Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o² is unknown. Consider the estimate for the variance. a) Show that 7² == n n i=1 (Xi − µ)² - - TL C:= = 2 ( x ₁ = 1) ²³ ~ x² (n). i=1 b) For a € (0, 1), denote the a-quantile of x²(n) by qn(a). Using this notation, find numbers an, bn ER such that P(02 b₂0²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o2 = 0 against the alternative H₁:0² +0. d) Assume we have n = 100,= 0 and we have observed ² = 4.81. Test the hypothesis Ho: 024 vs. H₁: 024 at significance level 5%. =

The length (in mm) of a spring subjected to a force FNewtons is of the form L = a + BF+ ɛ, where ɛ is assumed normally distributed with mean 0, variance o2. Given the sample statisticsn=D13, E F;= 138, E L; = 577, SEL = -43, SEF= 1735, estimate the expected length of the spring when a force of 11.2 Newtons is applied. (Give your answer correct to 2 decimal places.) Answer: Ti Check 江一 prime viden Type here to search 19 hp ins prt fu f12 f8 f9 f10 f7 f5 f6 『米 " JOI JDI & 6 7 8 5 96 %24 3

Let X be a random variable normally distributed with mean μx 1. Find the mean μy of Y. 2. Find the standard deviation oy of Y. 3. Find the PDF fy of Y and evaluate fy (5). (My, oy, fy (5)) = ___________). = 6 and standard deviation ox = 4. Let Y = 4X + 5.

An experiment to compare the tension bond strength of polymer latex modified mortar (Portland cement mortar to which polymer latex emulsións have been added during mixing) to that of unmodified mortar resulted in x = 18.11 kgf/cm2 for the modified mortar (m = 42) and y = 16.88 kgf/cm2 for the unmodified mortar (n = 31). Let ₁ and ₂ be the true average tension bond strengths for the modified and unmodified mortars, respectively. Assume that the bond strength distributions are both normal. (a) Assuming that o₁ = 1.6 and ₂ = 1.3, test Ho: ₁ - ₂ = 0 versus H₂: H₁ - H₂> 0 at level 0.01. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) Z = P-value = State the conclusion in the problem context. O Fail to reject Ho. The data suggests that the difference in average tension bond strengths exceeds 0. Fail to reject Ho. The data does not suggest that the difference in average tension bond strengths…