Problem 3. Consider two independent samples, X1, . . . , Xm from a N(µ1, σ12 ) distribution and Y1, . . . , Yn from a N(µ2, σ22 ) distribution. Here µ1, µ2, σ12 and σ2 are unknown. Consider testing th...
Problem 3. Consider two independent samples, X1, . . . , Xm from a N(µ1, σ12 ) distribution and Y1, . . . , Yn from a N(µ2, σ22 ) distribution. Here µ1, µ2, σ12 and σ2 are unknown. Consider testing the null hypothesis that the two population variance are equal, H0 : σ12 = σ22 , against the alternative that these variances are different, H1 : σ12 ≠ σ12 .
(a) Derive the LR test statistic Λ
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Problem 3. Consider two independent samples, X1, . . . , Xm from a N(µ1, σ12 ) distribution and Y1, . . . , Yn from a N(µ2, σ22 ) distribution. Here µ1, µ2, σ12 and σ2 are unknown. Consider testing th...
Let X1, . . . , Xn be i.i.d. from N(µ1, σ2 ), and Y1, ....
Let X1, . . . , Xn be i.i.d. from N(µ1, σ2 ), and Y1, . . . , Ym be i.i.d. from N(µ2, σ2 ). If the two samples are independent, find the maximum likelihood estimates for µ1, µ2, and the common variance σ 2 .
Let X1, ...., Xm be iid N(μ1,σ2) and Y1, ..., Yn be iid N(μ2,σ2), and X's...
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.
please answer both with an explanation (14) For testing Ho : µ1 = µ2 versus H1...
please answer both with an explanation
(14) For testing Ho : µ1 = µ2 versus H1 : µ1 # µ2, a completely randomized design used two indpendent samples of sizes n = m = 10 and obtained t-statistics = -2.87. If the assumption of equal but unknown population variances is justified, what is the p-value? The following "answers" have been proposed. (a) Approximately 0.005 (b) Approximately 0.01 (c) Approximately 0.025 (d) Approximately 0.05 (e) None of the above. The correct...
Suppose that X1, X2, . . . , Xn is an iid sample of N (0, σ2 ) observations, where σ 2 > 0 is unknown. Consider testing H0 : σ 2 = σ 2 0 versus H1 : σ 2 6= σ 2 0
Suppose that X1, X2, . . . , Xn is an iid sample of N (0, σ2
) observations, where σ
2 > 0 is
unknown. Consider testing
H0 : σ
2 = σ
2
0 versus H1 : σ
2
6= σ
2
0
;
where σ
2
0
is known.
(a) Derive a size α likelihood ratio test of H0 versus H1. Your rejection region should
be written in terms of a sufficient statistic.
(b) When the null...
12 marks Let independent random samples of sizes n and n2 be taken respectively from two normal distributions with unkn...
12 marks Let independent random samples of sizes n and n2 be taken respectively from two normal distributions with unknown means 1 and 2 and unknown variances oand o. Denote the two samples by . . ,Jn, and y,... , yn2: Which have means T and T, and sample variances s and s2, respectively (a) 4 marks Show that when of = o2, the likelihood ratio test statistic for testing Ho 12 against H 2 can be written as T2...
Concept Check: Terminology 0/3 points (graded) Suppose you observe iid samples X1,…,Xn∼P from some unknown distribution...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
Consider independent random samples from two populations that are normal or approximately normal, or the case...
Consider independent random samples from two populations that are normal or approximately normal, or the case in which both sample sizes are at least 30. Then, if σ1 and σ2 are unknown but we have reason to believe that σ1 = σ2, we can pool the standard deviations. Using sample sizes n1 and n2, the sample test statistic x1 − x2 has a Student's t distribution where t = x1 − x2 s 1 n1 + 1 n2 with degrees...
eBook Video Exercise 10.1 (Algorithmic)) Consider the following results for two independent random samples taken from...
eBook Video Exercise 10.1 (Algorithmic)) Consider the following results for two independent random samples taken from two populations. Sample 1 Sample 2 n 50 n2 35 1-1=13.6 X2= 11.1 a. What is the point estimate of the difference between the two population means? | b. Provide a 90% confidence interval for the difference between the two population means (to 2 decimals). c Provide a 95% confidence interval for the difference between the two population means to 2 decimals eBook Video...
please answer both with an explanation
(14) For testing Ho : µ1 = µ2 versus H1 : µ1 # µ2, a completely randomized design used two indpendent samples of sizes n = m = 10 and obtained t-statistics = -2.87. If the assumption of equal but unknown population variances is justified, what is the p-value? The following "answers" have been proposed. (a) Approximately 0.005 (b) Approximately 0.01 (c) Approximately 0.025 (d) Approximately 0.05 (e) None of the above. The correct...
12 marks Let independent random samples of sizes n and n2 be taken respectively from two normal distributions with unknown means 1 and 2 and unknown variances oand o. Denote the two samples by . . ,Jn, and y,... , yn2: Which have means T and T, and sample variances s and s2, respectively (a) 4 marks Show that when of = o2, the likelihood ratio test statistic for testing Ho 12 against H 2 can be written as T2...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
eBook Video Exercise 10.1 (Algorithmic)) Consider the following results for two independent random samples taken from two populations. Sample 1 Sample 2 n 50 n2 35 1-1=13.6 X2= 11.1 a. What is the point estimate of the difference between the two population means? | b. Provide a 90% confidence interval for the difference between the two population means (to 2 decimals). c Provide a 95% confidence interval for the difference between the two population means to 2 decimals eBook Video...
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