A random sample of X1,X,, X,...,X, have mean E(X,)= µ and Var(X,)=o². Given the value of §² ¿(x, - x} = £x;-nX² | to estimate the value k=1 k=1 of o?. Determine the bias for this estimator B(S²)= E[S³]-o². %3D
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![A random sample of X,,X2, X3..,X, have mean E(X,)= µ and Var(X,)=o².
Given the value of 5 Σ(x,-X -Σχ-η
¿(x, - x} = - £x; -nX | to estimate the value
n k=!
n
k=1
of o?. Determine the bias for this estimator B(5²)= E[S²]-o².
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- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.A random sample of n, 20 winter days in Denver gave a sample mean pollution index x, = 43. Previous studles show that o,- 10. For Englewood (a suburb of Denver), a random sample of n, - 19 winter days gave a sample mean pollution Index of x, = 37. Previous studies show that g, -13. Assume the pollution Index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance. (a) what is the level of significance? State the null and alternate hypotheses. (b) what sampling distribution will you use? What assumptions are you making? O The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's t. We assume that both population distributions are approximately normal with known standard devtations. O The standard normal. We assume that both population…Suppose that f(x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.
- Suppose that f (x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.Compute co-efficient of correlation from the following given results: x =10, Edx = 0. Edy = 0, E(x-K)? = 1350, Ety-7}²= 2180 and E(x-X) (y-7) =1590Suppose X~ Binom (6,p) and define an estimator T=X/6 to estimate p. Then variance of T is
- We know that the point estimator o` is an unbiased estimator for the parameter 0 if E(e') = 0.If the estimator is biased (not unbiased), then the difference E(O) – 0 is called the bias of the estimator e and is denoted by Bias(e'). That is,Suppose X~ N(2,6). What value of a has a z-score of three? x =Show that the conditional variance can be written as σ^2 (X) = E { Y^2 | X} −(E[Y | X])^2 no handwritten solution.
- 9. (8') Let X,, X,, X, be a sample of size n=3 from a distribution with unknown mean u, -0 < u<0, where the variance o is a known positive number. Show that both à =i and ê, = 2X, + X, +5X, are unbiased estimators for H.The random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−3/4. Calculate Var(X+Y) with a full explanationSuppose we have random samples (X₁, X2, X3, X₁, X5) and it is known that X; Exponential (0). Th observations are (x1, X2, X3, X4, X5)=(0.32, 2.37, 1.26, 2.18, 1.8) When typing in formulae, use the notation below xi theta x_i When typing in formulae, don't forget to use "*" for multiplication Likelihood Find the likelihood function by filling in the gaps below. Express the Likelihood functions through and X1, X2, X3, X4, X5. 5 = The formula for calculating the likelihood is L(0; x) = [] ƒ(X; ; 0); i=1