Statistics 101, Section 2: Practice Exam for Midterm IIInstructions: Write your answers on the exam in the spaces after the questions. For maximum credit, show all work. Writing an answer withoutshowing work may not receive full credit.You are permitted to use two sheets of paper filled with whatever information you put on them. Other notes, texts, or pieces of paper arenot permitted. You cannot work with or ask questions of others. If you need clarification on any part of the exam, contact Prof. Reiter.1. Isolation and brain-wave activityAn experiment was performed to see whether sensory deprivation over an extended period of time has any effect on the alpha-wave frequencies produced by the brain. To determine this, 20 inmates in a Canadian prison were randomly split into two groups of 10. Members of one group were placed in solitary confinement; those in the other group were allowed to remain in their own cells. Seven days later, alpha-wave frequencies were measured for all subjects. The data are displayed in the table below. For each column in the table,the variable Differences equals the frequency in the Non-confined row minus the frequency in the Confined row.Non-confined 10.7 10.7 10.9 10.3 9.6 11.1 11.1 11.2 10.4 10.4Confined 9.5 10.5 10.3 9.2 9.3 9.9 9.1 10.9 9.4 9.7Differences 1.2 0.2 0.6 1.1 0.3 1.2 2.0 0.3 1.0 0.7Here are the summary statistics: Variable Mean Standard Deviation Non-confined 10.58 .46 Confined 9.78 .61 Differences 0.80 .55 a) To assess differences in the population average alpha-wave frequencies of confined prisoners and non-confined prisoners, would you use a matched pairs analysis or a two separate samples analysis? In two sentences, explain why you chose your analysis, and what, if anything, is wrong with the analysis that you did not choose. b) Give the 95% confidence interval for the difference in the population average alpha-wave frequencies for non-confined prisoners andconfined prisoners. Use 17 degrees of freedom if you do a two-sample analysis and 9 degrees of freedom if you do a matched pairs analysis.c) Is there sufficient evidence in these data to conclude that the population average alpha-wave frequency for non-confined prisoners differs from the population average alpha-wave frequency for confined prisoners? Justify your conclusion using confidence intervals or hypothesis tests. d) What assumptions are you using in the methods in parts b and c? How would you go about checking these assumptions with a statistical software program? Be very brief and very specific with your answers.e) Based on the study design, do you think valid causal conclusions about the effect of isolation on all alpha-wave frequencies can be drawnfor Canadian prisoners? How about for all people? Explain why or why not in three or less sentences. 2. Chucky CheeseIn 1876, Charles Darwin published results of his experiments on the effects of cross-fertilization and self-fertilization in plants. For each run of his experiments, he took one offspring produced from cross-fertilization and one offspring produced from self-fertilization, and heplanted them in the same pot at the same time. When the two plants were fully matured, he measured their heights in inches. Darwin repeated thisprocess 15 times, so that he measured the heights of 15 cross-fertilizedplants and fifteen self-fertilized plants. It took him eleven years to complete his experiments. Darwin's data are in the table below. In thetable, Diffs is the difference in heights (cross - self) in each pot.Cross 23.5 12.0 21 22 19.1 21.5 21.1 20.4 18.2 21.7 23.2 21 21.1 23.1 12.1Self 17.4 20.4 20 20 18.3 18.7 18.7 15.2 16.5 18.0 16.2 18 12.8 15.5 18.1Diffs 6.1 -8.4 1 2 0.8 2.8 3.5 5.1 1.8 3.7 7.0 3 9.4 7.5 -6.0Here are the summary statistics for Darwin's data:Variable Mean Standard Deviation Cross-fertilized 20.21 3.63 Self-fertilized 17.58 2.06 Diffs 2.63 1.97 Darwin suspected that cross-fertilization was more successful than self-fertilization, where success was measured by height of the plant. Darwinhad minimal knowledge of statistics, so he asked his statistician friendGalton for help. (FYI--Galton is the guy who invented regression analysis.)Pretend we're back in the 1870s and that you rather than Galton are the consulting statistician for Charles Darwin (pretty hefty responsibility,eh?).a) Darwin wants to know if there is sufficient evidence in his data to conclude that the population average height of cross-fertilized plants is greater than the population average height of self-fertilized plants.Would you use a matched pairs or two separate groups analysis? In two sentences, explain why you chose your analysis, and what, if anything, is wrong with the analysis that you did not choose. b) Darwin wants a sense of the magnitude of the difference in populationaverage heights of cross-fertilized and self-fertilized plants. Give hima 95% confidence interval for this difference. Use 25 degrees of freedomif you decided on a two-sample analysis and 14 degrees of freedom if youdecided on a matched pairs analysis.c) Test the hypothesis that there is sufficient evidence in his data to conclude that the population average height of cross-fertilized plants is greater than the population average height of self-fertilized plants.Show your hypotheses, test statistic, p-value, and conclusions.d) What assumptions are you using in the methods in parts b and c? How would you go about checking these assumptions with a statistical software program? Be very brief and very specific with your answers.e) Actually, Galton analyzed these data as follows. First, he ordered all fifteen cross-fertilized plants from largest to smallest height. Next, he ordered all fifteen self-fertilized plants from largest to smallest height. Finally, he took the differences between these orderedheights, and he did a t-test of the null hypothesis that the population average of this difference variable is less than or equal to zero. A table showing Galton's manipulations of the Darwin data is shown below. The variable Dif contains the differences in the cross-fertilized and self-fertilized plants by Galton's height order pairings.Data for Galton's Analysis for Problem 2Cross 23.5 23.3 23 22.3 22.3 22.0 21.7 21.5 21 21.0 20.4 19.2 18.3 12.0 12.0Self 20.4 20.0 20 18.7 18.7 18.4 18.0 18.0 18 17.4 16.5 16.3 15.5 15.3