Cal Poly, STAT 301
Excerpt: ... STAT 301 Exam 2 Preparation Logistics: Wed Feb 18 from 10:10-11am Open-book, open-notes Bring calculator No computer use Material from Tues Jan 27 Fri Feb 13, HW9-14 Optional review problems o Ch1: 20, 21, 23, 24, 26, 43, 44, 46, 47, 51 o Ch5: 1, 3, 5, 8, 10, 13, 16, 17, 20, 22 Overview: We have analyzed studies that involve two categorical variable, for which the results can be organized in a two-way table. We have studied how to conduct inferences depending on whether the data were collected from: A randomized experiment Independent random samples Neither We have also considered how the scope of conclusions to be drawn depends on how the data were collected. More specifically, random assignment allows for drawing cause/effect conclusions, and random sampling allows for generalizing to a larger population. We learned three ways to conduct statistical inference in this situation: Simulation o Randomization model (for randomized experiment) o Binomial sampling mode ...
Cal Poly, STAT 301
Excerpt: ... Stat 301 Day 37 Bootstrapping, cont (5.5) Last Time Bootstrapping A simulation tool for exploring the sampling distribution of a statistic, using only the information in the obtained sample! In particular, standard error of the statistic Median, "trimmed mean," (heart transplant data). Confidence interval: statistic + t SE(statistic) ^ - * , 2 - * ) 95% Percentile CI:( 2 ^.975 ^ ^.025 No "technical conditions" other than random sample! Two samples? Investigation 5.5.1 (p. 454) Transplants.mtw What was random in this study? Independent random samples from two populations? Random assignment to treatment groups? Investigation 5.5.1 (d) Treat as random samples but consider the difference in medians Bootstrap each sample, calculate difference in medians How would you describe this distribution? Percentile confidence interval (f-h) Observed difference statistically significant? Investigation 5.5.1 (j) Treat as randomly assigned groups How would you describe this distribution? O ...
Penn State, STAT 544
Excerpt: ... Stat 544 - Assignment 1 Due Thursday, September 7 This assignment pertains to material covered in the first lecture. If you do not find this assignment to be straightforward, then perhaps Stat 544 is not the best course for you. Please bring hard cop ...
Ohio State, STAT 623
Excerpt: ... STAT 623 HOMEWORK 8 DUE Friday, March 3, 2006 Do the following problems in Chapter 11. 1- Exercise 1. 2- Exercise 10 (consider the case where the variances are unknown but assumed to be equal) 2- Exercise 11 (Justify your answer). 3- Exercise 15. 4- Exercise 16. 5- Exercise 31, parts a and b (Justify your answer). 5- Exercise 34 (To answer last part, carry out an appropriate hypothesis test at the = 0.05 level). Additional problems: 1- Let X1 , , Xn and Y1 , , Ym be independent random samples from exponential distributions with unknown parameter X and Y , respectively. Derive the Generalized Likelihood Ratio test of H0 : x = Y versus HA : x = Y . Note that the exponential density has the form f (x/) = exp(-x), x > 0. 2- Let X1 , , Xn and Y1 , , Ym be independent random samples from the normal (, 2 ) and the Gamma (2, 1/ ) distributions, respectively. Note that the pdf of Y is f (y| ) = (1/ 2 )y exp(-y/ ) if y > 0, zero otherwise. Here, , 2 , and are all unknown. a- Find a good pi ...
University of Hawaii - Hilo, MATH 373
Excerpt: ... Math 373 Hw 33 Name _ Score _/15 Hw 593: 14.24, 14.26. Rec 593: 14:23. 602: 14.39, 14.41'. Page 593. 14.24(8). Independent random samples of 100 observations each are selected from each of three binomial populations A, B, C, with the following results. A B C Total 76 succes 24 19 33 s _ _._ _ _ _._ _ _ _._ _ fail 76 81 67 _ _._ _ _ _._ _ _ _._ _ Total 100 100 100 224 300 14.26(7). A poultry disease is thought to be noncommunicable. To test this, 30,000 chickens are divided into three groups of 10,000 each. One group has no contact with infected chickens, one group has some moderate contact, and the third had heavy contact. After a 6-month period, the number of diseased chickens in each group was counted. Test for a significant difference between the three groups w.r.t. the incidence of the disease. Note, this is really equivalent to asking if the incidence of the disease depends on the amount of contact. The tests for a difference (Lecture 33) and for dependence (Lectu ...
Cal Poly, STAT 301
Excerpt: ... ction 3.3) Approximate test/CI procedure: normal (section 4.3) Test conditions: n0 10, n(1-0) 10 CI conditions: n p 10, n(1- p ) 10 Above tests/CIs require random sample or random process 2. Comparing two groups on a categorical (binary) response Graph: Segmented bar graph (section 1.1) Statistics: Difference in sample proportions, relative risk, odds ratio (sections 1.1, 1.2) Approximate test procedure: simulating randomization test (Jan 27 handout) Exact test procedure: hypergeometric (section 1.7) Above tests require randomized experiment Approximate test/CI procedure: normal (sections 5.1, 5.2) Test/CI conditions: At least 5 successes and 5 failures in each group Above tests/CIs require independent random samples or randomized experiment 3. Comparing two groups on a quantitative response Graphs: Boxplots, histograms, dotplot, stemplots (section 2.1) Statistics: Difference in means, difference in medians (section 2.1) Approximate test procedure: simulating randomizat ...
Austin Peay, MATH 3260
Excerpt: ... ' $ Math 3260 Statistical Methods II & www.apsu.edu/wuh/Math3260/Math3260.htm 1 % ' $ Lecture 6: Inferences About Population Proportions-Continued 1 Inferences About Differences Between Population Proportions 2 CIs for Population Proportion Differences 3 Hypothesis Tests for Population Proportion Differences 3 5 6 & www.apsu.edu/wuh/Math3260/Math3260.htm 2 % ' $ 1 Inferences About Differences Between Population Proportions We are often concerned with making inferences about differences between population proportions. For example, a director of marketing may wish to compare the public awareness of a new product recently launched and that of a competitor's production. Let p1 and p2 denote actual proportions of two different populations (called 1 and 2) having a particular attribute. Let n1 and n2 be sizes of independent random samples taken from populations 1 and 2, respectively, and let y1 and y2 denote the numbers of the respective samples exhibiting the attribute. An unbiased estimator for ...
Purdue, STAT 503
Excerpt: ... Stat 503 Lecture 35 Mon. Nov. 27, 2006 1. Review the 2x2 contingency table 2. Quiz 7 on Wed. (ANOVA F-test) 2x2 Contingency table chi-squared test of independence Two main contexts (sometimes blurred): 1. Two independent random samples ; one binomial variable observed in each 2. One sample; observe two different binomial variables on each Examples of context 1.: - Samples are "mammography" and "non-mammography"; observed variable "breast cancer death" or " not death" - samples are "drug" and "placebo" (any two treatments); observed variable "improve" or "don't improve" Examples of context 2.: - observe "eye color" (dark/light) and "hair color" (dark/light) - observe whether people smoke, exercise Example 10.21. 4 Categories; observations in 2x2 table: Hair color dark Light cye dark 729 131 color light 3129 2814 Total 3855 2945 Total 857 5943 6800 Test for independence of row and column variables. Fill in E's in the table: E = (row total)(column total)/(grand total) hair color dark light Eye color dark 7 ...
Middle Tennessee State University, FRANK 6603
Excerpt: ... Chi-Square Tests Using !(O E)2 /E Two major tests: for independence and for homogeneity 3 major scenarios single sample with no fixed margins (independence) r independent random samples (homogeneity) fixed margin totals (independence or homogeneity) I. Test for independence assumptions single random sample of size N each observation fits in exactlty one cell of contigency table H0 : row designation is independent of column designation (pij pi. p.j ) H1 : " is not independent " Test statistic: T !(O E)2 /E, which under H0 is approximately distributed as a ;2 with (r 1)(c 1) deg.free. Decision rule: Reject H0 if T is too large. Note that the exact distribution of T under H0 may be obtained by using the multinomial distribution. (See Conover text). N O11 O P(Table Outcome) O11 ,O12 ,.,Orc p11 pO12 prcrc 12 II. Test for homogeneity assumptions r mutually independent random samples (sample size ni fixed, i 1,.,r) each observation fits ...
Delaware, FREC 408
Excerpt: ... Difference of Means Problem for Homework 6 The data are two independent random samples Sample 1 Sample 2 52 52 33 43 42 47 44 56 41 62 50 53 44 61 51 50 45 56 38 52 37 53 40 60 44 50 50 48 43 60 55 ...
East Los Angeles College, MAS 2302
Excerpt: ... 1 MAS2302/MAS3302: Introduction to Statistical Inference Exercises 4 Hand in your solutions by 4pm on Thursday 13th December. Note that Exercises 1 consists of three parts: A. Standard exercises; B. Computer project; C. Computer Based Assessment (CBA) to be completed online within the same time period. A. Standard exercises 1. Use Neave tables or R to find the following: (i) P (T < 1.476) where T t5 ; (ii) P (T < -3.25) where T t9 ; (iii) P (T > 1.617) where T t7 ; (iv) t23 (0.05); (v) t40 (0.01); (vi) t17 (0.025); 2. A random sample of n observations from a N(, 2 ) random variable yielded values x = 10.7 and s = 1.45. Given that 2 is unknown, test the null hypothesis H0 : = 10.0 versus a two-sided alternative hypothesis when (i) n=10, (ii) n=40. 3. The following data consist of scores of mathematical ability from two independent random samples of economists and statisticians. Statisticians n1 = 12 408 448 136 344 332 348 472 344 452 248 321 346 Economists n2 = 10 284 173 241 170 203 344 337 89 ...
University of Florida, STA 4322
Excerpt: ... dispensing machine is said to be out of control if the variance of the contents exceeeds 1.15 (the unit of volume is deciliter). If a random sample of 25 drinks from this machine has a variance of 2.03, does this indicate at the 0.05 level of significance that the machine is out of control? Assume that the contents are approximately normally distributed. What can you say about the p-value of this test? 4. Two groups of elementary school children were taught to read by using different methods. The number of students in each group was 50. At the conclusion of the instructional period, a reading test yielded the results y1 = 74, y2 = 71, and s1 = 9, and s2 = 10. (a) Is there a difference between the population means for the two groups at the = 0.05 level? Please state what case is relevant here. (b) What is the p-value for this test? 5. A psychological study was conducted to compare the reaction times for men and women to a stimulus. Independent random samples of 50 men and 50 women were employed in the ex ...
Iowa State, STAT 101
Excerpt: ... Stat 101 Lecture 26 Interpretation Getting a value of the sample proportion of 0.12 is consistent with random sampling from a population with population proportion p=0.10. This sample result does not contradict the null hypothesis. The P-value is n ...
Iowa State, STAT 101
Excerpt: ... Stat 101L: Lecture 29 Interpretation Getting a value of the sample proportion of 0.12 is consistent with random sampling from a population with population proportion p = 0.10. This sample result does not contradict the null hypothesis. The P-value is ...
Penn State, AJW 13
Excerpt: ... Solutions - ANOVA 14.1 Hotel satisfaction: a) The response variable is the performance gap, the factor is which hotel the guest stayed in, and the categories are the five hotels. b) H 0 : 1 = 2 = 3 = 4 = 5 ; H a : at least two of the population means are unequal. c) df 1 = g - 1 = 5 - 1 = 4 because there are five groups; df 2 = N - g = 125 - 5 = 120 because there are 125 people in the study and five groups d) from a table or software, F = 2.45 and higher 14.3 What's the best way to learn French?: a) i) Assumptions: Independent random samples , normal population distributions with equal standard deviations ii) Hypotheses: H 0 : 1 = 2 = 3 ; H a : at least two of the population means are unequal iii) Test statistic: F = 2.50 ( df 1 = 2, df 2 = 5) iv) P-value: 0.177 (rounds to 0.18) v) Conclusion: If the null hypothesis were true, the probability would be 0.18 of getting a test statistic at least as extreme as the value observed. We cannot reject the null hypothesis at a standard significance level o ...
UIllinois, STAT 400
Excerpt: ... Statistics 400 Section 7.3 Confidence Intervals for Difference of Two Means If X1, X2, ., Xn are observations of a random sample of size n from a normal distribution N( x, 2 x ), then we have X is N( x, 2 x /n) If Y1, Y2, ., Ym are observations of ...
University of Texas, ECON 329
Excerpt: ... Practice Questions 6 1. C 2. B 3. D 4. C 5. B 6. B 7. D 8. C 9. B 10. A 11. D 12. B 13. C 14. C 15. A 16. A 17. D 18. B 19. B 20. C 21. D 22. X tn 1 , / 2 s / n = 63.57 2.032(17.32) / 5.92 = 63.57 Then, UCL = 69.52 and LCL = 57.62. 35 = 63.57 5.95. 23. If independent random samples of size 35 are repeatedly selected from the population and 95% confidence intervals for each of these samples are determined, then over a very large number of repeated trials, 95% of these intervals will contain the value of the true average amount of money a typical college student spends per day during spring break. 24. E ( X ) [ E ( X 1 ) E ( X 2 )] / 2 ( ) / 2 E (Y ) [ E ( X 1 ) 3E ( X 2 )] / 4 ( 3 ) / 4 E ( Z ) [ E ( X 1 ) 2 E ( X 2 )] / 3 ( 2 ) / 3 Since E ( X ) E (Y ) E ( Z ) , then all three estimators are unbiased. 25. Var ( X ) [Var ( X1 ) Var ( X 2 )]/ 4 ( 2 2 ) / 4 2 / 2 Var (Y ) [Var ( X1 ) 9Var ( X 2 )]/16 ( 2 9 2 ) /16 5 2 / 8 Var (Z ) [Var ( X1 ) 4Var ( X ...
The University of Oklahoma, ECON 2843
Excerpt: ... ECON 2843 Problem Set 8 1. Problem 9.6 in Newbold on p. 323. A company which receives shipments of batteries tests a random sample of nine of them before agreeing to take a shipment. The company is concerned that the true mean lifetime for all batter ...