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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. 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.
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
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.
what is the
5. 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.
3. Explain the logic of the comparison distribution that is used with a oneway analysis of variance (the F distribution).
2. Describe the core logic of hypothesis testing in this situation. Be sure to mention that the analysis of variance involves comparing the results of two ways of estimating the population variance. One population variance estimate (the within-groups estimate) is based on the variation within each
1. Explain that the one-way analysis of variance is used for hypothesis testing when you have scores from three or more entirely separate groups of people. Be sure to explain the meaning of the research hypothesis and the null hypothesis in this situation.
5. When there is both a main and an interaction effect, (a) under what conditions must you be careful in interpreting the main effect, and (b)under what conditions can you still be confident in the overall main effect?
4. For a two-way factorial design, what are the possible combinations of main and interaction effects?
3. Make two bar graphs of these results: One graph should show vividness(low and high) on the horizontal axis, with bars for short and long example length; another graph should show example length (short and long) on the horizontal axis, with bars for low and high vividness.
2. Explain the pattern of results in terms of numbers.
1. Describe the pattern of results in words.
2. In a factorial research design, (a) what is a main effect, and (b) what is an interaction effect?
1. (a) What is a factorial research design? (b) and (c) Give two advantages of a factorial research design over doing two separate experiments.
Source: Aron, E., & Aron, A. (1997). Sensory-processing sensitivity and its relation to introversion and emotionality. Journal of Personality and Social Psychology, 73, 345–368. Published by the American Psychological Association. Reprinted with permission.
6. Are you bothered by intense stimuli, like loud noises and chaotic scenes?Note: Each item is answered on a scale from 1 “Not at all” to 7“Extremely.”
5. Do changes in your life shake you up?
4. Do you get rattled when you have a lot to do in a short amount of time?
3. Are you easily overwhelmed by things like bright lights, strong smells, coarse fabrics, or sirens close by?
2. Do you find yourself wanting to withdraw during busy days, into bed or into a darkened room or any place where you can have some privacy and relief from stimulation?
1. Do you find it unpleasant to have a lot going on at once?
3. About how many participants do you need (a) in each group, and (b) in total, for 80% power in a planned study with five groups in which you predict a medium effect size and will be using the .05 significance level?
2. What is the power of a study with four groups of 40 participants each to be tested at the .05 significance level, in which the researchers predict a large effect size?
1. (a) Write the formula for effect size in analysis of variance; (b) define each of the symbols; and (c) figure the effect size for a study in which SBetween2=12.22, SWithin2=7.20, dfBetween=2, and dfWithin=8.
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