Problems Not to Be Turned In, But Worked On Your Own

PROBLEMS NOT TO BE TURNED IN, BUT WORKED ON YOUR OWN 1. The distributions of incomes in two cities follow two Pareto type pdf’s: f (x) = 2/x3 , 1 < x < ∞, and g(y) = 3/y 4 , 1 < y < ∞. Here one unit represents 20,000 dollars. One person with income is selected at random from each city and let X and Y be their respective incomes. Compute P (X < Y ). Comment: Ok, so this is basically a 2D random variable problem. You will need to find the jpdf h(x, y), and then to integrate over the given region. 2. Three components are placed in series. The time in hours to failure of each has the pdf f (x) = xe−x/500 /5002 , 0 < x < ∞. Since they are in series, we are concerned with the minimum time Y to failure of the three. Assuming independence, find the cdf and the pdf for Y and compute P (Y ≤ 300). Comment: This problem is kind of like that problem from the Maximum Likelihood Estimate and Unbiased Estimator notes about esimating the length t of the interval [0, t] by choosing numbers at random from the interval. The idea is to convert the the problem at hand into a probability calculation involving three independent events. 3. Bowl A contains 100 red balls and 200 white balls; bowl B contains 200 red balls and 100 white balls. let p denote the probability of drawing a red ball from a bowl, but say that p is unknown, since it is unknown whether bowl A or bowl B is being used. We shall test the simple null hypothesis H0 : p = 1/3 against the simple alternate hypothesis Ha : p = 2/3. Draw three balls at random, one at a time and with replacement from the selected bowl. Let X equal the number of red balls drawn. Then let the critical region (what we call the ‘rejection region’) be C = {x : x = 2, 3}. What are the values of α and β, the probabilities of Type I and Type II errors, respectively? 4. It was claimed that 75% of all dentists recommend a certain brand of gum for their gum-chewing patients. A consumer group doubted this claim and decided to test H0 : p = 0.75 against the alternative hypothesis H1 : p < 0.75, where p is the proportion of dentists who recommend this brand of gum. A survey of 390 dentists found that 273 recommended this brand of gum. Which hypothesis would you accept if the significance level is a. α = 0.05? b. α = 0.01? c. Find the p-value for this test. 5. Let X equal the thickness of spearmint gum manufactured for vending machines. Assume that the distribution of X is N (µ, σ2 ). The target thickness is 7.5 hundredths of an inch. We shall test the null hypothesis H0 : µ = 7.5 against a two-sided alternate hypothesis using 10 observations. a. Define the test statistic and rejectioin region for an α = 0.05 significance level. Sketch a figure illustrating this region. b. Calculate the value of the test statistic and clearly give your decision using the following n = 10 thickness in hundredths of an inch for pieces of gum that were selected randomly from the production line: 7.65, 7.60, 7.65, 7.70, 7.55, 7.55, 7.40, 7.40, 7.50, 7.50. c. Is µ = 7.50 contained in a 95% confidence interval for µ? 6. A 1-pound bag of candy-coated chocolate-covered peanuts contained 224 pieces of candy colored brown, green, and yellow. Test the null hypothesis that the machine filling these bags treats the four colors of candy equally likely; that is, test H0 : pB = pO = pG = pY = 1/4. The observed values were 42 brown, 64 orance, 53 green, and 65 yellow. You may select the significance level or give the approximate p-value. Comment: Obviously you want to use a χ2 test. 7. While testing a used tape for bad records, a computer operator counted the number of flaws per 100 feet of tape. Let X equal this random variable. Test the null hypothesis that X has a Poisson distribution with a mean of λ = 2.4 given that 40 observations of X yielded 5 zeros, 7 ones, 12 two, 9 threes, 5 fours, 1 five and 1 six. Let α = 0.05. Hint: Combine five and six into one set; that is, the last set would be all x values ≥ 5. 8. A random sample X1 , ..., Xn of size n is taken from a Poisson distribution with a mean of λ, 0 < λ < ∞. a. Show that the maximum likelihood estimator for λ is λ̂ = X. b. Let X equal the number of flaws per 100 feet of a used computer tape. Assume that X has a Poisson distribution with a mean of λ. If 40 observations of X yielded 5 zeros, 7 ones, 9 threes, 5 fours, 1 five, and 1 six, find the MLE estimate for λ. 9. Let X1 , ..., Xn be a random sample of size n from the exponential distribution whose pdf is f (x; θ) = (1/θ)e−x/θ , 0 < x < ∞, 0 < θ < ∞. a. Show that X is an unbiased estimator of θ. b. Show that the variance of X is θ2 /n. c. What is a good estimate of θ if a random sample of size 5 yielded the sample values 3.5, 8.1, 0.9, 4.4, and 0.5?