1) Suppose tha Y is a random variable having an gamma distribution with a = 1, find the value of k such that an interval from 0 to ky is a (1 - a)100% confidence interval for the parameter ß.
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is a random variable having a Gamma distribution with .
confidence interval for parameter is .That is ,
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- 3. Assume that the distribution of the length X of male spider is N(ux,o}) and the length Y of female spiders is N(µy,o?), and X and Y are independent. Construct approximated 95% one-sided confidence intervals for o? /o? when n = m = 30, and = 0.4399 and S? = 1.4100.1) Suppose tha Y is a random variable having an gamma distribution with a = 1, find the value of k such that an interval from 0 to ky is a (1 - a) 100% confidence interval for the parameter B.Q2: Let x1,X2, . , Xn and y1, y2, ..., Ym represent two independent random samples from the respective normal distributions N(H1,07) and N (H2, 03). It is given that of = 30, but ožis unknown. Then 1. A random variable that can be used to find a 95% confidence interval for - is (x- y) - (H1 - H2) А. ns3 + ms n+m. 3(n+m-2)"tm, nm O A (x – y) – (41 – H2) В. ns? + ms; n + m. (n + m – - 2) пт (x – y) – (41 - H2) С. ns3 + mS;_n+3 3(n+ m-2) ("+3m nm Ос
- 6). A random sample of 90 observations produced a mean of x¯=21 from a population with a normal distribution and a standard deviation σ=4.1.Suppose X1,..., Xn is a random sample from an exponential distribution with mean e. If X = 17.9 with n = 50, find (a) a one-sided 95% confidence interval for 0, and (b) a two-sided 95% confidence interval for 0.(Mathematical Statistics) Let X1,X2,...,X5 be a random sample of the Gamma distribution with parameters a = and ß = 0, e > 0. Determine the 95% confidence interval for the parameter e based on the statistic E(i=1 - 5) Xi. 2
- 2. We performed a linear regression using 25 observations. From the regression output we find that bo = 5.7, bị = 12.9, x = 11.4, Sz = 3.2 and MSE = 12.25. a. From the least squares line, what is the predicted response when x* = 10.65? y = b. What is the 95% confidence interval for the mean response when x* = 10.65? c. What is the 95% prediction interval for an individual response when x* = 10.65? d. Which interval is wider? The confidence interval or the prediction interval? O a. Confidence Interval b. Prediction IntervalSuppose that a random sample X₁, X2, X20 follows an exponential distribution with parameter B. Check whether or not a pivotal quantity exixts, if it exists, find a 100(1 - α)% confidence interval for B. www.You have taken a random sample of sizen & 95of a normal population that has a population mean ofμ = 140and a population standard deviation ofo = 23. Your sample, which is Sample 1 in the following table, has a mean ofx = 141.3. (In the table, Sample 1 is indicated by "M1", Sample 2 by "M2", and so on.) (to) Based on Sample 1, plot the confidence intervals of80%and95%for the population mean. Use1,282as the critical value for the confidence interval of 80%and use1960 as the critical value for the confidence interval of95%. (If necessary, you can refer to a list of formulas .) • Write the upper limit and the lower limit on the graphs to indicate each confidence interval. Write the answers with one decimal place. • For the points (♦and ◆), write the population mean, μ = 140. 128.0 128.0 80% confidence interval 139.0 X Ś 150.0 150.0 128.0 128.0 95% confidence interval 139.0 X Ś 150.0 150.0
- IF n < 30 AND T = Ý−µ -H, √n SHOW HOW THE 100(1-2x)% CONFIDENCE INTERVAL FOR U IS DERIVED1) Suppose tha Y is a random variable having an gamma distribution with a = 1, find the value of k such that an interval from 0 to ky is a (1 - a) 100% confidence interval for the parameter B.Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally distributed populations. Also assume that the population standard deviations are equal (Η1 = Η2), so that the standard error of the difference between means is obtained by pooling the sample variances. 1) A paint manufacturer wanted to compare the drying times of two different types of paint.Independent simple random samples of 11 cans of type A and 9 cans of type B were selected andapplied to similar surfaces. The drying times, in hours, were recorded. The summary statistics areas follows.Type A Type Bx1 = 71.5 hr x2= 68.5 hrs1 = 3.4 hr s2 = 3.6 hrn1 = 11 n2 = 9a). Construct a 99% confidence interval for µ1- µ2, the difference between the mean drying timefor paint type A and the mean drying time for paint type B b). Run a hypt. test to determine the results*Alt is not equal*Make sure to select pooled…
