STAB57: Quiz-1 Tutorial 1 (Show your work clearly) 1. random variable X has a continuous distribution for which the p.d.f.

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1 STAB57: Quiz-1 Tutorial 1 1. random variable X has a continuous distribution for which the p.d.f. is as follows: { kx < x < 1 f(x) = 0 otherwise where k > 0 is a constant. (a) (4 points) Determine the prediction of for a future X, if we have decided to predict using the mean of the distribution. Solution: This is a beta(3.5, 1) distribution and so EX = (b) (4 points) Determine the prediction of for a future X, if we have decided to predict using the median of the distribution. (Hint: median is the value m such that P (X > m) = 0.5) Solution: k = 3.5 (ingrate the density and set it equal to 1 and solve for k). 1 f(x)dx = 0.5 = m = m 0 (c) (2 points) Calculate the mean-squared error of the predictor in part (b) above. ( ) Solution: MSE = E(X m) 2 = V X+(µ m) = + ( ( )(3.5+1) ) 2 4.5

2 STAB57: Quiz-2 1. Suppose that we have a finite population Π and a measurement X : Π {1, 2, 3} where Π = 10 and {π : X(π) = 2} = {π : X(π) = 3} = 2. (a) (3 points) Calculate F X (1.5) Solution: F X (1.5) = = 0.6 (b) (3 points) We are planing to select a simple random sample, S = {π 1, π 2 } of two individuals from this population. Calculate P (X (π 2 ) < 2 X(π 1 ) = 1). Solution: P (X (π 2 ) < 2 X(π 1 ) = 1) = (c) (4 points) For a simple random sample of 5 individuals from this population, determine the probability that ˆF X (1) = F X (1) Solution: P ( ˆF X (1) = F X (1)) = P (n ˆF X (1) = nf X (1) = = 3) = ( )( 4 2) ( 10 5 )

3 STAB57: Quiz-3 1. (4 points) Suppose that a statistical model for a random variable X is given by the family of Beta(θ, 1) distributions where θ R +. If our interest is in making inferences about the first quartile of the distribution, then determine the characteristic of interest ψ(θ) Solution: f(x) = θx θ 1 and Q1 θx θ 1 dx = 0.25, i.e. (Q1) θ = 0.25 and ψ(θ) = 0 ln 0.25 Q1 = e θ. 2. (6 points) The data set below gives the survival times in days of 31 guinea pigs in a medical experiment Use the 1.5 IQR rule to determine if there are any outliers present (list all outliers). Show your work clearly. Solution: Q1 = (31+1)/4th value in the ordered data set = 8th value = 66. Q3 = 91 IQR = = IQR = = Q IQR = = 128.5(Q1 1.5 IQR = 28.5(148, 152 are greater than and so outliers. No low outliers

4 STAB57: Quiz-4 1. Suppose that a statistical model is comprised of two distributions given by the following table: s = 1 s = 2 s = 3 f 1 (s) f 2 (s) Find a sufficient statistic (T ) that makes a reduction in the data. statistic (T ) is sufficient. Show all your work clearly. Prove that your Solution: Notice that the likelihood ratios are the same for s = 2 and s = 3 and define T s. t. T (1) = 1, T (2) = 2, and T (3) = 2. T (s 1 ) = T (s 2 ) can happen in two ways: i) with s 1 = s 2 ii)s 1 s 2 T (s 1 ) = T (s 2 ) and s 1 = s 2 = L(θ s 1 ) = L(θ s 2 ) T (s 1 ) = T (s 2 ) and s 1 s 2 = s 1 = 2 and s 2 = 3 or s 1 = 3 and s 2 = 2 s 1 = 2 and s 2 = 3 = L(θ s 1 ) = 4L(θ s 2 ) s 1 = 3 and s 2 = 2 = L(θ s 1 ) = 0.25L(θ s 2 ) i.e. in any case T (s 1 ) = T (s 2 ) = L(θ s 1 ) = kl(θ s 2 ) for some k > 0 and so T is a sufficient statistic.

5 STAB57: Quiz-5 1. Let (x 1, x 2,..., x n ) be an observed sample from a distribution with p.d.f. given by: f(x) = { 1 θ θ x 2θ 0 otherwise where θ > 0 is an unknown parameter. Determine the MLE of θ Solution: L(θ s) = n i=1 I 1 1 (0,2θ+1](x i ) = I (2θ+1) n (0,2θ+1] (x (n) ) = I (2θ+1) n x (n) 1 (θ). [, ) 2 This is maximized when θ = x (n) 1 and so the MLE of θ is x (n)

6 STAB57: Quiz-6 1. (6 points) A random sample of 150 fish were caught in Lake Woebegone. These fish had a mean length of 35.3 cm. Assume that the lengths have a Normal distribution with standard deviation 6.2 cm. (a) Find a 95% confidence interval for the population mean length. Solution: 35.3 ± (b) Calculate the minimum sample size required to make the margin of error of the 95% confidence interval no larger than 0.5 cm.? Solution: Minimum sample size = ( Zσ m ) 2 ( = ) (4 points) Let X 1, X 2,..., X n be a random sample form the Beta(θ, 2), θ > 0. Find the method of moments estimator of θ. Solution: EX = θ θ+2 set = x = θ MOM = 2 x 1 x.

7 STAB57: Quiz-7 1. A It has been reported that 40% of the adult population participate in computer-related hobbies of some kind (for example, chat, web-surfing, programming). A random sample of 180 adults found that 65 participated in computer-related hobbies. We want to assess the evidence that the 40% figure is incorrect. Let p be the proportion of the adult population participate in computer-related hobbies of some kind. Assess the hypothesis H 0 : p = 0.4. Also calculate a 95% CI for (p). Solution: H 0 : p = 0.4 ( against H a : p 0.4 ) Test statistic: z = x p 0 p0 (1 p 0 = 65/ ) 0.4(1 0.4) = 1.07 and p-value = 0.29 > 0.05 and so n 180 no evidence against the null hypothesis. 0.36(1 0.36) CI: 0.36 ± 1.96 = (0.29, 0.43) 180

8 STAB57: Quiz-8 1. I throw a tetrahedral die (i.e. a four-sided die) 30 times and have counts (10,7,5,8) of outcomes 1, 2, 3, 4. Assume that this is an observed sample from Multinomial(30, θ 1, θ 2, θ 3, θ 4 ) and the prior distribution of (θ 1, θ 2, θ 3, θ 4 ) is Dirichlet(1, 1, 1, 1). (a) Find the posterior distribution of θ 1. Solution: θ s Dirichlet(1 + 10, 1 + 7, 1 + 5, 1 + 8) and θ 1 s Beta(11, 23) (b) Find the posterior mean, posterior variance and the posterior mode of θ 1. Solution: E(θ 1 s) = V (θ 1 s) = ) ) ( )(11+23) 2

Review. December 4 th, Review

Review. December 4 th, Review December 4 th, 2017 Att. Final exam: Course evaluation Friday, 12/14/2018, 10:30am 12:30pm Gore Hall 115 Overview Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 6: Statistics and Sampling Distributions Chapter