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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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