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Statistics 100B University of California, Los Angeles Department of Statistics Instructor: Nicolas Christou Homework 6 EXERCISE 1 The following 16...

three statistics problems from my attached PDF.
question number is 5, 8,9
please done by due date and time.

University of California, Los Angeles Department of Statistics Statistics 100B Instructor: Nicolas Christou Homework 6 EXERCISE 1 The following 16 numbers came from a normal random number generator on a computer 5.3299 4.2537 3.1502 3.7032 1.6070 6.3923 3.1181 6.5941 3.5281 4.7433 0.1077 1.5977 5.4920 1.7220 4.1547 2.2799 Enter these data in R using: x <- c(5.3299, 4.2537, 3.1502, . ..) mean(x) var(x) #Note: this is the unbiased estimator! a. Find the mle estimates for the mean and variance ( μ and σ 2 ). b. Give 90% , 95%, and 99% confidence intervals for μ and σ 2 . In constructing the confidence interval for the population variance use the unbiased estimate for σ 2 , not the mle. c. Using the results from part (b) give 90% , 95%, and 99% confidence intervals for σ . d. How much larger a sample do you think you would need to halve the length of the interval for μ ? EXERCISE 2 In homework 2, exercise 8, you were asked to show that ¯ X - ¯ Y , follows the normal distribution with mean μ 1 - μ 2 and variance σ 2 1 m + σ 2 2 n . Show the steps needed to construct a 1 - α confidence interval for μ 1 - μ 2 . Assume that σ 1 and σ 2 are known. EXERCISE 3 If X 1 and X 2 are independent random variables having, respectively, binomial distributions with parameters n 1 ,p 1 , and n 2 ,p 2 , construct a 1 - α confidence interval for p 1 - p 2 . Hint: What is the distribution of X 1 n 1 - X 2 n 2 when the sample sizes are large? EXERCISE 4 The manager of a supermarket would like to know the average time that a person spends at the checkout counter. Using a stopwatch, he observes 100 customers. He computed the sample mean to be ¯ x = 15 . 35 minutes and the sample standard deviation to be s = 6 . 1 minutes. a. Construct a 95% confidence interval for the population mean μ . b. Suppose that the manager wants a smaller error in estimation (smaller than what you found in (a)). He wants his error to be ± 1 minute with 95% confidence. How many customers will he need? For this question assume σ = 6 . 1 minutes. EXERCISE 5 A precision instrument is guaranteed to read accurately to within 2 units. A sample of 4 instrument readings on the same object yielded the measurements 353, 351, 351, and 355. Find a 90% confidence interval forn the population variance. What assumptions are necessary? Does the guarantee seem reasonable? EXERCISE 6 Recently there have been discussions about constructing a subway system that would run from Downtown Los Angeles to Santa Monica through Wilshire Boulevard. Suppose a random sample of 900 voters in Hollywood indicates that 600 support such an idea. a. Construct a 95% confidence interval for the Hollywood population proportion of residents who would support this idea. b. Suppose that the City of Los Angeles wants to estimate with 95% confidence the percentage of residents who would support this idea in Hollywood. The city wants the error of estimation to be ± 2% of the population proportion. What is the minimum sample size required? c. Suppose that the City of Los Angeles wants to estimate with 95% confidence the percentage of residents who would support this idea in Westwood. The city wants the error of estimation to be ± 2% of the population proportion. What is the minimum sample size required? Assume that there is no prior information about the population proportion. EXERCISE 7 Suppose that two independent random samples of n 1 and n 2 observations are selected from normal populations with means μ 1 2 and variances σ 2 1 2 2 respectively. Find a confidence interval for the variance ratio σ 2 1 σ 2 2 with confidence level 1 - α .
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EXERCISE 8 The sample mean ¯ X is a good estimator of the population mean μ . It can also be used to predict a future value of X inde- pendently selected from the population. Assume that you have a sample mean ¯ x and a sample variance s 2 , based on a random sample of n measurements from a normal population. Construct a prediction interval for a new observation x , say x p . Use 1 - α confidence level. Hint: Start with the quantity X p - ¯ X and then use the definition of the t distribution. EXERCISE 9 (from Mathematical Statistics and Data Analysis ), by J. Rice, 2nd Edition. In a study done at the National Institute of Science and Technology (Steel et al. 1980), asbestos fibers on filters were counted as part of a project to develop measurement standards for asbestos concentration. Asbestos dissolved in water was spread on a filter, and punches of 3-mm diameter were taken from the filter and mounted on a transmission electron microscope. An operator counted the number of fibers in each of 23 grid squares, yielding the following counts: 31 29 19 18 31 28 34 27 34 30 16 18 26 27 27 18 24 22 28 24 21 17 24 Assume that the Poisson distribution with unknown parameter λ would be a plausible model for describing the variability from grid square to grid square in this situation. a. Use the method of maximum likelihood to estimate the parameter λ . b. Use the asymptotic properties of the maximum likelihood estimates to construct a 95% confidence interval for λ . As a reminder, for large samples the distribution of ˆ θ - θ p 1 nI ( θ ) is approximately standard normal, where I ( θ ) is the Fisher information. EXERCISE 10 Use R to access the data form the Maas river (see data points below). These data contain the concentration of lead and zinc in ppm at 155 locations at the banks of the Maas river in the Netherlands. You can read the data in R as follows: soil <- read.table("http://www.stat.ucla.edu/~nchristo/soil.txt", header=T) a. Use R to compute the sample mean and sample standard deviation of lead. b. Construct a 95% confidence interval for the population mean of lead in this area. c. The level of risk for surface soil based on lead concentration in ppm is given on the table below: Mean concentration (ppm) Level of risk Below 150 Lead-free Between 150-400 Lead-safe Above 400 Significant environmental lead hazard Based on your confidence interval form (b) in which category does the soil of this area fall in terms of the ppm concentration of lead? d. Do you see any problem in these calculations (meaning by just using the averages)? Concentration of lead and zinc: Lead concentration (ppm) ●● 37 72.5 123 207 654 Zinc concentration (ppm) ●● 113 198 326 674.5 1839
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