Probability and Statistics Lecture 9: 1 and 2-Sample Estimation

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Probability and Statistics Lecture 9: 1 and 2-Sample Estimation"

Transcription

1 Probability and Statistics Lecture 9: 1 and -Sample Estimation to accompany Probability and Statistics for Engineers and Scientists Fatih Cavdur

2 Introduction A statistic θ is said to be an unbiased estimator of the parameter θ if E θ = θ We can show that E X = μ and E S = σ

3 Introduction If we consider all possible unbiased estimators of θ, the one with the smallest variance is said to be the most efficient estimator of θ.

4 Introduction

5 1-Sample: Estimating the Mean If x is the mean of a random sample of size n from a population with known variance σ, a α % confidence interval on μ: x z α/ σn < μ < x + z α/ σ n

6 1-Sample: Estimating the Mean

7 1-Sample: Estimating the Mean

8 1-Sample: Estimating the Mean

9 1-Sample: Estimating the Mean Example 9.: The average zinc concentration recovered from a sample of measurements taken in 36 different locations in a river is found to be.6 grams per milliliter. Find the 95% and 99% confidence intervals for the mean zinc concentration in the river. Assume that the population standard deviation is 0.3 grams per milliliter.

10 1-Sample: Estimating the Mean The 95% confidence interval is x z α/ σn < μ < x + z σ α/ n z < μ <.6 + z < μ < < μ <.70

11 1-Sample: Estimating the Mean The 99% confidence interval is x z α/ σn < μ < x + z σ α/ n z < μ <.6 + z < μ < < μ <.73

12 1-Sample: Estimating the Mean If x is the mean of a random sample of size n from a population with known variance σ, a 1-sided α % confidence interval on μ: μ x + z α μ x z α σ n σ n

13 1-Sample: Estimating the Mean Example 9.4: In an experiment, 5 subjects are selected randomly and their reaction time to a particular stimulus is measured. Past experience suggests that the standard deviation of these types of stimuli is seconds and that the distribution of reaction times is approximately normal. The average time for the subjects is 6. seconds. Give an upper 95% bound for the mean reaction time.

14 1-Sample: Estimating the Mean μ x + z α σn = 6. + z =

15 1-Sample: Estimating the Mean If x and s are the mean and standard deviation of a random sample of size n from a population with unknown variance σ, a α % confidence interval on μ: x t α/;n 1 sn < μ < x + t α/;n 1 s n

16 1-Sample: Estimating the Mean

17 1-Sample: Estimating the Mean Example 9.: The contents of seven similar containers of sulfuric acid are 9.8, 10., 10.4, 9.8, 10.0, 10. and 9.6 liters. Find a 95% confidence interval for the mean contents of all such containers, assuming an approximately normal distribution.

18 1-Sample: Estimating the Mean We have x = 10.0, s = 0.83 and t 0.05;6 =.447. Hence, the 95% confidence interval is x t α/;n 1 sn < μ < x + t s α/;n 1 n t 0.05;6 7 < μ < t ; < μ < < μ < 10.6

19 -Samples: Estimating the Difference of Means If x 1 and x are the means of independent random sample of size n 1 and n from a population with known variances σ 1 and σ, a α % confidence interval for μ 1 μ : x 1 x z α/ σ 1 + σ < μ n 1 n 1 μ < x 1 x + z α/ σ 1 n 1 + σ n

20 -Samples: Estimating the Difference of Means Example 9.10: A study was conducted to compare two types of engines, A and B. Gas mileages, in miles per gallon, was measured. 50 experiments were conducted for engine type A and 75 experiments were conducted for engine type B. All conditions were held constant. The average gas mileages were 36 miles per gallon for type A and 4 miles per gallon for type B. Find a 96% confidence interval on μ B μ A. Assume that the population standard deviations are 6 and 8 for type A and B, respectively.

21 -Samples: Estimating the Difference of Means We have, x 1 x z α/ 4 36 z 0.0 σ 1 + σ < μ n 1 n 1 μ < x 1 x + z α/ < μ B μ A < z < μ B μ A < 8.57 σ 1 n 1 + σ n

22 -Samples: Estimating the Difference of Means If x 1 and x are the means of independent random sample of size n 1 and n from a population with unknown variances σ 1 = σ, a α % confidence interval for μ 1 μ : x 1 x ± t α/;n1 +n s p 1 n n where s p is the pooled estimate of the population standard deviation s p = n 1 1 s 1 + n 1 s n 1 + n

23 -Samples: Estimating the Difference of Means Example 9.11: Assume that we have two samples with n 1 = 1, x 1 = 3.11, s 1 = and n = 10, x =.04, s = Find a 99% confidence interval for the difference between the population means assuming that the populations are approximately normally distributed with equal variances.

24 -Samples: Estimating the Difference of Means We have s p = = n 1 1 s 1 + n 1 s n 1 + n = 0.646

25 -Samples: Estimating the Difference of Means and x 1 x ± t α/;n1 +n s p ± t 0.05; n n < μ 1 μ < 1.55

26 -Samples: Estimating the Difference of Means If x 1 and x are the means of independent random sample of size n 1 and n from an approximately population with unknown variances σ 1 and σ, a α % confidence interval for μ 1 μ : x 1 x ± t s 1 α/;v + s n 1 n where v is the degrees of freedom of the t distribution and defined as v = s 1 n + s 1 n s 1 n 1 n s n 1 n

27 -Samples: Estimating the Difference of Means We have, v = s 1 n + s 1 n s 1 n 1 n s 1 n 1 n 1 = = 16.3 x 1 x ± t s 1 α/;v + s n 1 n ± t 0.05; < μ 1 μ < 4.1

28 -Samples: Estimating the Difference of Means If d and s d are the mean and standard deviation of the normally distributed differences of n random pairs of measurements, a α % confidence interval for μ 1 μ : d t α/;n 1 s d n < μ 1 μ < d + t α/;n 1 s d n

29 -Samples: Estimating the Difference of Means Example 9.13: Assume that we have n = 0, d = 0.87 and s d = We also assume that the distribution of the differences is approximately normal. Find a 95% confidence interval for the difference. We have s d d t α/;n 1 n < μ s d 1 μ < d + t α/;n 1 n t 0.05;19 0 < μ μ < t 0.05; < μ 1 μ < 0.534

30 1-Sample: Estimating a Proportion If p is the proportion of successes in a random sample of size n and q = 1 p, an approximate α % confidence interval for p: or p z α/ p q n < p < p + z p q α/ n p + z α/ n 1 + z α/ n ± z α/ 1 + z α/ n p q n + z α/ 4n

31 -Samples: Estimating the Difference of Proportions If p 1 and p are the proportions of successes in random samples of sizes n 1 and n, respectively, and q 1 = 1 p 1 and q = 1 p, an approximate α % confidence interval for p 1 p : p 1 p ± z α/ p 1 q 1 n 1 + p q n

32 1-Sample: Estimating the Variance If s is the variance of a random sample of size n from a normal population, a α % confidence interval for σ : n 1 s < σ < χ α/;n 1 n 1 s χ 1 α/;n 1

33 1-Sample: Estimating the Variance

34 -Samples: Estimating the Ratio of Variances If s 1 and s are the variances of independent samples of sizes n 1 and n from normal populations, respectively, a α % confidence interval for σ 1 /σ : s 1 s 1 f α/ n 1 1, n 1 < σ 1 σ < s 1 s f α/ n 1 1, n 1

35 -Samples: Estimating the Ratio of Variances

36 End of Lecture Thank you! Questions?

8.2 Confidence Intervals for One Population Mean When σ is Known

8.2 Confidence Intervals for One Population Mean When σ is Known 8.2 Confidence Intervals for One Population Mean When σ is Known Tom Lewis Fall Term 2009 8.2 Confidence Intervals for One Population Mean When σ isfall Known Term 2009 1 / 6 Outline 1 An example 2 Finding

More information

The calculations lead to the following values: d 2 = 46, n = 8, s d 2 = 4, s d = 2, SEof d = s d n s d n

The calculations lead to the following values: d 2 = 46, n = 8, s d 2 = 4, s d = 2, SEof d = s d n s d n EXAMPLE 1: Paired t-test and t-interval DBP Readings by Two Devices The diastolic blood pressures (DBP) of 8 patients were determined using two techniques: the standard method used by medical personnel

More information

Wording of Final Conclusion. Slide 1

Wording of Final Conclusion. Slide 1 Wording of Final Conclusion Slide 1 8.3: Assumptions for Testing Slide 2 Claims About Population Means 1) The sample is a simple random sample. 2) The value of the population standard deviation σ is known

More information

The t Test. April 3-8, exp (2πσ 2 ) 2σ 2. µ ln L(µ, σ2 x) = 1 σ 2. i=1. 2σ (σ 2 ) 2. ˆσ 2 0 = 1 n. (x i µ 0 ) 2 = i=1

The t Test. April 3-8, exp (2πσ 2 ) 2σ 2. µ ln L(µ, σ2 x) = 1 σ 2. i=1. 2σ (σ 2 ) 2. ˆσ 2 0 = 1 n. (x i µ 0 ) 2 = i=1 The t Test April 3-8, 008 Again, we begin with independent normal observations X, X n with unknown mean µ and unknown variance σ The likelihood function Lµ, σ x) exp πσ ) n/ σ x i µ) Thus, ˆµ x ln Lµ,

More information

Lecture Topic 6: Chapter 9 Hypothesis Testing

Lecture Topic 6: Chapter 9 Hypothesis Testing Lecture Topic 6: Chapter 9 Hypothesis Testing 9.1 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should

More information

CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS

CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS LESSON SEVEN CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS An interval estimate for μ of the form a margin of error would provide the user with a measure of the uncertainty associated with the point estimate.

More information

Exam 3 - Math Solutions - Spring Name : ID :

Exam 3 - Math Solutions - Spring Name : ID : Exam 3 - Math 3200 Solutions - Spring 2013 Name : ID : General instructions: This exam has 14 questions, each worth the same amount. Check that no pages are missing and promptly notify one of the proctors

More information

Comparing Two Populations OPRE 6301

Comparing Two Populations OPRE 6301 Comparing Two Populations OPRE 6301 Introduction... In many applications, we are interested in hypotheses concerning differences between the means of two populations. For example, we may wish to decide

More information

Prob & Stats. Chapter 9 Review

Prob & Stats. Chapter 9 Review Chapter 9 Review Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally

More information

Section 7.2 Confidence Intervals for Population Proportions

Section 7.2 Confidence Intervals for Population Proportions Section 7.2 Confidence Intervals for Population Proportions 2012 Pearson Education, Inc. All rights reserved. 1 of 83 Section 7.2 Objectives Find a point estimate for the population proportion Construct

More information

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6. Math 144 tutorial April, 2010

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6. Math 144 tutorial April, 2010 Math 144 tutorial 7 12 April, 2010 1. Let us define S 2 = 1 n n i=1 (X i X) 2. Show that E(S 2 ) = n 1 n σ2. 1. Let us define S 2 = 1 n n i=1 (X i X) 2. Show that E(S 2 ) = n 1 n σ2. 1. Let us define S

More information

1 Hypotheses test about µ if σ is not known

1 Hypotheses test about µ if σ is not known 1 Hypotheses test about µ if σ is not known In this section we will introduce how to make decisions about a population mean, µ, when the standard deviation is not known. In order to develop a confidence

More information

1. Tips for Translation

1. Tips for Translation BIOSTATS 540 Fall 2015 7. Hypothesis Testing Supplement - Tips Page 1 of 8 Introduction This is a supplement to help you with three of the most common challenges in understanding and performing statistical

More information

4) The role of the sample mean in a confidence interval estimate for the population mean is to: 4)

4) The role of the sample mean in a confidence interval estimate for the population mean is to: 4) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Assume that the change in daily closing prices for stocks on the New York Stock Exchange is a random

More information

Math Chapter Seven Sample Exam

Math Chapter Seven Sample Exam Math 333 - Chapter Seven Sample Exam 1. The cereal boxes coming off an assembly line are each supposed to contain 12 ounces. It is reasonable to assume that the amount of cereal per box has an approximate

More information

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Practice for Chapter 9 and 10 The acutal exam differs. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the number of successes x suggested by the

More information

Statistics revision. Dr. Inna Namestnikova. Statistics revision p. 1/8

Statistics revision. Dr. Inna Namestnikova. Statistics revision p. 1/8 Statistics revision Dr. Inna Namestnikova inna.namestnikova@brunel.ac.uk Statistics revision p. 1/8 Introduction Statistics is the science of collecting, analyzing and drawing conclusions from data. Statistics

More information

6.1 The Elements of a Test of Hypothesis

6.1 The Elements of a Test of Hypothesis University of California, Davis Department of Statistics Summer Session II Statistics 13 August 22, 2012 Date of latest update: August 20 Lecture 6: Tests of Hypothesis Suppose you wanted to determine

More information

Statistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

Statistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage 7 Statistical Intervals Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to

More information

Math 62 Statistics Sample Exam Questions

Math 62 Statistics Sample Exam Questions Math 62 Statistics Sample Exam Questions 1. (10) Explain the difference between the distribution of a population and the sampling distribution of a statistic, such as the mean, of a sample randomly selected

More information

Confidence level. Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%)

Confidence level. Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%) Confidence Interval A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated

More information

Stat 210 Exam Four. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Stat 210 Exam Four. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Stat 210 Exam Four Read these directions carefully. As usual, you may pick any part of one problem to omit by writing OMIT in the answer blank (any one multiple choice or any one part of the other problems).

More information

Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar

Sampling (cont d) and Confidence Intervals Lecture 9 8 March 2006 R. Ryznar Sampling (cont d) and Confidence Intervals 11.220 Lecture 9 8 March 2006 R. Ryznar Census Surveys Decennial Census Every (over 11 million) household gets the short form and 17% or 1/6 get the long form

More information

Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

More information

EC 112 SAMPLING, ESTIMATION AND HYPOTHESIS TESTING. One and One Half Hours (1 1 2 Hours) TWO questions should be answered

EC 112 SAMPLING, ESTIMATION AND HYPOTHESIS TESTING. One and One Half Hours (1 1 2 Hours) TWO questions should be answered ECONOMICS EC 112 SAMPLING, ESTIMATION AND HYPOTHESIS TESTING One and One Half Hours (1 1 2 Hours) TWO questions should be answered Statistical tables are provided. The approved calculator (that is, Casio

More information

Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

More information

Estimation; Sampling; The T distribution

Estimation; Sampling; The T distribution Estimation; Sampling; The T distribution I. Estimation A. In most statistical studies, the population parameters are unknown and must be estimated. Therefore, developing methods for estimating as accurately

More information

Hypothesis Testing. Concept of Hypothesis Testing

Hypothesis Testing. Concept of Hypothesis Testing Quantitative Methods 2013 Hypothesis Testing with One Sample 1 Concept of Hypothesis Testing Testing Hypotheses is another way to deal with the problem of making a statement about an unknown population

More information

For eg:- The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random

For eg:- The yield of a new paddy variety will be 3500 kg per hectare scientific hypothesis. In Statistical language if may be stated as the random Lecture.9 Test of significance Basic concepts null hypothesis alternative hypothesis level of significance Standard error and its importance steps in testing Test of Significance Objective To familiarize

More information

6. Duality between confidence intervals and statistical tests

6. Duality between confidence intervals and statistical tests 6. Duality between confidence intervals and statistical tests Suppose we carry out the following test at a significance level of 100α%. H 0 :µ = µ 0 H A :µ µ 0 Then we reject H 0 if and only if µ 0 does

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com 1. A machine fills jars with jam. The weight of jam in each jar is normally distributed. To check the machine is working properly the contents of a random sample of 15 jars are weighed in grams. Unbiased

More information

Need for Sampling. Very large populations Destructive testing Continuous production process

Need for Sampling. Very large populations Destructive testing Continuous production process Chapter 4 Sampling and Estimation Need for Sampling Very large populations Destructive testing Continuous production process The objective of sampling is to draw a valid inference about a population. 4-

More information

Statistics Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Statistics Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Statistics Final Exam Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Assume that X has a normal distribution, and find the indicated

More information

Chapter 9: Hypothesis Tests of a Single Population

Chapter 9: Hypothesis Tests of a Single Population Chapter 9: Hypothesis Tests of a Single Population Department of Mathematics Izmir University of Economics Week 12 2014-2015 Introduction In this chapter we will focus on Example developing hypothesis

More information

Confidence Intervals and Hypothesis Testing

Confidence Intervals and Hypothesis Testing Name: Class: Date: Confidence Intervals and Hypothesis Testing Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The librarian at the Library of Congress

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question Stats: Test Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question Provide an appropriate response. ) Given H0: p 0% and Ha: p < 0%, determine

More information

A) to D) to B) to E) to C) to Rate of hay fever per 1000 population for people under 25

A) to D) to B) to E) to C) to Rate of hay fever per 1000 population for people under 25 Unit 0 Review #3 Name: Date:. Suppose a random sample of 380 married couples found that 54 had two or more personality preferences in common. In another random sample of 573 married couples, it was found

More information

Provide an appropriate response. Solve the problem. Determine the null and alternative hypotheses for the proposed hypothesis test.

Provide an appropriate response. Solve the problem. Determine the null and alternative hypotheses for the proposed hypothesis test. Provide an appropriate response. 1) Suppose that x is a normally distributed variable on each of two populations. Independent samples of sizes n1 and n2, respectively, are selected from the two populations.

More information

Do the following using Mintab (1) Make a normal probability plot for each of the two curing times.

Do the following using Mintab (1) Make a normal probability plot for each of the two curing times. SMAM 314 Computer Assignment 4 1. An experiment was performed to determine the effect of curing time on the comprehensive strength of concrete blocks. Two independent random samples of 14 blocks were prepared

More information

Inference, Sampling, and Confidence. Inference. Inference

Inference, Sampling, and Confidence. Inference. Inference Inference, Sampling, and Confidence Inference defined Sampling Statistics and parameters Sampling distribution Confidence and standard error Estimation Precision and accuracy Estimating sample size 1 Inference

More information

MCQ TESTING OF HYPOTHESIS

MCQ TESTING OF HYPOTHESIS MCQ TESTING OF HYPOTHESIS MCQ 13.1 A statement about a population developed for the purpose of testing is called: (a) Hypothesis (b) Hypothesis testing (c) Level of significance (d) Test-statistic MCQ

More information

Chapter 11: Two Variable Regression Analysis

Chapter 11: Two Variable Regression Analysis Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

More information

CONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN

CONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN CONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN A. Introduction 1. this situation, where we do not know anything about the population but the sample characteristics, is far and away the most common circumstance

More information

Hypothesis testing. Null hypothesis H 0 and alternative hypothesis H 1.

Hypothesis testing. Null hypothesis H 0 and alternative hypothesis H 1. Hypothesis testing Null hypothesis H 0 and alternative hypothesis H 1. Hypothesis testing Null hypothesis H 0 and alternative hypothesis H 1. Simple and compound hypotheses. Simple : the probabilistic

More information

Confidence Intervals about a Population Mean

Confidence Intervals about a Population Mean Confidence Intervals about a Population Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Motivation Goal: to estimate a population mean µ based on data collected

More information

HYPOTHESIS TESTING (TWO SAMPLE) - CHAPTER 8 1. how can a sample be used to estimate the unknown parameters of a population

HYPOTHESIS TESTING (TWO SAMPLE) - CHAPTER 8 1. how can a sample be used to estimate the unknown parameters of a population HYPOTHESIS TESTING (TWO SAMPLE) - CHAPTER 8 1 PREVIOUSLY estimation how can a sample be used to estimate the unknown parameters of a population use confidence intervals around point estimates of central

More information

Basic Statistics. Probability and Confidence Intervals

Basic Statistics. Probability and Confidence Intervals Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the

More information

Seminar paper Statistics

Seminar paper Statistics Seminar paper Statistics The seminar paper must contain: - the title page - the characterization of the data (origin, reason why you have chosen this analysis,...) - the list of the data (in the table)

More information

Inference in Regression Analysis. Dr. Frank Wood

Inference in Regression Analysis. Dr. Frank Wood Inference in Regression Analysis Dr. Frank Wood Inference in the Normal Error Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters

More information

PROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful.

PROBLEM SET 1. For the first three answer true or false and explain your answer. A picture is often helpful. PROBLEM SET 1 For the first three answer true or false and explain your answer. A picture is often helpful. 1. Suppose the significance level of a hypothesis test is α=0.05. If the p-value of the test

More information

Hence, multiplying by 12, the 95% interval for the hourly rate is (965, 1435)

Hence, multiplying by 12, the 95% interval for the hourly rate is (965, 1435) Confidence Intervals for Poisson data For an observation from a Poisson distribution, we have σ 2 = λ. If we observe r events, then our estimate ˆλ = r : N(λ, λ) If r is bigger than 20, we can use this

More information

OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance

OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance Lecture 5: Hypothesis Testing What we know now: OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique - ie the estimator has the smallest variance (if the Gauss-Markov

More information

Lecture 13: Some Important Continuous Probability Distributions (Part 2)

Lecture 13: Some Important Continuous Probability Distributions (Part 2) Lecture 13: Some Important Continuous Probability Distributions (Part 2) Kateřina Staňková Statistics (MAT1003) May 14, 2012 Outline 1 Erlang distribution Formulation Application of Erlang distribution

More information

Chapter 7. Section Introduction to Hypothesis Testing

Chapter 7. Section Introduction to Hypothesis Testing Section 7.1 - Introduction to Hypothesis Testing Chapter 7 Objectives: State a null hypothesis and an alternative hypothesis Identify type I and type II errors and interpret the level of significance Determine

More information

5.1 Identifying the Target Parameter

5.1 Identifying the Target Parameter University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

More information

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests Spring 2014 Jeremy Orloff and Jonathan Bloom

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests Spring 2014 Jeremy Orloff and Jonathan Bloom Null Hypothesis Significance Testing Signifcance Level, Power, t-tests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is

More information

Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

More information

Stats Review chapters 7-8

Stats Review chapters 7-8 Stats Review chapters 7-8 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

More information

Lecture 13: Kolmogorov Smirnov Test & Power of Tests

Lecture 13: Kolmogorov Smirnov Test & Power of Tests Lecture 13: Kolmogorov Smirnov Test & Power of Tests S. Massa, Department of Statistics, University of Oxford 2 February 2016 An example Suppose you are given the following 100 observations. -0.16-0.68-0.32-0.85

More information

STA218 Introduction to Hypothesis Testing

STA218 Introduction to Hypothesis Testing STA218 Introduction to Hypothesis Testing Al Nosedal. University of Toronto. Fall 2015 October 29, 2015 Who wants to be a millionaire? Let s say that one of you is invited to this popular show. As you

More information

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for

More information

F and t-test Statistical Method for HMA Voids Acceptance

F and t-test Statistical Method for HMA Voids Acceptance Page 1 Colorado Procedure 14-03 Standard Practice for F and t-test Statistical Method for HMA Voids Acceptance 1. SCOPE 1.1 Use this procedure as required by the project specifications to provide a method

More information

This HW reviews the normal distribution, confidence intervals and the central limit theorem.

This HW reviews the normal distribution, confidence intervals and the central limit theorem. Homework 3 Solution This HW reviews the normal distribution, confidence intervals and the central limit theorem. (1) Suppose that X is a normally distributed random variable where X N(75, 3 2 ) (mean 75

More information

Math 58. Rumbos Fall Solutions to Assignment #5

Math 58. Rumbos Fall Solutions to Assignment #5 Math 58. Rumbos Fall 2008 1 Solutions to Assignment #5 1. (Typographical Errors 1 ) Typographical and spelling errors can be either nonword errors or word errors. A nonword error is not a real word, as

More information

N Mean Std. Deviation Std. Error of Mean

N Mean Std. Deviation Std. Error of Mean DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 815 SAMPLE QUESTIONS FOR THE FINAL EXAMINATION I. READING: A. Read Agresti and Finalay, Chapters 6, 7, and 8 carefully. 1. Ignore the

More information

Determining the Sample Size: One Sample

Determining the Sample Size: One Sample Sample Sie Laboratory Determining the Sample Sie: One Sample OVERVIEW In this lab, you will be determining the sample sie for population means and population proportions. This lab is intended to help you

More information

Applications of the Central Limit Theorem

Applications of the Central Limit Theorem Applications of the Central Limit Theorem October 23, 2008 Take home message. I expect you to know all the material in this note. We will get to the Maximum Liklihood Estimate material very soon! 1 Introduction

More information

The written Master s Examination

The written Master s Examination The written Master s Examination Option Statistics and Probability Fall Full points may be obtained for correct answers to 8 questions. Each numbered question (which may have several parts) is worth the

More information

Estimating and Finding Confidence Intervals

Estimating and Finding Confidence Intervals . Activity 7 Estimating and Finding Confidence Intervals Topic 33 (40) Estimating A Normal Population Mean μ (σ Known) A random sample of size 10 from a population of heights that has a normal distribution

More information

An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters)

An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters) An example ANOVA situation Example (Treating Blisters) 1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Subjects: 25 patients with blisters Treatments: Treatment A, Treatment

More information

Unit 21 Student s t Distribution in Hypotheses Testing

Unit 21 Student s t Distribution in Hypotheses Testing Unit 21 Student s t Distribution in Hypotheses Testing Objectives: To understand the difference between the standard normal distribution and the Student's t distributions To understand the difference between

More information

Simple Linear Regression

Simple Linear Regression Inference for Regression Simple Linear Regression IPS Chapter 10.1 2009 W.H. Freeman and Company Objectives (IPS Chapter 10.1) Simple linear regression Statistical model for linear regression Estimating

More information

Hypothesis Testing with One Sample. Introduction to Hypothesis Testing 7.1. Hypothesis Tests. Chapter 7

Hypothesis Testing with One Sample. Introduction to Hypothesis Testing 7.1. Hypothesis Tests. Chapter 7 Chapter 7 Hypothesis Testing with One Sample 71 Introduction to Hypothesis Testing Hypothesis Tests A hypothesis test is a process that uses sample statistics to test a claim about the value of a population

More information

Practice Exam. 1. What is the median of this data? A) 64 B) 63.5 C) 67.5 D) 59 E) 35

Practice Exam. 1. What is the median of this data? A) 64 B) 63.5 C) 67.5 D) 59 E) 35 Practice Exam Use the following to answer questions 1-2: A census is done in a given region. Following are the populations of the towns in that particular region (in thousands): 35, 46, 52, 63, 64, 71,

More information

SCHOOL OF MATHEMATICS AND STATISTICS

SCHOOL OF MATHEMATICS AND STATISTICS RESTRICTED OPEN BOOK EXAMINATION (Not to be removed from the examination hall) Data provided: Statistics Tables by H.R. Neave MAS5052 SCHOOL OF MATHEMATICS AND STATISTICS Basic Statistics Spring Semester

More information

Elements of Hypothesis Testing (Summary from lecture notes)

Elements of Hypothesis Testing (Summary from lecture notes) Statistics-20090 MINITAB - Lab 1 Large Sample Tests of Hypothesis About a Population Mean We use hypothesis tests to make an inference about some population parameter of interest, for example the mean

More information

Statistics 104: Section 7

Statistics 104: Section 7 Statistics 104: Section 7 Section Overview Reminders Comments on Midterm Common Mistakes on Problem Set 6 Statistical Week in Review Comments on Midterm Overall, the midterms were good with one notable

More information

Adverse Impact Ratio for Females (0/ 1) = 0 (5/ 17) = 0.2941 Adverse impact as defined by the 4/5ths rule was not found in the above data.

Adverse Impact Ratio for Females (0/ 1) = 0 (5/ 17) = 0.2941 Adverse impact as defined by the 4/5ths rule was not found in the above data. 1 of 9 12/8/2014 12:57 PM (an On-Line Internet based application) Instructions: Please fill out the information into the form below. Once you have entered your data below, you may select the types of analysis

More information

Sampling Distribution of a Normal Variable

Sampling Distribution of a Normal Variable Ismor Fischer, 5/9/01 5.-1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,

More information

Statistical Inference

Statistical Inference Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

More information

Statistical foundations of machine learning

Statistical foundations of machine learning Machine learning p. 1/45 Statistical foundations of machine learning INFO-F-422 Gianluca Bontempi Département d Informatique Boulevard de Triomphe - CP 212 http://www.ulb.ac.be/di Machine learning p. 2/45

More information

Connexions module: m

Connexions module: m Connexions module: m17025 1 Hypothesis Testing: Two Population Means and Two Population Proportions: Comparing Two Independent Population Means with Unknown Population Standard Deviations Susan Dean Barbara

More information

CONFIDENCE INTERVALS I

CONFIDENCE INTERVALS I CONFIDENCE INTERVALS I ESTIMATION: the sample mean Gx is an estimate of the population mean µ point of sampling is to obtain estimates of population values Example: for 55 students in Section 105, 45 of

More information

Hypothesis Testing. Bluman Chapter 8

Hypothesis Testing. Bluman Chapter 8 CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

More information

Third Midterm Exam (MATH1070 Spring 2012)

Third Midterm Exam (MATH1070 Spring 2012) Third Midterm Exam (MATH1070 Spring 2012) Instructions: This is a one hour exam. You can use a notesheet. Calculators are allowed, but other electronics are prohibited. 1. [40pts] Multiple Choice Problems

More information

Lecture 1: t tests and CLT

Lecture 1: t tests and CLT Lecture 1: t tests and CLT http://www.stats.ox.ac.uk/ winkel/phs.html Dr Matthias Winkel 1 Outline I. z test for unknown population mean - review II. Limitations of the z test III. t test for unknown population

More information

Pooling and Meta-analysis. Tony O Hagan

Pooling and Meta-analysis. Tony O Hagan Pooling and Meta-analysis Tony O Hagan Pooling Synthesising prior information from several experts 2 Multiple experts The case of multiple experts is important When elicitation is used to provide expert

More information

Chapter 7 Sampling and Sampling Distributions. Learning objectives

Chapter 7 Sampling and Sampling Distributions. Learning objectives Chapter 7 Sampling and Sampling Distributions Slide 1 Learning objectives 1. Understand Simple Random Sampling 2. Understand Point Estimation and be able to compute point estimates 3. Understand Sampling

More information

t-statistics for Weighted Means with Application to Risk Factor Models

t-statistics for Weighted Means with Application to Risk Factor Models t-statistics for Weighted Means with Application to Risk Factor Models Lisa R. Goldberg Barra, Inc. 00 Milvia St. Berkeley, CA 94704 Alec N. Kercheval Dept. of Mathematics Florida State University Tallahassee,

More information

Sampling & Confidence Intervals

Sampling & Confidence Intervals Sampling & Confidence Intervals Mark Lunt Arthritis Research UK Centre for Excellence in Epidemiology University of Manchester 18/10/2016 Principles of Sampling Often, it is not practical to measure every

More information

MAT X Hypothesis Testing - Part I

MAT X Hypothesis Testing - Part I MAT 2379 3X Hypothesis Testing - Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating

More information

Demonstrating Systematic Sampling

Demonstrating Systematic Sampling Demonstrating Systematic Sampling Julie W. Pepe, University of Central Florida, Orlando, Florida Abstract A real data set involving the number of reference requests at a university library, will be used

More information

Hypothesis testing: Examples. AMS7, Spring 2012

Hypothesis testing: Examples. AMS7, Spring 2012 Hypothesis testing: Examples AMS7, Spring 2012 Example 1: Testing a Claim about a Proportion Sect. 7.3, # 2: Survey of Drinking: In a Gallup survey, 1087 randomly selected adults were asked whether they

More information

Review Solutions, Exam 3, Math 338

Review Solutions, Exam 3, Math 338 Review Solutions, Eam 3, Math 338. Define a random sample: A random sample is a collection of random variables, typically indeed as X,..., X n.. Why is the t distribution used in association with sampling?

More information

An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0.

An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0. Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.

More information

Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

More information

CHAPTER 9 One- and Two-Sample Estimation Problems

CHAPTER 9 One- and Two-Sample Estimation Problems CHAPTER 9 One- and Two-Sample Estimation Problems 4/252) An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40

More information

11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE

11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE 11. CONFIDENCE INTERVALS FOR THE MEAN; KNOWN VARIANCE We assume here that the population variance σ 2 is known. This is an unrealistic assumption, but it allows us to give a simplified presentation which

More information