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Statistics For The Behavioral And Social Sciences A Brief Course 6th Edition Arthur Aron Elliot J Blows Elaine N Aron - Solutions
3. The significance level cutoff from the t table is not accurate.
2. The population of individuals’ difference scores is assumed to be a normal distribution.
1. An assumption is a requirement that you must meet for the results of the hypothesis-testing procedure to be accurate.
5. Decide whether to reject the null hypothesis. The sample’s t score of .67 is not more extreme than the cutoff t of ± 2.776. Therefore, do not reject the null hypothesis.
4. Determine your sample’s score on the comparison distribution. t=(4−0)/6=.67.
3. Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. For a two-tailed test at the .05 level, the cutoff sample t scores are 2.776 and−2.776.
2. Determine the characteristics of the comparison distribution. The mean of the distribution of means of difference scores (the comparison distribution) is 0. The standard deviation of the distribution of means of difference scores is 6. It is a t distribution with 4 degrees of freedom.
5. Assumptions for the significance test of a correlation coefficient are that(a) the populations for both variables are normally distributed, and (b) in the population, the distribution of each variable at each point of the other variable has about equal variance. However, as with the t test,
4. Note that significance tests of a correlation, like a t test, can be either one-tailed or two-tailed. A one-tailed test means that the researcher has predicted the sign (positive or negative) of the correlation.
3. You figure the correlation coefficient’s score on that t distribution using the formula t=(r)(N−2)1−r2(8–9)The t score for a correlation is the correlation coefficient multiplied by the square root of 2 less than the number of people in the study, divided by the square root of 1 minus
2. If the data meet assumptions (explained below), the comparison distribution is a t distribution with degrees of freedom equal to the number of people minus 2 (that is, df=N−2).
1. Usually, the null hypothesis is that the correlation in a population like that studied is no different from a population in which the true correlation is 0.
6. Explain how and why the scores from Steps and of the hypothesis-testing process are compared. Explain the meaning of the result of this comparison with regard to the specific research and null hypotheses being tested.
5. Describe why and how you figure the t score of the sample mean on the comparison distribution.
4. Describe the logic and process for determining the cutoff sample score(s) on the comparison distribution at which the null hypothesis should be rejected.
3. Describe the comparison distribution (the t distribution) that is used with a t test for a single sample, noting how it is different from a normal curve and why. Explain why a t distribution (as opposed to the normal curve) is used as the comparison distribution.
2. Outline the logic of estimating the population variance from the sample scores. Explain the idea of biased and unbiased estimates of the population variance, and describe the formula for estimating the population variance and why it is different from the ordinary variance formula.
5. Decide whether to reject the null hypothesis. The t of 2.54 is more extreme than the needed t of 2.365. Therefore, reject the null hypothesis; the research hypothesis is supported. The experimental procedure does make a difference.
4. Determine your sample’s score on the comparison distribution. t=(M− Population M)/SM=(9−6)/1.18=3/1.18=2.54.
3. Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. From Table A–2, the cutoffs for a two-tailed t test at the .05 level for df=7 are 2.365 and −2.365.
2. Determine the characteristics of the comparison distribution. The mean of the distribution of means is 6 (the known population mean). To figure the estimated population variance, you first need to figure the sample mean, which is(14+8+6 +5+13+10+10+6)/8=72/8=9. The estimated population variance
6. Power and needed sample size for 80% power for a t test for dependent means can be looked up using power software packages, an Internet power calculator, or special tables. The power of studies using difference scores is usually much higher than that of studies using other designs with the same
5. The effect size of a study using a t test for dependent means is the mean of the difference scores divided by the standard deviation of the difference scores.
4. An assumption of the t test is that the distribution of the population of individuals follows a normal curve. However, even when it does not follow it closely, the t test is usually fairly accurate.
3. You use a t test for dependent means in studies where each participant has two scores, such as a before score and an after score, or a score in each of two experimental conditions. In this t test, you first figure a difference score for each participant, then go through the usual five steps of
2. You use the standard five steps of hypothesis testing even when you don’t know the population variance. However, in this situation you have to estimate the population variance from the scores in the sample, using a formula that divides the sum of squared deviation scores by the degrees of
1. The hypothesis-testing procedures in this chapter, in which the population variance is unknown, are examples of t tests.
4. (a) Why do repeated-measures designs have so much power? (b) What is the main disadvantage of research designs in which participants are measured before and after some intervention, without any control group?
3. You are planning a study in which you predict an effect size of .50. You plan to test significance using a t test for dependent means, one-tailed, using a significance level of .05. (a) What is the power of this study if you carry it out with 20 participants? (b) How many participants would you
2. (a) Write the formula for estimated effect size in a t test for dependent means situation, and (b) describe each of its terms.
1. (a) What is an assumption in hypothesis testing? (b) Describe a specific assumption for a t test for dependent means. (c) What is the effect of violating this assumption? (d) When is the t test for dependent means likely to give a very distorted result?
6. What about the research situation makes the difference in whether you should carry out a t test for a single sample or a t test for dependent means?
5. What about the research situation makes the difference in whether you should carry out a Z test or a t test for a single sample?
4. Five individuals are tested before and after an experimental procedure;their scores are given in the following table. Test the hypothesis that there is no change, using the .05 significance level.
3. In a t test for dependent means, (a) what is usually considered to be the mean of the “known” population (Population 2). (b) Why?
2. When doing a t test for dependent means, what do you do with the two scores you have for each participant?
1. Describe the situation in which you would use a t test for dependent means.
2. Using the .05 significance level with 9 degrees of freedom, Table A–2 shows a cutoff t of 1.833. In Table 8–7, the difference score is figured as brain activation when viewing the beloved’s picture minus brain activation when viewing the neutral person’s picture.Thus, the research
1. We will use the standard .05 significance level. This is a one-tailed test because the researchers were interested only in a specific direction of difference.
3. Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
5. The shape is a t distribution with df=N−1. Therefore, the comparison distribution is a t distribution for 9 degrees of freedom.It is a t distribution because we figured its variance based on an estimated population variance. It has 9 degrees of freedom
3. Figure the standard deviation of the distribution of means of difference scores: SM=SM2=.044=.210.
2. Figure the variance of the distribution of means of difference scores: SM2=S2/N =.438/10=.044.
1. Figure the estimated population variance of difference scores:S2=[Σ (X−M)2]/ df=3.940/(10−1)=.438.
4. The standard deviation of the distribution of means of difference scores is figured as follows:
3. Assume a mean of the distribution of means of difference scores of 0: Population M=0.
2. Figure the mean of the difference scores. The sum of the difference scores (12.0) divided by the number of difference scores (10) gives a mean of the difference scores of 1.200. So, M=1.200.
2. Determine the characteristics of the comparison distribution.1. Make each person’s two scores into a difference score. This is shown in the column labeled “Difference” in Table 8–7. You do all the remaining steps using these difference scores.
4. Determine your sample’s score on the comparison distribution: t=(M−Population M)/SM
2. Look up the appropriate cutoff in a t table.
1. Decide the significance level and whether to use a onetailed or a two-tailed test.
3. Determine the cutoff sample score on the comparison
5. The shape is a t distribution with N−1 degrees of freedom.
3. Figure the standard deviation of the distribution of means of difference scores: SM=SM2
2. Figure the variance of the distribution of means of difference scores: SM2=S2/N
1. Figure the estimated population variance of difference scores: S2=[Σ(X−M)2]/df
4. The standard deviation of the distribution of means of difference scores is figured as follows:
3. Assume a mean of the distribution of means of difference scores of 0.
2. Figure the mean of the difference scores.
1. Make each person’s two scores into a difference score.Do all the remaining steps using these difference scores.
1. Restate the question as a research hypothesis and a null hypothesis about the populations. There are two populations:Population 1: Husbands who receive ordinary (very minimal)premarital counseling.Population 2: Husbands whose communication quality does not change from before to after marriage.
6. A population has a mean of 23. A sample of four is given an experimental procedure and has scores of 20, 22, 22, and 20. Test the hypothesis that the procedure produces a lower score. Use the .05 significance level. (a) Use the steps of hypothesis testing and (b) make a sketch of the
5. List three differences in how you do hypothesis testing for a t test for a single sample versus for the Z test you learned in Chapter 6.
4. (a) How does a t distribution differ from a normal curve? (b) How do degrees of freedom affect this? (c) What is the effect of the difference on hypothesis testing?
3. (a) What are degrees of freedom? (b) How do you figure the degrees of freedom in a t test for a single sample? (c) What do they have to do with estimating the population variance? (d) What do they have to do with the t distribution?
2. What is the difference between the usual formula for figuring the variance and the formula for estimating a population’s variance from the scores in a sample (that is, the formula for an unbiased estimate of the population variance)?
1. In what sense is a sample’s variance a biased estimate of the variance of the population from which the sample is taken? That is, in what way does a sample’s variance typically differ from the population’s?
1. The mean is the same as the known population mean.
2. Determine the characteristics of the comparison distribution.
1. Restate the question as a research hypothesis and a null hypothesis about the populations.
5. Decide whether to reject the null hypothesis. No difference in method.Table 8–3 Full Alternative Text That is, we are comparing the current situation in which you know the population’s mean but not its variance to the situation where you knew the population’s mean and variance. Table 8–4
4. Determine your sample’s score on the comparison distribution.No difference in method(but called a t score).
3. Determine the significance cutoff. Use the t table.
2. Determine the characteristics of the comparison distribution:Population mean No difference in method.Population variance Estimate from the sample.Standard deviation of the distribution of means No difference in method(but based on estimated population variance).Shape of the comparison
1. Restate the question as a research hypothesis and a null hypothesis about the populations.No difference in method.
Explain these results, including why it was especially important for the researchers in this study to give effect sizes, to a person who understands hypothesis testing but has never learned about effect size or power.
14. You read a study that just barely fails to be significant at the .05 level.That is, the result is not statistically significant. You then look at the size of the sample. If the sample is very large (rather than very small), how should this affect your judgment
13. What is meant by the statistical power of an experiment? (Write your answer for someone who has never had a class in statistics.)
Explain the purpose and results of this meta-analysis to someone who is familiar with effect size but has never heard of meta-analysis.
11. What is meant by effect size? (Write your answer for someone who has never had a class in statistics.)
10. Here is information about several possible versions of a planned study, each involving a single sample.
(e) 10? For each part, also indicate whether the predicted effect is approximately small, medium, or large.
9. In a planned study, there is a known population with a normal distribution, Population M=0, and Population SD=10. What is the predicted effect size if the researchers predict that those given an experimental treatment have a mean of (a) −8, (b) −5, (c) −2, (d) 0, and
8. In a completed study, there is a known population with a normal distribution, Population M=122, and Population SD=8. What is the estimated effect size if a sample given an experimental procedure has a mean of (a) 100, (b) 110, (c) 120, (d) 130, and (e) 140? For each part, also indicate whether
what the use is for each.
7. List two situations in which it is useful to consider power, indicating
5. Using a two-tailed test instead of a one-tailed test
4. Using a more extreme significance level (e.g., p
3. A larger sample size
2. A larger population standard deviation
1. A larger predicted difference between the means of the populations
6. How does each of the following affect the power of a planned study?
Explain this result to a person who understands hypothesis testing but has never learned about effect size or power.
3. Explain the relationship between effect size and power.
2. Explain the idea of power as the probability of getting significant results if the research hypothesis is true. Be sure to mention that the usual minimum acceptable level of power for a research study is 80%. Explain the role played by power when you are interpreting the results of a study(both
1. Explain the idea of effect size as the degree of overlap between distributions. Note that this overlap is a function of mean difference and population standard deviation (and describe precisely how it is figured and why it is figured that way). If required by the question, discuss the effect
6. Research articles commonly report effect size, and effect sizes are almost always reported in meta-analyses. Research articles sometimes include discussions of power, especially when evaluating nonsignificant results.
5. Statistically significant results from a study with high power (such as one with a large sample size) may or may not have practical importance.Results that are not statistically significant from a study with low power(such as one with a small sample size) leave open the possibility that
4. Determining power is important when planning a study. There is general agreement among researchers that a study should have 80% power to be worth doing. There are a number of ways to increase the power of a planned study, including increasing the predicted difference between the population
3. The larger the effect size is, the greater the power is. This is because the greater the difference between means or the smaller the population standard deviation is (the two ingredients in effect size), the less overlap there is between the known and predicted populations’ distributions of
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