We have a random sample of size 17 from the normal distribution N(u,02) where u and...
3. Suppose that ai . ,,an are a random sample from a N( ,02) distribution. Recall that the MLE in this case is [a, σ]T = [x, V (n-1)s2/n]T and the information matrix is Consider the data s2-4.84 with n 16 (a) Use the delta-method to obtain an approximate 95% confidence interval for log(o) (b) Obtain an approximate 95% confidence interval for σ2 using the confidence interval from (a). Compare to the exact interval, [2.21,15.77], and approximate interval [0.43, 10.50...
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)
We draw a random sample of size 25 from a normal population with a known variance of 2.4. If the sample mean is 12.5, what is the Lower Confidence Limit for the 95% confidence interval for the population mean? Include 1 decimal place in your answer
Example 3.6. Take a random sample of size n from an exponential distri- bution with rate parameter XA. 1. Derive an exact 95% confidence interval for X. 2. Suppose your sample is of size 9 and has sample mean 3.93. (a) What is your 95% confidence interval for λ? (b) What is your 95% confidence interval for the population mean? 3. Repeat the above using the CLT approximation (rather than an eract interval
A simple random sample of size 64 is drawn from a normal population with a known standard deviation σ. The 95% confidence interval for the population mean μ is found to be (12, 16). The approximate population standard deviation σ is:
1. (50 points) Suppose X1, ..., Xn form a random sample from a N(u,02) distribution with p.d.f. Fe 202, for – V2110 <x< . Assume that o = 2 is known. a) (10 points) Derive the 90% confidence interval for u that has the shortest length. You must show all details including the pivot you use. b) (8 points) Show that the sample mean is an efficient estimator for u. Assume in (c)- (f) that the prior distribution of u...
7.5 Suppose you draw a random sample of size n from a normal distribution with unknown mean u and known standard deviation o and construct a 95% confidence interval for u. If you want to halve the margin of error, how much larger would the sample size have to be?
(21) A sample of size 20 is drawn from a normal distribution with unknown variance and mean. The sample variance s2 = 0.012. Find a 95% two-sided confidence interval for the standard deviation o of the population. A. [0.0833,0.1600] B. (0.010,0.0130] C. (0.0069,0.0256] D. None of the above
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 which are both unknown. (a) Given observations xi, ,Xn, one would like to obtain a (1-a) x 100% one-sided confidence interval for u as a form of L E (-00, u) the expression of u for any a and n. (b) Based on part (a), use the duality between confidence interval and hypothesis testing problem, find a critical region of size...
A random sample of size n = 21, taken from a normal population with a standard deviation 04 =5, has a mean X4 = 90. A second random sample of size n2 = 37, taken from a different normal population with a standard deviation o2 = 4, has a mean X2 = 39. Find a 94% confidence interval for 11 - H2 Click here to view page 1 of the standard normal distribution table. Click here to view page 2...