describing the outcome of the hypothesis test and the measure of effect size as it would appear in a research report.
15. A random sample is selected from a normal population with a mean of _ _ 50 and a standard deviation of _ _ 12. After a treatment is administered to the individuals in the sample, the sample mean is found to be M _ 55.
a. If the sample consists of n _ 16 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with _ _ .05.
b. If the sample consists of n _ 36 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with _ _ .05.
c. Comparing your answers for parts a and b, explain how the size of the sample influences the outcome of a hypothesis test.
16. Miller (2008) examined the energy drink consumption of college undergraduates and found that males use energy drinks significantly more often than females. To further investigate this phenomenon, suppose that a researcher selects a random sample of n=36 male undergraduates and a sample of n=25 females. On average, the males reported consuming M=2.45 drinks per month and females had an average of M=1.28. Assume that the overall level of consumption for college undergraduates averages µ=1.85 energy drinks per month, and that the distribution of monthly consumption scores is approximately normal with a standard deviation of o=1.2.
a. Did this sample of males consume significantly more energy drinks than the overall population average? Use a one-tailed test with α=.01.
b. Did this sample of females consume significantlyfewer energy drinks than the overall populationaverage? Use a one-tailed test with α=.01
17. There is some evidence that REM sleep, associated with dreaming, may also play a role in learning and memory processing. For example, Smith and Lapp (1991) found increased REM activity for college students during exam periods. Suppose that REM activity for a sample of n=16 students during the final exam period produced an average score of M=143. Regular REM activity for the college population averages µ=110 with a standard deviation of o=50. The population distribution is approximately normal.
a. Do the data from this sample provide evidence for a significant increase in REM activity during exams? Use a one-tailed test with α=.01.
b. Compute Cohen's d to estimate the size of the effect.
c. Write a sentence describing the outcome of the hypothesis test and the measure of effect size as it would appear in a research report.
18. A psychologist is investigating the hypothesis that children who grow up as the only child in the household develop different personality characteristics than those who grow up in larger families. A sample of n=30 only children is obtained and each child is given a standardized personality test. For the general population, scores on the test from a normal distribution with a mean of µ=50 and a standard deviation of o=15. If the mean for the sample is M=58, can the researcher conclude that there is a significant difference in personality between only children and the rest of the population? Use a two tailed test with α=.05. of male college students, without any sports drink, the scores for this task average µ=50 with a standard deviation of o=12.
19. Montarello and Martins (2005) found that fifth-grade students completed more mathematics problems correctly when simple problems were mixed inwith their regular math assignments. To furtherexplore this phenomenon, suppose that a researcher selects a standardized mathematics achievement test that produces a normal distribution of scores with a mean of µ=100 and a standard deviation of o=.0118. The researcher modifies the test by inserting a set of very easy problems among the standardized questions, and gives the modified test to a sample of n=36 students. If the average test score for the sample is M=104, is this result sufficient to conclude that inserting the easy questions improves student performance? Use a one-tailed test with α= .01.
20. A researcher plans to conduct an experiment testing the effect of caffeine on reaction time during a driving simulation task. A sample of n=9 participants is selected and each person receives a standard dose of caffeine before being tested on the simulator. The caffeine is expected to lower reaction time by an average of 30 msec. Scores on the simulator task for the regular population (without caffeine) form a normal distribution with µ=240 msec. and o=30.
a. If the researcher uses a two-tailed test with α=.05, what is the power of the hypothesis test?
b. Again assuming a two-tailed test with α=.05, what is the power of the hypothesis test if the sample size is increased to n=25?
21. Briefly explain how increasing sample size influences each of the following. Assume that all other factors are held constant.
a. The size of the z-score in a hypothesis test.
b. The size of Cohen's d.
c. The power of a hypothesis test.
22. A researcher is investigating the effectiveness of a new medication for lowering blood pressure for individuals with systolic pressure greater than 140. For this population, systolic scores average µ=160 with a standard deviation of o=20, and the scores form a normal-shaped distribution. The researcher plans to select a sample of n=25 individuals, and measure their systolic blood pressure after they take the medication for 60 days. If the researcher uses a two-tailed test with α=.05,
a. What is the power of the test if the medication has a 5-point effect?
b. What is the power of the test if the medication has a 10-point effect?