Version A Problem 1/Version B Problem 3:Version A: Assume X1,…,X25 iid ¼ N(1,Ï2). (a) [5
Version A Problem 1/Version B Problem 3:Version A: Assume X1,…,X25 iid ¼ N(1,Ï2). (a) [5 pts] The MLE of Ï2 isË Ï2 =P25 i=1(Xi 1)2 25)Show that 25Ë Ï2/Ï2 has a Chi-squared distribution with 25 degrees of freedom. (b) [5 pts] Use the result in part (a) to derive a 99% conï¬dence interval for Ï2) (Your quantiles should have numerical valuesnot just symbols.) Your ï¬nal answer should be in interval form (L,U), for some lower bound L and upper bound U. Version B: Assume X1,…,X15 iid ¼ N(1,Ï2). (a) [5 pts] The MLE of Ï2 isË Ï2 =P15 i=1(Xi 1)2 15)Show that 15Ë Ï2/Ï2 has a Chi-squared distribution with 15 degrees of freedom. (b) [5 pts] Use the result in part (a) to derive a 90% conï¬dence interval for Ï2) (Your quantiles should have numerical valuesnot just symbols.) Your ï¬nal answer should be in interval form (L,U), for some lower bound L and upper bound U. Version A Problem 2/Version B Problem 4:Version A: Edgar Anderson collected data on sepal widths (in centimeters) of the Setosa species of iris ï¬ower (Bulletin for the America Iris Society, 59, 2-5, 1935). For a random sample of 18 ï¬owers, the sample mean sepal width was 3)228 cm with a sample standard deviation of 0.329) A normal quantile-quantile plot of the sample widths is below.-2 -1 0 1 22)42)62)83)03)23)43)6Normal Q-Q PlotTheoretical QuantilesSample QuantilesUsing a signiï¬cance level of .05, conduct a one-sample t-test to determine if the mean sepal width in the population of Setosa species of iris ï¬owers diï¬ers from 3)2 cm:(a) [2 pts] State hypotheses in terms of the population parameter. (Deï¬ne any symbols used.) (b) [3 pts] Check assumptions and calculate the test statistic. (c) [2 pts] Find the p-value. State which distribution table you used and where you looked on the table to ï¬nd the p-value. (d) [1 pt] State your decision (Reject H0 or Fail to reject H0). (e) [2 pts] Provide a conclusion in terms of the research study. Version B: Edgar Anderson collected data on sepal widths (in centimeters) of the Virginica species of iris ï¬ower (Bulletin for the America Iris Society, 59, 2-5, 1935). For a random sample of 18 ï¬owers, the sample mean sepal width was 2)994 cm with a sample standard deviation of 0.372) A normal quantile-quantile plot of the sample widths is below.-2 -1 0 1 22)62)83)03)23)43)63)8Normal Q-Q PlotTheoretical QuantilesSample QuantilesUsing a signiï¬cance level of .05, conduct a one-sample t-test to determine if the mean sepal width in the population of Virginica species of iris ï¬owers diï¬ers from 2)9 cm:4(a) [2 pts] State hypotheses in terms of the population parameter. (Deï¬ne any symbols used.)(b) [3 pts] Check assumptions and calculate the test statistic. (c) [2 pts] Find the p-value. State which distribution table you used and where you looked on the table to ï¬nd the p-value.(d) [1 pt] State your decision (Reject H0 or Fail to reject H0). (e) [2 pts] Provide a conclusion in terms of the research study.