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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
5. After getting a significant result with an analysis of variance, why do researchers usually go on to compare each population to each other population?
4. What is the general rule about when violations of the equal variance assumption are likely to lead to serious inaccuracies in results?
3. Why do we need the equal variance assumption?
2. Give the two main assumptions for the analysis of variance.
1. A study compares the effects of three experimental treatments, A, B, and C, by giving each treatment to 16 participants and then assessing their performance on a standard measure. The results on the standard measure are as follows: Treatment A group: M=20, S2=8; Treatment B group:M=22, S2=9;
2. Figure the estimated variance of the population of individual scores based on the variance of the distribution of means.
1. Estimate the variance of the distribution of means from the means of your samples.
5. (a) What is the F ratio? (b) Why is it usually about 1 when the null hypothesis is true? (c) Why is it usually larger than 1 when the null hypothesis is false?
4. What are two sources of variation that can contribute to the betweengroups population variance estimate?
3. (a) What is the between-groups population variance estimate based on?(b) How is it affected by the null hypothesis being true or not? (c) Why?
2. (a) What is the within-groups population variance estimate based on? (b)How is it affected by the null hypothesis being true or not? (c) Why?
1. When do you use an analysis of variance?
3. A researcher compares the resting heart rate of 15 individuals who have been taking a particular drug to the resting heart rate of 48 other individuals who have not been taking this drug.
2. A researcher tests performance on a math skills test of each of 250 individuals before and after they complete a one-day seminar on managing test anxiety.
1. A researcher measures the heights of 40 university students who are the firstborn in their families and compares the 15 who come from large families to the 25 who come from smaller families.
13. For each of the following studies, say whether (and why) you would use a t test for dependent means or a t test for independent means.
12. Make up two examples of studies (not from this book or your lectures)that would be tested with a t test for independent means.
8. Figure the approximate power of each of the following planned studies, all using a t test for independent means at the .05 significance level, onetailed, with a predicted small effect size:Study N1 N2(a) 3 57(b) 10 50(c) 20 40(d) 30 30
7. Figure the estimated effect size (and whether it is approximately small, medium, or large) for problems (a) 4, (b) 5, and (c) 6. (d) Explain what you have done in part (a) to someone who understands the t test for independent means but knows nothing about effect size.
6. A developmental researcher compares 4-year-olds and 8-year-olds on their ability to understand the analogies used in stories. The scores for the five 4-year-olds tested were 7, 6, 2, 3, and 8. The scores for the three 8-year-olds tested were 9, 2, and 5. Using the .05 level, do older children do
9.1-11 Full Alternative Text
(c) explain your answers to someone who has never had a course in statistics.Ordinary Story Own-Name Story Student Reading Time Student Reading Time A 2 G 4 B 5 H 8 C 7 I 10 D 9 J 9 E 6 K 8 F 7
5. A teacher was interested in whether using a student’s own name in a story affected children’s attention span while reading. Six children were randomly assigned to read a story under ordinary conditions (using names like Dick and Jane). Five other children read versions of the same story, but
4. A communication researcher randomly assigned 82 volunteers to one of two experimental groups. Sixty-one were instructed to get their news for a month only from television, and 21 were instructed to get their news for a month only from the Internet. (Why the researcher didn’t assign equal
3. For each of the following experiments, decide whether the difference between conditions is statistically significant at the .05 level (twotailed).Experimental Group Control Group Study N M S2 N M S2(a) 30 12.0 2.4 30 11.1 2.8(b) 20 12.0 2.4 40 11.1 2.8(c) 30 12.0 2.2 30 11.1 3.0 9.1-10 Full
2. Figure SDifference for each of the following studies:Study N1 S12 N2 S22(a) 20 1 20 2(b) 20 1 40 2(c) 40 1 20 2(d) 40 1 40 2(e) 40 1 40 4 9.1-9 Full Alternative Text
3. A researcher tests reaction time of each member of a group of 14 individuals twice, once while in a very hot room and once while in a normal-temperature room.
2. A marketing researcher measures 100 physicians’ reports of the number of their patients asking them about a particular drug during the month before and the month after a major advertising campaign for that drug.
1. A researcher randomly assigns a group of 47 English speakers to receive a new Spanish language program and 48 other English speakers to receive the standard Spanish language program, and then measures how well they all do on a Spanish language test.
1. For each of the following studies, say whether (and why) you would use a t test for dependent means or a t test for independent means.
7. 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
6. Describe why and how you figure the t score of the sample mean on the comparison distribution.
5. Describe the logic and process for determining the cutoff sample score(s) on the comparison distribution at which the null hypothesis should be rejected.
4. Explain why the shape of the comparison distribution that is used with a t test for independent means is a t distribution (as opposed to the normal curve).
3. Outline the logic of estimating the population variance and the variance of the two distributions of means. Describe how to figure the standard deviation of the distribution of differences between means.
2. Explain the entire complex logic of the comparison distribution that is used with a t test for independent means (the distribution of differences between means). Be sure to explain why you use 0 as its mean. (This point and point 3 will be the longest part of your essay.)
1. Describe the core logic of hypothesis testing in this situation. Be sure to mention that the t test for independent means is used for hypothesis testing when you have scores from two entirely separate groups of people. Be sure to explain the meaning of the research hypothesis and the null
5. Decide whether to reject the null hypothesis. The t of 3.37 is more extreme than the cutoffs of ±2.306. Thus, you can reject the null hypothesis. The research hypothesis is supported
4. Determine the sample’s score on the comparison distribution. t=(7−4)/.89=3.37.
3. Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. At the .05 significance level, for a two-tailed test, the cutoffs are 2.306 and −2.306.
8. The single sample t test is used for hypothesis testing when you are comparing the mean of a single sample to a known population mean.The t test for dependent means is the appropriate t test when each participant has two scores
7. Power for a t test for independent means can be determined using a table(see Table 9–4), a power software package, or an Internet power calculator. Power is greatest when the sample sizes of the two groups are equal. When they are not equal, you use the harmonic mean of the two sample sizes
6. Estimated effect size for a t test for independent means is the difference between the samples’ means divided by the pooled estimate of the population standard deviation.
5. The assumptions of the t test for independent means are that the two populations are each normally distributed and have the same variance.However, the t test gives fairly accurate results when the true situation is moderately different from the assumptions.
4. Considering the five steps of hypothesis testing, there are three new wrinkles for a t test for independent means: (1) the comparison distribution is now a distribution of differences between means (this affects Step ); (2) the degrees of freedom for finding the cutoff on the t table is based on
5. Figure the standard deviation of the distribution of differences between means (the square root of the variance of the distribution of differences between means).
4. Figure the variance of the distribution of differences between means (the sum of the variances of the two distributions of means).
3. Figure the variance of each distribution of means (the pooled variance estimate divided by each sample’s N).
2. Figure the pooled estimate of the population variance (the weighted average of the two individual population variance estimates, with the weighting by each sample’s proportion of the total degrees of freedom).
1. Figure the estimated population variances based on each sample.
3. The distribution of differences between means is a t distribution with the total of the degrees of freedom from the two samples. Its standard deviation is figured in several steps:
2. The comparison distribution for a t test for independent means is a distribution of differences between means of samples. This distribution can be thought of as being built up in two steps: Each population of individuals produces a distribution of means, and then a new distribution is created of
1. A t test for independent means is used for hypothesis testing with scores from two entirely separate groups of people.
5. Figure the standard deviation of the distribution of differences between means (the square root of the variance of the distribution
4. Figure the variance of the distribution of differences between means (the sum of the variances of the two distributions of means).
3. Figure the variance of each distribution of means (the pooled variance estimate divided by each sample’s N).
2. Figure the pooled estimate of the population variance (the weighted average of the two individual population variance estimates, with the weighting by each sample’s proportion of the total degrees of freedom).
1. Figure the estimated population variances based on each sample.
3. The distribution of differences between means is a t distribution with the total of the degrees of freedom from the two samples. Its standard deviation is figured in several steps:
2. The comparison distribution for a t test for independent means is a distribution of differences between means of samples. This distribution can be thought of as being built up in two steps: Each population of individuals produces a distribution of means, and then a new distribution is created of
1. A t test for independent means is used for hypothesis testing with scores from two entirely separate groups of people.
6. How many participants do you need in each group for 80% power in a planned study in which you predict a small effect size, will have equal numbers of participants in each group, and will be using a t test for independent means, one-tailed, at the .05 significance level?
5. What is the approximate power of a study using a t test for independent means, with a two-tailed test at the .05 significance level, in which the researchers predict a large effect size, and there are 6 participants in one group and 34 participants in the other group?
4. What is the power of a study using a t test for independent means, with a two-tailed test at the .05 significance level, in which the researchers predict a large effect size and there are 20 participants in each group?
3. What is the estimated effect size for a study in which the sample drawn from Population 1 has a mean of 17, Population 2’s sample mean is 25, and the pooled estimate of the population standard deviation is 20?
2. Why do you need to assume the populations have the same variance?
1. List two assumptions for the t test for independent means. For each, give the situations in which violations of these assumptions would be a serious problem.
1. Restate the question as a research hypothesis and a null hypothesis about the populations. There are two populations:Population 1: Individuals who could not hold a job who then participate in the special job skills program.Population 2: Individuals who could not hold a job who then participate
5. Decide whether to reject the null hypothesis: Compare the scores from Steps Table 9–2 Full Alternative Text 1The steps of figuring the standard deviation of the distribution of differences
4. Determine your sample’s score on the comparison distribution: t=(M
2. Look up the appropriate cutoff in a t table. If the exact df is not given, use the
1. Determine the degrees of freedom (dfTotal), desired significance level, and or two).
3. Determine the cutoff sample score on the comparison distribution at which the null should be rejected.
3. The comparison distribution will be a t distribution with dfTotal
. Figure the standard deviation of the distribution of differences between SDifference=SDifference2.
4. Figure the variance of the distribution of differences between means:SDifference2=SM12+SM22.
3. Figure the variance of each distribution of means: SM12=SM22=SPooled2/N2.
2. Figure the pooled estimate of the population variance:SPooled2=df1dfTotal(S12)+df2dfTotal(S22)(df1=N1−1 and
1. Figure the estimated population variances based on each sample.[Σ(X−M)2]/(N−1).
2. Figure its standard deviation:
1. Its mean will be 0.
2. Determine the characteristics of the comparison distribution.
1. Restate the question as a research hypothesis and a null hypothesis about the populations.
3. For a particular study comparing means of two samples, the first sample has 21 participants and an estimated population variance of 100; the second sample has 31 participants and an estimated population variance of 200. (a) What is the standard deviation of the distribution of differences
2. Explain (a) why a t test for independent means uses a single pooled estimate of the population variance, (b) why, and (c) how this estimate is“weighted.”
1. Write the formula for each of the following: (a) pooled estimate of the population variance, (b) variance of the distribution of means for the first population, (c) variance of the distribution of differences between means, and (d) t score in a t test for independent means. (e) Define all of the
4. Figure the variance of the distribution of differences between means:SDifference2=SM12+SM22
3. Figure the variance of each distribution of means:SM12=SPooled2/N1 and SM22=SPooled2/N2
2. Figure the pooled estimate of the population variance:SPooled2=df1dfTotal (S12)+df2dfTotal (S22)(df1=N1−1 and df2=N2−1; dfTotal=df1+df2)
1. Figure the estimated population variances based on each sample. That is, figure one estimate for each population using the formula S2=[Σ(X−M)2]/(N−1).
3. (a) In the context of the t test for independent means, explain the general logic of going from scores in two samples to an estimate of the variance of this comparison distribution. (b) Illustrate your answer with sketches of the distributions involved. (c) Why is the mean of this distribution 0
2. (a) What is the comparison distribution in a t test for independent means? (b) How is this different from the comparison distribution in a t test for dependent means?
1. (a) When would you carry out a t test for independent means? (b) How is this different from the situation in which you would carry out a t test for dependent means?
6. the t score for the difference between the particular two means being compared.
5. the shape of the distribution of differences between means, and
4. the variance and standard deviation of the distribution of differences between means,
3. the variance of the two distributions of means,
2. the estimated population variance,
1. the mean of the distribution of differences between means,
4. The t test for dependent means is likely to give a very distorted result when doing a one-tailed test and the population distribution is highly skewed
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