Answer the following using the R statistical computing platform. Your answer should include the code you wrote plus the

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Answer the following using the R statistical computing
platform. Your answer should include the code you wrote plus the
output of such code and English rhetoric / coding comments where
necessary.

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Instructions: Answer the following using the R statistical computing platform. Your answer should include the code you wrote plus the output of such code and English rhetoric / coding comments where necessary. In eachist any mathematical assumptions about what method you are using. N1. [Two-Sample Mean Test whenčı + [2] As an extension to Wednesday's discussion, what if the two samples have different variance structures. In a large sample problem, we can do the same procedure but with an approximate test. A confidence ellipsoid forui - uz is all ui - uz satisfying 1 1 [Xi - X2 - (ui - uz)]' S1 + S2] [X1 - X2 - (ui - uz)] = xbla) n2 Suppose two measurements of power usage for electric car driversnı = 56) and non-electric car drivers (n2 = 89).The summary statistics are: 20441 138.3 2381 Si = [ 238 731] -13001 186.3 1961 S2 = [196 560] Find 95% simultaneous confidence intervals for the differences in the mean components and discuss what can be concluded in this test[ 30 pts] N2. [Multiple-Sample Testing of Variance Matrix Problem 14.8, 14.9, and 14.10[20 pts each) X1 = 15566) X2 = (3550]
14.4.1 Testing for Mean Vector with known Variance-Covariance Matrix If the variance-covariance matrix known, the test statistic is a multivariate extension of the Z -statistic, and is given by 14.8 Z2 = n(X - Mo)'-'(- Ho). Under the hypothesis H : u = Ho, the test statistic Z2 is distributed as a chi-square variate with p degrees of freedom, and a random variable. The computations are fairly easy and we do that in the next example. Example 14.4.1. The Importance of Handling the Covariances Rencher (2002). Consider the height and weight of 20 college-age males. This dataset is available in the csv file Height_weight.csv. Assume that the variance-covariance matrix is known as I = [20, 100, 100, 1000), and that the hypothesis of interest is H : u = Mo = [70, 170), where the height is measured in inches and the weight in pounds. > data(w) mue n signa meanx 22 22 # the test statistic value [1] 8.4026 >qchis (1-.05,2) # 95% confidence level and 2 d.f. [1] 5.991465 Since the calculated x2 value is greater than the tabulated value, we reject the hypothesis that the average height and weight are at 70 and 170 units respectively. If we were to ignore the correlation between the height and weight, we have the following conclusions: > htest wtest as.nuneric(htest); as.nuneric(wtest) [1] 1.45 [1] -0.7495332 The absolute value of each of these tests is less than 1.96, and hence we would have failed to reject the hypothesis H, which is not the case when the correlations are adjusted for. Thus, we learn an important story that whenever the correlations are known, it is always better to adjust the statistical procedure for them..
14.4.2 Testing for Mean Vectors with Unknown Variance-Covariance Matrix It turns out that in many practical settings, the variance-covariance matrix is unknown. Therefore, we need to extend the test procedure for this important case. The Hotelling's T2-statistic is given by 14.9 T2 = n(X - Mo)'s-'(X-Mo), where S is the sampling covariance matrix. Under the hypothesis the test statistic T2 is distributed as Hotellings' T2 distribution with P and V = n - 1 degrees of freedom. Example 14.4.3. The Calcium in Soil and Turnip Greens Data of Rencher (2002) Kramer and Jensen (1969) collected data on three variables at ten different locations. The variables of interest are 0)*1: available calcium in the soil, m) X2: exchangeable soil calcium, and () 43: turnip green calcium. Suppose the hypothesis of interest is H : u = [15.0 6.0 2.85). > data(calciun) > n meanx varx @ t2 library(ICSNP) > HotellingsT2(calciun[,-1], nunue, test="+") Hotelling's one sample T2-test data: calciun, -1] T. 2 = 6.3671, df1 = 3, df2 = 7, p-value = 0.82868 alternative hypothesis: true location is not equal to c(15,6,2.85) The conclusion on using any of the methods does not vary and we are led to reject the hypothesis H.-