Statistics Review: Probability, Means, and Confidence Intervals - Prof. Janice L. Case, Exams of Mathematical Statistics

Download Statistics Review: Probability, Means, and Confidence Intervals - Prof. Janice L. Case and more Exams Mathematical Statistics in PDF only on Docsity! MS 404 Test 2 Review Chapters 7, 8 & 9 1. Scores on an aptitude test are symmetric with mean 50 and standard deviation 10. What is the probability that the average score of 100 students exceeds 52? 2. A study was conducted to compare the mean number of police emergency calls per 8- hour shift in two districts of a large city. Samples of 100 8-hour shifts were randomly selected from the police records for each of the two regions, and the number of emergency calls was recorded for each shift. The sample statistics are given in the following table. Region 1 Region 2 Sample size 100 100 Sample mean 2.4 3.1 Sample variance 1.44 2.64 Estimate the difference in the mean number of police emergency calls per 8-hour shift between the two districts in the city. Find a bound for the error of estimation. 3. For a comparison of the rates of defectives produced by two assemble lines, independent random samples of 100 items were selected from each line. Line A yielded 18 defectives in the sample, and line B yielded 12 defectives. a. Find a 98% confidence interval for the true difference in proportions of defectives for the two lines. b. Is there enough evidence here to suggest that one line produces a higher proportion of defectives than the other? c. If we wanted to keep the 98% confidence, but cut the margin of error in half, how large a sample would be needed? 4. The EPA set a maximum noise level for heavy trucks at 83 decibels (dB). A random sample of six heavy trucks of a certain type produced the following noise level in dB. 83.4 86.8 86.1 85.3 84.8 82.0 Construct a 90% confidence interval for the mean dB level of all trucks of this type and determine the possibility that this type of truck is in compliance. (Assume that the dB levels are approximately normally distributed.) 5. Given Y1,…, Yn a random sample from an exponential distribution with parameter θ a. Find the maximum likelihood estimator for θ, θ b. Is θ biased? c. Is θ consistent? 6. Given Y1, …, Yn a random sample from a uniform distribution over the interval (0, 3θ). Find the method of moments estimator for θ. 7. Given Y1, …, Yn a random sample from a Poisson distribution with parameter λ. a. Show that U = Σyyi is a sufficient statistic for λ. b. Find the minimum variance unbiased estimator for λ