Proof:
If X1 and X2 are independent, then E[(X1-μ1) (X2-μ2)] = 0.
If X1 and X2 are independent,
, by definition ...equation(i)
Now,
, since and are constants and we know E(constant) = constant
Let X1 and X2 be independent random variables with means μ1 and μ2, and variances σ21 and σ22, respectively. Find the correlation of X1 and X1 + X2. Note that: The covariance of random variables X; Y is dened by Cov(X; Y ) = E[(X - E(X))(Y - E(Y ))]. The correlation of X; Y is dened by Corr(X; Y ) =Cov(X; Y ) / √ Var(X)Var(Y )
Consider the following hypothesis test. H0: μ1 − μ2 = 0 Ha: μ1 − μ2 ≠ 0 The following results are for two independent samples taken from the two populations. Sample 1 Sample 2 n1 = 80 n2 = 70 x1 = 104 x2 = 106 σ1 = 8.4 σ2 = 7.2 (a) What is the value of the test statistic? (Round your answer to two decimal places.) (b) What is the p-value? (Round your answer to four decimal places.)...
Consider the following hypothesis test. H0: μ1 − μ2 = 0 Ha: μ1 − μ2 ≠ 0 The following results are from independent samples taken from two populations. Sample 1 Sample 2 n1 = 35 n2 = 40 x1 = 13.6 x2 = 10.1 s1 = 5.9 s2 = 8.5 (a) What is the value of the test statistic? (Use x1 − x2. Round your answer to three decimal places.) (b) What is the degrees of freedom for the t...
Let X1, ...., Xm be iid N(μ1,σ2) and Y1, ..., Yn be iid N(μ2,σ2), and X's and Y's are independent. Here -∞<μ1,μ2<∞ and 0<σ<∞ are unknown. Derive the MLE for (μ1,μ2,σ2). Is the MLE sufficient for (μ1,μ2,σ2)? Also derive the MLE for (μ1-μ2)/σ.
Considering two Gaussian distributions N1~(μ1,σ1^2) and N2~(μ2,σ2^2), we pick two random variables x1 and x2 in order to compute the sum x3=x1+x2. We want to prove that: a) x3 follows a gaussian distribution b) estimate mean value μ3 and variance σ3^2 c) repeat the above steps for multivariate Gaussian distributions N1~(μ1,Σ1) and N2~(μ2,Σ2)
Find the standardized test statistic to test the claim that μ1 ≠ μ2. Assume the two samples are random and independent. Population statistics: σ1 = 0.76 and σ2 = 0.51 Sample statistics: x1 = 3.6, n1 = 51 and x2 = 4, n2 = 38
Ho: μ1 - μ2 = 0 Ha: μ1 - μ2 ≠ 0 When testing above hypotheses the test statistic t was found to be 5.15. If the degrees of freedom = 40, then, the p-value for this test would be Question options: a) less than zero b) between 0 and 0.5 c) less than .005 d) greater than 1
Give a 95% confidence interval, for μ1−μ2μ1-μ2 given the following information. n1=45n1=45, ¯x1=2.67x¯1=2.67, s1=0.69s1=0.69 n2=20n2=20, ¯x2=2.8x¯2=2.8, s2=0.61s2=0.61 <μ1−μ2
Find the standardized test statistic to test the claim that μ1 < μ2. Two samples are random, independent, and come from populations that are normally distributed. The sample statistics are given below. Assume that σ 2 /1 = σ 2 /2 . n1 = 15 n2 = 13 x1 = 27.88 x2 = 30.43 s1 = 2.9 s2 = 2.8
Find the critical value to test the claim that μ1 < μ2. Two samples are random, independent, and come from populations that are normal. The sample statistics are given below. Assume that σ 2/1= σ2/2. Use α = 0.05. n1 = 15 n2 = 15 x1 = 25.74 x2 = 28.29 s1 = 2.9 s2 = 2.8