Let X1, X2, . , X25 be a simple random sample from a population having distri- bution N(0, o2), where o² is an unknown parameter. For the hypotheses Ho : o? < 1, H1 : o? > 1, consider the test that rejects Ho when 25 x? > c, where the constant c is chosen to have level of significance a = = 0.05. • Obtain the constant c. [Hint: The sampling distribution of zLE1 X is X(25): Now find c by solving max,2<1 P(E X? > c) = 0.05.] Li=1 25

from each population with bivariate normal distribution, samples were taken with each size n = 4, as follows. 6 5 8 5 Population | X₁ = 10 6 9 8 Population II X₂ = (₁ 5 45, 11 10 12 11' By using a = 5% a). in population 1, test the hypothesis Ho : μ = (₁ Is this Ho rejected? 5 b). in population II, test the hypothesis Ho:μ=(₁). Is this Ho rejected? 10 c). test the hypothesis Ho: E₁ = ₂. Is this Horejected?

For this study, we should use     The null and alternative hypotheses would be: z-test for a population proportion or t-test for a population mean Ho: (p or u)           (please enter a decimal)    H1:  (p or u)          (Please enter a decimal) The test statistic z =  (please show your answer to 3 decimal places.) The critical value =       (Please show your answer to 3 decimal places.) The test statistic (is or is not)    in the rejection region. Based on this, we should  (reject or accept or fail to accept)    the null hypothesis. Thus, the final conclusion is that ... The data suggest the population proportion is not significantly larger 50% at � = 0.05, so there is not sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is larger 50%. The data suggest the populaton proportion is significantly larger 50% at � = 0.05, so there is sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is…

3. (Sample size determination for hypothesis testing) Suppose that X₁,..., Xn N(μ, o2) and o² is known. We would like to test We consider the test statistic Ho: μ = po, and the critical region at significance level a 1- {=; C = T = X x; x > µo + %α² H : μ > μο· B(μ') = = (a) We write ß(µ') to be the probability of type II error given the true value of µ = µ'. Prove that 0 (za + 10 = 1²). where Þ(z) = P(Z ≤ z) is the cdf of Z ~ N(0,1). 1 n = = O 2.2.d. (b) We would like to determine the sample size n so that the probability of type I error is a and the probability of type II error is ß. Using part (a), prove that 2 - ( 0¹ (²0 + zp)) ². Po - (c) Let u denote the true average tread life of a certain type of tire. Consider testing Ho : μ = 25,000 versus H₁ : μ> 25, 000 from a normal population distribution with o = 1300. Assuming the true value of μ is μ' = 24,000. What is the sample size n so that the probability of type I error is a = 0.05 and the probability of type II error is = 0.025?