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- A random sample of 25 students obtained a mean of 78 and a variance of s2 = 15 on a college entrance exam in engineering. Assuming the scores are normally distributed, construct a 98% confidence interval of σ2Find the first two moments about the mean for the random variable X defined by X = -2 with prob. 1/3 3 with prob. 1/2 = 1 with prob. 1/6. %3DSAT scores in one state is normally distributed with a mean of 1526 and a standard deviation of 112. Suppose we randomly pick 46 SAT scores from that state.a) Find the probability that one of the scores in the sample is less than 1529.P(X<1529)P(X<1529) = b) Find the probability that the average of the scores for the sample of 46 scores is less than 1529.P(¯¯¯X<1529)P(X¯<1529) =
- Q1.[5 points]: A researcher wants to assess if there is a difference in the average age of onset of a certain disease for men and women who get the disease. Let µ = average age of onset for women and µ2 = average age of onset for men. A random sample of 30 women with the disease showed an average age of onset of 83 years, with a sample standard deviation of 11.5 years. A random sample of 20 men with the disease showed an average age of onset of 77 years, with a sample standard deviation of 4.5 years. Assume that ages at onset of this disease are normally distributed for each gender, and assume the population variances are equal. i. What are the appropriate null and alternative hypothesis? ii. What is the value of the test statistics? iii. Why you decided to use the test statistics in part ii ? For a significant level of 0.05, what is the critical value (or critical region)? What is the appropriate conclusion to this test? iv. V.If the random variable X as a mean of u and a variance of o2 what is the P((X-u)² < 20): O less that equal to 1/4 O greater than or equal to 3/4 cannot tell O equal to 5/4If X is a negative binomial rv, then Y= r+Xis the total number of trials necessary to obtainr S’s. Obtain the mgf of Y and then its mean valueand variance. Are the mean and variance intuitively consistent with the expressions for E(X)and V(X)? Explain
- SAT scores in one state is normally distributed with a mean of 1532 and a standard deviation of 200. Suppose we randomly pick 50 SAT scores from that state. a) Find the probability that one of the scores in the sample is less than 1492. P(X < 1492) = b) Find the probability that the average of the scores for the sample of 50 scores is less than 1492. P(X < 1492) =Let X1, X2, X3, ..., X, be a random sample from a distribution with known variance Var(X,) = o², and unknown mean EX, = 0. Find a (1 – a) confidence interval for 0. Assume that n is large.Q3: The effective life of a component used in a jet-turbine aircraft engine is a random variable with mean 5000 hours and standard deviation 40 hours. The distribution of effective life is fairly close to a normal component that increases the mean life to 5030 hours and decreases the standard deviation to 30 hours. distribution. The engine manufacturer introduces an improvement into the manufacturing process for this Suppose that a random sample of n₁ = 16 components is selected from the "Old" process and a random sample of n₂ = 25 components is selected from the "Improved" process. What is the probability that difference in the two-sample means that the old and improved processes can be regarded as independent populations. 5 X₂X₁ is at least 15 hours? Assume
- The average wait time to get seated at a popular restaurant in the city on a Friday night is 12 minutes. Is the mean wait time greater for men who wear a tie? Wait times for 13 randomly selected men who were wearing a tie are shown below. Assume that the distribution of the population is normal. 13, 13, 11, 13, 13, 13, 13, 10, 12, 13, 13, 10, 13 What can be concluded at the the αα = 0.05 level of significance level of significance? The null and alternative hypotheses would be: H0 = (p,u) (<,>,=,not equal) to _____ H1 = (p,u) (<,>,=,not equal) to _____ The test statistic (t,z) = ___ The p-value = _____ Thus, the final conclusion is that ... The data suggest that the population mean wait time for men who wear a tie is not significantly more than 12 at αα = 0.05, so there is statistically insignificant evidence to conclude that the population mean wait time for men who wear a tie is more than 12. The data suggest the population mean is not significantly more than 12…DCX Suppose that an airline quotes a flight time of 2 hours, 10 minutes between two cities. Furthermore, suppose that historical flight records indicate that the actual flight time between the two cities, x, is uniformly distributed between 2 hours and 2 hours, 20 minutes. Letting the time unit be one minute, (a) Calculate the mean flight time (x) and the standard deviation (ax) of the flight time. (Round your answers to 4 decimal places.) ux OX (b) Find the probability that the flight time will be within one standard deviation of the mean. (Round your answer to 5 decimal places.) P = 18 Fo MAR LO 5 li Ⓒ A WD ShowX is distributed as a Normal random variable with mean of 100 and a standard deviation of 10 (i.e X~N(100,10). You take a random sample of 20 items from X and you calculate the average of this sample. What is the expected value of this sample average? (i.e. what value would you expect to get for the average of this small sample?
