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## Statistics- What conditions for the population and the study design

Color and cognition. In a randomized comparative experiment on the effect of color on the performance of a cognitive task, researchers randomly divided 69 subjects (27 males and 42 females ranging in age from 17 to 25 years) into three groups. Participants were asked to solve a series of six anagrams. One group was presented with the anagrams on a blue screen, one group saw them on a red screen, and one group had a neutral screen. The time, in seconds, taken to solve the anagrams was recorded. The paper reporting the study gives = 11.58 and s = 4.37 for the times of the 23 members of the neutral group.18(a)Give a 95% confidence interval for the mean time in the population from which the subjects were recruited.(b)What conditions for the population and the study design are required by the procedure you used in part (a)? Which of these conditions are important for the validity of the procedure in this case?The first child has higher IQ. Does the birth order of a familyâs children influence their IQ scores? A careful study of 241,310 Norwegian 18- and 19-year-olds found that firstborn children scored 2.3 points higher on the average than second children in the same family. This difference was highly significant (P < 0.001). A commentator said,âOne puzzle highlighted by these latest findings is why certain other within-family studies have failed to show equally consistent results. Some of these previous null findings, which have all been obtained in much smaller samples, may be explained by inadequate statistical power.â10 Explain in simple language why tests having low power often fail to give evidence against a null hypothesis even when the hypothesis is really false.How Power Behaves. In the setting of Exercise 18.14, use the Statistical Powerapplet to find the power in each of the following circumstances.(a)Standard deviation σ = 55, significance level = 0.05, alternative μ = 255, and sample sizes n = 25, n = 50, and n = 100. How does increasing the sample size with no other changes affect the power?(b)Standard deviation σ = 55, significance level = 0.05, sample size n = 25, and alternatives μ = 255, μ = 260, and μ = 265. How do alternatives more distant from the hypothesis (larger effect sizes) affect the power?(c)Standard deviation σ = 55, sample size n = 25, alternative μ = 255, and significance levels = 0.05, = 0.10, and = 0.20. How does increasing the desired significance level affect the power?

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