Question 8. Consider a sequence of n independent trials, each resulting in one of k + 1 outcomes 1, 2, ……· ‚k + 1. Outcome j occurs with probability p; on any given trial. Let Y; be the number of trials resulting in outcome j. Consider testing the simple null hypothesis 1, 2, …… · ‚ k + 1. Find the likelihood ratio statistic An and show that 0 where Pj = Пј for j -2 log(n) - Qn = P k+1 Qn => (Yj – nπj)² Nπ j Using the asymptotic equivalence of -2 log(n) and Qn find the asymptotic distribution of Qn. (Hint: Use the Taylor expansion f(x) = x log(x/x0) = (x − x0) + (x − xo)²/2xo + o[(x − x0)²]. ) - Solution: -

A consumer group plans a comparative study of the mean life of four different brands of batteries. Ten batteries of each brand will be randomly selected and the time until the energy level falls below a pre-specified level is measured. a) Which of the following is the appropriate alternative hypothesis for the null hypothesis: H0: μa = μb = μc = μd? (In this problem, μa = mean time of Brand A, μb = mean time of Brand B, etc.) Ha: none of the means are equal Ha: μa ≠ μb ≠ μc ≠ μd Ha: μa ≠ μb, μa ≠ μc, μa ≠ μd, μb ≠ μc, μb ≠ μd, μc ≠ μd Ha: at least one of the means is different b)In order to analyze the data with ANOVA, we need to satisfy the condition of randomization. How do we know that this condition has been met? -One SRS of batteries is selected from a collection of batteries of all brands. -The batteries are randomly allocated to the four brands. -Separate random samples of batteries are selected from each brand.

A researcher administers a treatment to a sample of n=36 selected from a population with u=30 and o=2. If the sample mean after treatment is M=32, then Cohen’s d=1.00. True or False?

Two separate samples are selected from the same population. Each have an n=8. Both samples receive a different treatment. After the treatment, the first sample has SS=16 and the second sample has SS=24.   With a mean of 3 for the first group and mean of 6 for the second group, compute the t statistic. State the conclusion of these findings.

2. Suppose we have a random sample X1,..., X, from a normal distribution N(u, o?), with known mean µ and unknown variance o?.Suppose we are interested in o and not o?. Calculate the Cramer-Rao Lower Bound for estimators of o.

Prove that the events Es, E2... En are independent iff their corresponding indicator variates I,, I₂.. In are independent.

An event is biased such that an outcome "x" is thrice as likely to occur as "y". Find the expected value of "y" outcomes if the event happened twice given that "x" and "y" are discrete random variables with a unit value of 1 and both of them are the only possible outcomes of the said event. Select the correct response.: 1/2 none of these 1/4 1/3 1/5

Let X1, X2, ... be a sequence of random variables with finite mean u and non-zero variance o?. Let S, = X;. Which random variable has meanequal to 0 and variance - equal to 1? Select one: S-np O a. Zn Vno Sn-H O b. Zn Vno O c. Zn Vño O d. Zn Vno