A report states that the mean yearly salary offer for students graduating with mathematics and statistics degrees Is $52,965. Suppose that a random sample of 50 mathematics and statistics graduates at a large university who received job offers resulted In a mean offer of $63,600 and a standard deviation of $3,600. Do the sample data provide strong support for the claim that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,9657 Test the relevant hypotheses using a = 0.05. In USE SALT Find the test statistic and P-value. (Use technology to calculate the Pivalue. Round your test statistic to two decimal places and your P-value to three decimal places.) P-value m State your conclusion. Fail to reject Me. We have convincing evidence that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,965. Reject Me. We do not have convincing evidence that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,965. @Fail to reject N. We do not have convincing evidence that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,965. Reject No. We have convincing evidence that the mean salary offer for mathematics and statistics graduates of this university is greater than the national average of $62,965.1. Fill in the details for the proof of the Cauchy-Schweitz inequality for random variables EXY 5 VIEXZEYZ. 2. Show that the \"main" theorem on Galton-Watson processes cannot be extended to cases where the offspring disnibution changes over time. That is, give random variables Xle , . .. each of mean EX,- 2 2 but such that if individuals in generation I each have a number of children EX} (independently of one another} then PtUdZ: =0})=1. 3. Show that, for every tr > 1 and I] 1. Let T denote the correSponding Gabon-Watson tree. and let '1\" denote (the distribution of] T conditioned on extinction. Show that T' is distributed like a Salton-Watson tree with offspring distribution X\" d:P0l:q.u} where q is the extintion probability of T. 1. Each year, ratings are compiled concerning the performance of new cars during the first 90 days of use. Suppose that the cars have been categorized according to whether the car needs warranty-related repair (yes or no) and the country in which the company manufacturing the car is based (U.S. or not U.S.). Based on the data collected, the probability that the new car needs a warranty repair is 0.04, the probability that the new car is manufactured by a U.S. based company is 0.60, and the probability that the new car needs a warranty repair and was manufactured by a U.S. based company is 0.025. a. Construct a contingency table to evaluate the probabilities. b. Give an example of a simple event. c. Give an example of a joint probability. d. What is the complement of "a car is manufactured by a company based in the United States"? e. What is the probability that a randomly chosen car needs a warranty repair? f. What is the probability that a randomly chosen car was not manufactured by a company based in the United States? g. What is the probability that a randomly chosen car does not need a warranty repair? h. What is the probability that a randomly chosen car was manufactured by a company based in the United States? i. What is the probability that a randomly chosen car was manufactured by a company based in the United States and does not need a warranty repair? i. What is the probability that a randomly chosen car needs a warranty repair and was not manufactured by a company based in the United States? k. What is the probability that a randomly chosen car was manufactured by a company based in the United States and needs a warranty repair? I. What is the probability that a randomly chosen car needs a warranty repair or was manufactured by a company based in the United States? m. What is the probability that a randomly chosen car was not manufactured by a company based in the United States or does not need a warranty repair? n. Suppose we choose a car that does not need a warranty repair, then what is the probability that it was not manufactured by a company based in the United States? o. Does the country of manufacture influence whether the car needs to be repaired or not?Suppose the following is known about students in their first year of a Mathematics program at university, regarding the introductory Calculus (C), Linear Algebra (L). and Statistics (S) courses: . 80% of students take Calculus . 76% of students take Linear Algebra . 22% of students take Statistics . 65% of students take Calculus and Linear Algebra . 11% of students take Linear Algebra and Statistics . 10% of students take Calculus and Statistics . 4% of students take Linear Algebra, Calculus, and Statistics. Suppose we randomly select a citizen in this city. Answer the following questions: 1. What is the probability that this student is studying Calculus or Linear Algebra? 2. What is the probability that, out of these three courses, this student is only studying Statistics? 3. Are C and S independent? Are S and L mutually exclusive? | Choose
