Suppose the random variable, X, follows an exponential distribution with mean 8 (0 ≤ @<∞). Let X₁, X₂, X, be a random sample of size n from the population of X. (d) Let 6 = 2 and n = 1. Determine the value of k (that is used in the specification of the rejection region) so that a = 0.05. (e) If Y₁, Y₂..... Ym is a random sample from another exponential distribution with mean 6, find the likelihood ratio criterion for testing Ho: 0= 6 versus H₁: 06. (Note here you need to use both samples-X₁, X2, X and Y₁, Y2.... Ym to test the equality of the two population means.)

50. Let X₁, X2, ..., X50 be i.i.d. with E(X) = 25 and V(X₂) = 10. Consider the random variable defined by 50 X₁ the sample average X = Σ Apply the Central Limit Theorem to answer the following: n i=1 (a) What is the approximate distribution of X (include parameter(s))? (b) Approximate P(X> 25.5). (c) Approximate P(|X-25 <0.8). (d) Approximate the 0.975 quantile (97.5th%tile) of X.

If we assume that the distribution of means is normal, then 95% of all sample means  a) will fall in the interval m ± 1.96(σm) b) will fall in the interval m ± 1.645(σm) c) will be no more than 1.645(σm) from μ d) will be no more than from 1.96(σm) from μ

A simple random sample of size n =66,  is obtained from a population that is skewed left with =33 and =3. . Does the population need to be normally distributed for the sampling distribution of x  to be approximately normally​ distributed? Why? What is the sampling distribution of x?     Does the population need to be normally distributed for the sampling distribution of x to be approximately normally​ distributed? Why?(A) Yes. The central limit theorem states that the sampling variability of nonnormal populations will increase as the sample size increases. (B) Yes. The central limit theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x  ​normal, regardless of the sample​ size, n. (C)No. The central limit theorem states that only if the shape of the underlying population is normal or uniform does the sampling distribution of x,  become approximately normal as the sample​ size, n, increases. (D) No. The central limit theorem…

5. Let (*1, x2, .., rn) be n independent random sample of size n from a uniform distribution on the interval [01,02]; (a) find the c.d.f of, X(1), the minimum order statistic (b) find the c.d.f of, X(n), the maximum order statistic (c) Hence, find the mean and variance of X(1) and X(n)-