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statistics for engineers and scientists
Statistics For The Behavioral And Social Sciences A Brief Course 6th Edition Arthur Aron Elliot J Blows Elaine N Aron - Solutions
2. Figure the effect size.
1. Figure the chi-square yourself (your results should be the same, within rounding error).
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.
what was found in the study and what would be expected based on some null hypothesis idea (such as the groups all being equal). To get this number you figure, for each group, the difference between the observed frequency and the expected frequency, square it (because otherwise the sign of the
4. Describe how to figure the chi-square value for the sample. The key idea is to get a single number that indicates the overall discrepancy between
how to determine the degrees of freedom and the cutoff chi-square value.
3. Explain that the comparison distribution in this situation is a chi-square distribution. Be sure to mention that the shape of the chi-square distribution depends on the number of degrees of freedom. Describe
2. Describe the core logic of hypothesis testing in this situation. Be sure to mention that the hypothesis testing involves comparing observed frequencies (that is, frequencies found in the actual study) with expected frequencies (that is, frequencies that you would expect based on a particular
if the null hypothesis is true. Be sure to explain the meaning of the research hypothesis and the null hypothesis in this situation.
1. Explain that chi-square tests are used for hypothesis testing with nominal variables. The chi-square test for goodness of fit is used to test hypotheses about whether a distribution of frequencies over two or more categories of a single nominal variable matches an expected distribution
5. If conditions are not met for a parametric test, (a) what are the advantages and (b) disadvantages of data transformation over rank-order tests, and what are the (c) advantages and (d) disadvantages of rankorder tests over data transformation?
4. (a) What happens if you change your scores to ranks and then figure an ordinary parametric test using the ranks? (b) Why will this not be quite as accurate, even assuming that the transformation to ranks is appropriate? (c) Why will this result not be quite as accurate as using the standard
3. (a) If you wanted to use a standard rank-order test instead of a t test for independent means, what procedure would you use? (b) What are the steps of doing such a test?
2. Transform the following scores to ranks: 5, 18, 3, 9, 2.
1. (a) What is a rank-order transformation? (b) Why is it done? (c) What is a rank-order test?
4. Consider the following distribution of scores: 4, 16, 25, 25, 25, 36, 64.(a) Are these scores roughly normally distributed? (b) Why? (c) Carry out a square-root transformation for these scores (that is, list the squareroot transformed scores). (d) Are the square-root transformed scores roughly
3. When is this legitimate?
2. Why is it done?
1. What is a data transformation?
3. (a) What is an outlier? (b) Why are outliers likely to have an especially big distorting effect in most statistical procedures?
2. (a) How do researchers typically check to see if they have met the assumptions? (b) Why is this problematic?
1. What are the two main assumptions for t tests and the analysis of variance?
6. About how many participants do you need for 80% power in a planned 2×2 study in which you predict a medium effect size and will be using the .05 significance level?
5. What are three factors that affect the power of a study using a chi-square test of independence?
4. What is the power of a planned 3×3 chi-square with 50 participants total and a predicted medium effect size?
3. (a) What is the measure of effect size for a chi-square test for independence for a contingency table that is larger than 2×2? (b) Write the formula for this measure of effect size and define each of the symbols. (c) What is Cohen’s convention for a small effect size for a 4×6 contingency
2. (a) What is the measure of effect size for a 2×2 chi-square test for independence? (b) Write the formula for this measure of effect size and define each of the symbols. (c) What are Cohen’s conventions for small, medium, and large effect sizes? (d) Figure the effect size for a 2×2 chisquare
1. What are the assumptions for chi-square tests?
3. The larger the expected effect size is, the greater the power. Thus, for maximum power, you want as many participants as possible, as simple a contingency table (that is, as few categories in each direction) as possible, and a large expected effect size.
2. The more degrees of freedom there are (the more different categories are crossed with each other), the less power there is.
1. Like all other hypothesis-testing situations, the more participants there are in the study, the more power there will be.
3. (a) Write the formula for figuring degrees of freedom in a chi-square test for independence and define each of its symbols. (b) Explain the logic behind this formula.
2. (a) List the steps for figuring the expected frequencies in a contingency table. (b) Write the formula for expected frequencies in a contingency table and define each of its symbols.
1. (a) In what situation do you use a chi-square test for independence? (b)How is this different from the situation in which you would use a chisquare test for goodness of fit?
4. For each cell, multiply its row’s percentage by its column’s total. The column total for the male characters is 153; 26.1% of 153 comes out to 39.9 (that is, .261×153=39.9).These steps can also be stated as a formula,
3. Find each row’s percentage of the total. The 58 characters in the child row is 26.1% of the overall total of 222 (that is, 58/222=26.1%).
2. For each cell, multiply its row’s percentage by its column’s total.Applying these steps to the top left cell in Table 11–4 (child characters who are male),
1. Find each row’s percentage of the total.
5. Use the steps of hypothesis testing to carry out a chi-square test for goodness of fit (using the .05 significance level) for a sample in which one category has 15 people, the other category has 35 people, and the first category is expected to have 60% of people and the second category is
4. (a) What is a chi-square distribution? (b) What is its shape? (c) Why does it have that shape?
3. Write the formula for the chi-square statistic and define each of the symbols.
2. List the steps for figuring the chi-square statistic, and explain the logic behind each step.
1. In what situation do you use a chi-square test for goodness of fit?
6. Add up the results of Step for all the categories.
5. Divide each squared difference by the expected frequency for its category.
4. Square each of these differences.
3. In each category, take observed minus expected frequencies.
2. Determine the expected frequencies in each category.
1 Determine the actual, observed frequencies in each category.
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.
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.
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