Hypothesis testing. 1 Principle of hypothesis testing 2
|
|
- Kelley Fleming
- 5 years ago
- Views:
Transcription
1 Hypothesis testing Contents 1 Principle of hypothesis testing One sample tests 3.1 Tests on Mean of a Normal distribution Tests on Variance of a Normal distribution Large Sample Test on Mean Test on a Population Proportion Two Samples Tests Testing Equality of Variances Testing Equality of Means (σ 1 = σ ) Testing Equality of Means (σ 1 σ ) Large Samples Tests on Means Testing Equality of Means Paired Samples Hypothesis Testing of Distribution Chi-Square Goodness of Fit Test Tests of Skewness and Kurtosis Excercises 11
2 1 Principle of hypothesis testing A statistical hypothesis is a claim (or statement) about a population parameters (µ σ π λ... ) a distribution (normal Poisson... ). A null hypothesis H is a claim (or statement) about a population parameter that is assumed to be true until it is declared false... for example H : µ = µ 0. An alternative hypothesis A is a claim about a population parameter that will be true if the null hypothesis is false A : µ µ 0 both-sided test A : µ > µ 0 one-sided test A : µ < µ 0 one-sided test. reality H is true H is false decision about H prob. prob. not reject correct decision 1 type II error β reject type I error correct decision 1 β If we reject the null hypothesis which is true we call this type I error. The probability of this error is a significance level. A number 1 is probability that we do not reject the true hypothesis H. If we accept the null hypothesis although it is false we call this type II error. The probability of this error is β. A number 1 β a power of the test is the probability that we reject the null hypothesis H if it is false. To test the null hypothesis we use a function of a random sample T = T (x 1 x... x n ) so called test statistic which has under the null hypothesis H known distribution (usually t u χ F ). We divide the all possible values of the test statistic into W 1 - non of H the set of values connected with the hypothesis H W - of H the set of values connected with the hypothesis A. Steps to Perform a Test of Hypothesis 1. State the null and alternative hypothesis H and A.. Select a significance level (usually 0.05 a 0.01). 3. Choose the test statistic. 4. Determine the W. 5. Calculate the value of the test statistic. 6. Make a decision: If the value of the test statistic falls in the we reject the null hypothesis H and say that we accept the alternative hypothesis A with the probability 1. If the value of the test statistic falls in the non we do not reject the null hypothesis H.
3 alternative hypothesis A : µ > µ 0 A : µ < µ 0 A : µ µ 0 W = {t t t 1 (ν)} W = {t t t 1 (ν)} W = { t t t 1 (ν)} Steps to Perform a Test of Hypothesis via p-value 1. State the null and alternative hypothesis H and A.. Select a significance level (usually 0.05 a 0.01). 3. Calculate the p-value statistical software e.g. STAT1 R etc. 4. Make a decision: If > p-value we reject the null hypothesis H with the probability 1 i.e. significance level. If < p-value we do not reject the null hypothesis H. at One sample tests.1 Tests on Mean of a Normal distribution Let X 1 X... X n be a random sample from a normal distribution N(µ σ ). A statistic T = X µ n S has a Student distribution with ν = n 1 degrees of freedom. We use this statistic for testing of the parameter µ. Let x 1 x... x n be values of a random sample (measured data) x denotes an arithmetic mean and s a sample standard deviation. We test the hypothesis that the parameter µ is equal to a constant µ 0 : H : µ = µ 0 the test statistic t = x µ 0 n s has under the null hypothesis H a Student t-distribution with ν = n 1 degrees of freedom. According to the alternative hypothesis we construct following regions of rejection: where t 1 t 1 are quantiles of the Student distribution.. Tests on Variance of a Normal distribution Let X 1 X... X n be a random sample from a normal distribution N(µ σ ). A statistic χ = (n 1)S σ 3
4 alternative hypothesis A : σ > σ0 W = { χ χ χ 1 (ν) } A : σ < σ0 W = {χ χ χ { (ν)} } A : σ σ0 W = χ χ χ (ν) or χ χ 1 (ν) alternative hypothesis A : µ > µ 0 W = {u u u 1 } A : µ < µ 0 W = {u u u 1 } A : µ µ 0 W = { } u u u 1 has Pearson distribution with ν = n 1 degrees of freedom. We use this statistic for testing of the parameter σ. Let x 1 x... x n be values of a random sample (measured data) s denotes sample variance. We test the hypothesis that the parameter σ is equal to a constant σ 0: the test statistic H : σ = σ 0 χ = (n 1)s σ 0 has under the null hypothesis H a Pearson χ -distribution with ν = n 1 degrees of freedom. According to an alternative hypothesis we construct following regions of rejection: where χ 1 χ 1 χ χ are quantiles of the Pearson χ -distribution..3 Large Sample Test on Mean Let X 1 X... X n be a random sample from any distribution with the mean µ. A statistic U = X µ n S has for large n approximately a normal distribution N(0 1) see the central limit theorems. We use this statistic for testing of the parameter µ. Let x 1 x... x n be values of a random sample (measured data) x denotes an arithmetic mean and s a sample standard deviation. We test the hypothesis that the parameter µ is equal to a constant µ 0 : H : µ = µ 0 the test statistic u = x µ 0 n s has under the null hypothesis H asymptotically a normal distribution N(0 1). According to an alternative hypothesis we construct following regions of rejection: where u 1 u 1 are quantiles of N(0 1). 4
5 alternative hypothesis.4 Test on a Population Proportion A : π > π 0 W = {u u u 1 } A : π < π 0 W = {u u u 1 } A : π π 0 W = { } u u u 1 Suppose that a random sample of size n has been taken from a large (possibly infinite) population and that m observations in this sample belong to a class of interest. Then p = m is a point n estimator of the proportion of the population π that belongs to this class. A random variable U = p π π(1 π)/n has for n approximately a normal distribution N(0 1) see central limit theorems. We use this statistic for testing of the population proportion. Let ˆπ = m be a point estimator of population proportion. We test the hypothesis that the n parameter π is equal to a constant π 0 : H : π = π 0 a test statistic u = ˆπ π 0 π0 (1 π 0 )/n has under the null hypothesis H asymptotically a normal distribution N(0 1). According to an alternative hypothesis we construct following regions of rejection: where u 1 u 1 are quantiles of N(0 1). 3 Two Samples Tests 3.1 Testing Equality of Variances Let X 1 X... X n1 be a random sample from N(µ 1 σ1) and Y 1 Y... Y n be a random sample from N(µ σ). SX and S Y are corresponding sample variances. The statistic F = S X S Y σ σ 1 has a Fisher-Snedecor distribution with ν 1 = n 1 and ν = n 1 degrees of freedom. Let x 1 x... x n1 be values of a random sample from N(µ 1 σ 1) y 1 y... y n be values of a random sample from N(µ σ ) s x and s y corresponding values of sample variances. We test a hypothesis that the parametr σ 1 is equal to the parameter σ : H : σ 1 = σ the test statistic is F = s x s y 5
6 alternative hypothesis A : σ 1 > σ W = {F F F 1 (ν 1 ν )} A : σ1 < σ W = {F F F (ν 1 ν )} A : σ1 σ W = { F F F (ν 1 ν ) F F 1 (ν 1 ν ) } which has under the null hypothesis H a Fisher-Snedecor distribution with ν 1 = n 1 1 and ν = n 1 degrees of freedom. According to the alternative hypothesis we construct following regions of rejection: where F F 1 F F 1 are quantiles of the Fisher-Snedecor distribution ν 1 = n 1 1 ν = n Testing Equality of Means (σ 1 = σ ) Let X 1 X... X n1 be a random sample from N(µ 1 σ 1) and Y 1 Y... Y n be a random sample from N(µ σ ). We assume that these random samples are independent. X Y S X a S Y are corresponding sample means and variances. If σ 1 = σ then a statistic where T = X Y (µ 1 µ ) n1 n S n 1 + n S = [ (n1 1)S X + (n 1)S Y n 1 + n (it is co called pooled estimator of the common σ) has a Student distribution with ν = n 1 +n degrees of freedom. Let x 1 x... x n1 be values of a random sample from N(µ 1 σ 1) y 1 y... y n be values of a random sample from N(µ σ ) x y s x a s y are corresponding values of sample means and variances. We test a hypothesis that the parameter µ 1 is equal to the parameter µ (σ 1 = σ ): a test statistic is where H : µ 1 = µ t = x y n1 n S n 1 + n [ (n1 1)s x + (n 1)s y S = n 1 + n has under the null hypothesis H the Student distribution with ν = n 1 + n degrees of freedom. According to the alternative hypothesis we construct following regions of rejection: where t 1 t 1 are quantiles of the Student distribution ν = n 1 + n. 3.3 Testing Equality of Means (σ 1 σ ) Let X 1 X... X n1 be a random sample from N(µ 1 σ 1) and Y 1 Y... Y n be a random sample from N(µ σ ). We assume that these random samples are independent. 6 ] 1/ ] 1/
7 alternative hypothesis A : µ 1 > µ A : µ 1 < µ A : µ 1 µ alternative hypothesis A : µ 1 > µ A : µ 1 < µ A : µ 1 µ W = {t t t 1 (ν)} W = {t t t 1 (ν)} W = { t t t 1 (ν)} W = {t t t 1 (ν)} W = {t t t 1 (ν)} W = { t t t 1 (ν)} X Y S X a S Y are corresponding sample means and variances. If σ 1 = σ then a statistic T = X Y (µ 1 µ ) SX n1 + S Y n has approximately a Student distribution with ν degrees of freedom. Let x 1 x... x n1 be values of a random sample from N(µ 1 σ 1) y 1 y... y n be values of a random sample from N(µ σ ) x y s x a s y are corresponding values of sample means and variances. We test a hypothesis that the parameter µ 1 is equal to the parameter µ (σ 1 σ ): a test statistic H : µ 1 = µ t = x y s x n 1 + s y n has under the null hypothesis H approximately a Student distribution with ν degrees of freedom. Degrees of freedom are given by a formula ν ( ) s x n 1 + s y n ( ) 1 s ( ) x n 1 1 n s y n 1 n rounded down to the nearest integer number. According to the alternative hypothesis we construct following regions of rejection: where t 1 t 1 are quantiles of the Student distribution with ν degrees of freedom (see the previous page). 3.4 Large Samples Tests on Means Let X 1 X... X n1 be a random sample from a distribution with the mean µ 1 and Y 1 Y... Y n be a random sample from a distribution with the mean µ. We assume that these random samples are independent and samples are large enough. 7
8 X Y S X a S Y alternative hypothesis A : µ 1 > µ W = {u u u 1 } A : µ 1 < µ W = {u u u 1 } A : µ 1 µ W = { } u u u 1 are corresponding sample means and variances. A statistic U = X Y (µ 1 µ ) SX n1 + S Y n has approximately a normal distribution N(0 1). Let x 1 x... x n1 are values of a random sample from the first distribution y 1 y... y n are values of a random sample from the second distribution x y s x and s y are corresponding values of sample means and variances. We test a hypothesis that the parameter µ 1 is equal to the parameter µ : H : µ 1 = µ a test statistic u = x y s x n 1 + s y n has under the null hypothesis H approximately a normal distribution N(0 1). According to the alternative hypothesis we construct following regions of rejection: where are quantiles of N(0 1). u 1 u Testing Equality of Means Paired Samples Let us have two dependent samples two data values one for each sample are collected from a source (or an element). These are also called paired or matched samples. We assume two dependent random variables X and Y with means µ 1 and µ the difference D = X Y is a random variable too. Let D 1 D... D n be a random sample where differences D i = X i Y i have a normal distribution N(µ σ ) where µ = µ 1 µ (σ is not needed). A statistic T = D µ S D n where D is a sample mean of differences and S D is a sample standard deviation of differences has a Student distribution with ν = n 1 degrees of freedom. Let d 1 = x 1 y 1 d = x y... d n = x n y n be measure valued of differences d is its sample mean and s d is its sample standard deviation. We test a hypothesis that the parameter µ 1 is equal to the parameter µ : a test statistic H : µ 1 = µ t = d s d n has under the null hypothesis H a Student distribution with ν = n 1 degrees of freedom. According to the alternative hypothesis we construct following regions of rejection: where t 1 t 1 are quantiles of the Student distribution ν = n 1. 8
9 alternative hypothesis A : µ 1 > µ A : µ 1 < µ A : µ 1 µ W = {t t t 1 (ν)} W = {t t t 1 (ν)} W = { t t t 1 (ν)} 4 Hypothesis Testing of Distribution 4.1 Chi-Square Goodness of Fit Test We divide values of a random sample x 1 x... x n into k disjoint classes where n j j = 1... k is frequency of the class j and π j is a probability that the random variable X has value from the class j calculated under the condition that X has an assumed distribution. The main idea of the test is to compare relative frequencies n j /n with theoretical probabilities π j. We state the null and alternative hypothesis: H : the random variable X follows an assumed distribution A : the random variable X does not follow an assumed distribution. The test statistic is k χ (n j nπ j ) = nπ j j=1 which has under the null hypothesis H for large n (asymptotically) a Pearson χ -distribution with ν = k c 1 degrees of freedom where c is a number of estimated parameters of the assumed distribution. A is W = { χ χ χ 1 (ν) } where χ 1 (ν) is a quantile of the Pearson χ -distribution. Recommendation: nπ j > 5 j = 1... k. If this condition is not satisfied it is necessary to join the classes. 4. Tests of Skewness and Kurtosis The normal distribution has 3 = 0 a 4 = 0. We can use these properties to test normality. We calculate a sample skewness and kurtosis (they are estimates of 3 and 4 ) We state hypothesis: ˆ 3 = a 3 = 1 ns 3 n n i=1 H 1 : 3 = 0 A 1 : 3 0 Test statistic is u 3 = (x i x) 3 ˆ 3 = a 4 = 1 ns 4 n a 3 D(a3 ) where D(a 3) = n (x i x) 4 3. i=1 6(n ) (n + 1)(n + 3) 9
10 which has under the null hypothesis H 1 asymptotically normal distribution N(0 1). A is W = { } u 3 u 3 u 1 where u 1 is a quantile of N(0 1). H : 4 = 0 A : 4 0 Test statistic is u 4 = a n+1 D(a4 ) where D(a 4) = 4n(n )(n 3) (n + 1) (n + 3)(n + 5) which has under the null hypothesis H asymptotically normal distribution N(0 1). A is W = { } u 4 u 4 u 1 where u 1 is a quantile of N(0 1). Compound Tests of Skewness and Kurtosis We state hypothesis: H : a random variable X has a normal distribution A : a random variable X has not a normal distribution. Test statistic is C = u 3 + u 4 which has under the null hypothesis H approximately χ distribution with two degrees of freedom. u 3 and u 4 are test statistics defined above. A is W = { C C χ 1 () } where χ 1 () is a quantile of the Pearson χ -distribution. 10
11 5 Excercises 1. Repeat all three basic procedures 1 from section 1 for testing of statistical hypotheses (use the application STAT1). Use your own data sets.. The desired average moisture content of roasted coffee is 4.% and a standard deviation of 0.4%. In 0 samples were measured following actual values of moisture in percents: Assume that a random sample is from a normal distribution (check). Use the significance level of 0.05 to determine whether the underlying file from which the samples come exhibits the desired (a) average moisture (b) variability of moisture. 3. Representatives of the environmental movement actively oppose the construction of new factories in the area which environment is already quite marked by industrial activities. They assume that one of the consequences of an unhealthy environment could be also the low birth weight of newborns in the given area. Does it make sense to use a lower birth weight as an argument against the construction of a new factory knowing that the birth weight of the healthy population has a normal distribution with a mean of 3500 g? Their claim would be supported via the sample of 50 randomly selected newborns born in this area measured an average weight is 3310 g and a standard deviation of 500 g. Perform the test at a significance level of The operation was reduced in a factory which largely employs commuters from the area. The transport company is afraid of the fact that it would decreas the average number of passengers in one bus on certain lines. For this reason an investigation was carried out in 40 randomly selected buses and the relevant lines in rush hours with the following results: Number of passengers Number of cases For the past years it is known that the average number of passengers per one bus (under comparable conditions) was 36 persons. In the case that the survey will prove that the number of passengers decreased the transport company will have to reduce traffic. What is the decision ( = 0.05)? 5. According to an unnamed company that is engaged in the survey of consumer habits 33% of households prefer shopping in hypermarkets. The company anticipates further growth in their popularity. To verify that idea they randomly selected 50 people and found that hypermarkets prefer 93 of all respondents. Is the result of a survey in accordance with that assumption ( = 005)? 6. For two technicians we are supposed to verify the accuracy of their measurement of chemical concentration of given substance. The first technician performed 15 measurements with a standard deviation of 03 and the second technician performed 0 measurements with a standard deviation of With 95% and 99% confidence level make sure 1 1. via the confidence interval. via the 3. via the p-value 11
12 that the first technician performs measurements more accurately than the second technician. Measurement errors have a normal distribution. 7. Two years ago in June a chemical analysis of 85 samples of water from different parts of the town lake was conducted. The main reason was to investigate the amount of chlorine in the water. The use of salt for road maintenance in winters was significantly reduced over the last two years. This year also in June 110 water samples from the lake was a re-analysed. Following results were obtained: years ago This year Average Sample standard deviation Verify the claim that lower salt use in winter reduces the amount of chlorine in the lake ( = respectively). 8. As a part of the pre-election survey there was the key question whether voters in the upcoming elections give priority to the Left or Right. The survey was conducted in two regions. It is possible on the basis of the data insist that the Left has in the region A significantly more supporters than in region B? Perform the test at a significance level of The survey results are listed in the table below: Respondents Left Right Does not know Region A Region B The thickness of two tree types were compared (the thickness of the tree trunk at a height of 1.3 m above the heel strain). Test at the 0.05 level of significance that both tree types are well advanced in thickness. Assume that samples have a normal distribution. Stand A Stand B During the test of the reliability of the altimeter 15 control measurements were done when compared with other standard altimeter. Perform the paired test that both altimeters measure well. Assume that these samples are from normal distributions. Altimeter A Altimeter B Altimeter A Altimeter B In the group of 110 students pocket money was monitored. From the identified data were calculated sample skewness a 3 = and sample kurtosis a 4 = Using tests of zero skewness zero kurtosis and C-test verify at the significance level of 0.05 that the sample comes from a normal distribution. Do the same using the modified variants of the tests. 1
13 1. The shooter shoots at a target and it is recorded the number of hits in 3 shots. The shooter performed 50 triples shots with the following results: Number of hits Number of cases Estimate the probability ˆπ of hit in one shot. Is it possible to describe the distribution of hits in 3 shots by the binomial distribution? Verify the hypothesis using χ -goodness of fit test at a significance level of Use the fact that the sample mean is an unbiased estimate of the mean which in the case of the binomial distribution is equal to nπ where n is the number of shots and π is the probability of hitting in one shot measurements of the milk fat were performed (in g/100 g milk) with the following results: Verify the normality at the significance level of 0.05 use the Kolmogorov-Smirnov s test. 14. For the sample of 50 civil servants was find out the amount of the monthly salary. From the collected data were determined the sample skewness a 3 = 0.73 and the sample kurtosis a 4 = Use the C-test modified C-test and verify whether a random variable indicating the monthly salary of civil servants could be described by a normal distribution. 13
Hypothesis Testing One Sample Tests
STATISTICS Lecture no. 13 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 12. 1. 2010 Tests on Mean of a Normal distribution Tests on Variance of a Normal
More informationTHE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 004 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER II STATISTICAL METHODS The Society provides these solutions to assist candidates preparing for the examinations in future
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 4.0 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationChapter 9 Inferences from Two Samples
Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations Review
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Chapter 9 Hypothesis Testing: Single Population Ch. 9-1 9.1 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population
More information+ Specify 1 tail / 2 tail
Week 2: Null hypothesis Aeroplane seat designer wonders how wide to make the plane seats. He assumes population average hip size μ = 43.2cm Sample size n = 50 Question : Is the assumption μ = 43.2cm reasonable?
More informationSTAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots. March 8, 2015
STAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots March 8, 2015 The duality between CI and hypothesis testing The duality between CI and hypothesis
More informationSalt Lake Community College MATH 1040 Final Exam Fall Semester 2011 Form E
Salt Lake Community College MATH 1040 Final Exam Fall Semester 011 Form E Name Instructor Time Limit: 10 minutes Any hand-held calculator may be used. Computers, cell phones, or other communication devices
More informationSummary of Chapters 7-9
Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two
More informationClassroom Activity 7 Math 113 Name : 10 pts Intro to Applied Stats
Classroom Activity 7 Math 113 Name : 10 pts Intro to Applied Stats Materials Needed: Bags of popcorn, watch with second hand or microwave with digital timer. Instructions: Follow the instructions on the
More informationCIVL /8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8
CIVL - 7904/8904 T R A F F I C F L O W T H E O R Y L E C T U R E - 8 Chi-square Test How to determine the interval from a continuous distribution I = Range 1 + 3.322(logN) I-> Range of the class interval
More informationAs an example, consider the Bond Strength data in Table 2.1, atop page 26 of y1 y 1j/ n , S 1 (y1j y 1) 0.
INSY 7300 6 F01 Reference: Chapter of Montgomery s 8 th Edition Point Estimation As an example, consider the Bond Strength data in Table.1, atop page 6 of By S. Maghsoodloo Montgomery s 8 th edition, on
More informationHypothesis Tests and Estimation for Population Variances. Copyright 2014 Pearson Education, Inc.
Hypothesis Tests and Estimation for Population Variances 11-1 Learning Outcomes Outcome 1. Formulate and carry out hypothesis tests for a single population variance. Outcome 2. Develop and interpret confidence
More informationThe Multinomial Model
The Multinomial Model STA 312: Fall 2012 Contents 1 Multinomial Coefficients 1 2 Multinomial Distribution 2 3 Estimation 4 4 Hypothesis tests 8 5 Power 17 1 Multinomial Coefficients Multinomial coefficient
More informationT.I.H.E. IT 233 Statistics and Probability: Sem. 1: 2013 ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS
ESTIMATION AND HYPOTHESIS TESTING OF TWO POPULATIONS In our work on hypothesis testing, we used the value of a sample statistic to challenge an accepted value of a population parameter. We focused only
More informationAMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
More informationThe Chi-Square Distributions
MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation σ of a normally distributed measurement and to test the goodness
More informationCOVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS
COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS COURSE: CBS 221 DISCLAIMER The contents of this document are intended for practice and leaning purposes at the undergraduate
More informationThe University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80
The University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80 71. Decide in each case whether the hypothesis is simple
More informationAsymptotic Statistics-VI. Changliang Zou
Asymptotic Statistics-VI Changliang Zou Kolmogorov-Smirnov distance Example (Kolmogorov-Smirnov confidence intervals) We know given α (0, 1), there is a well-defined d = d α,n such that, for any continuous
More informationReview: General Approach to Hypothesis Testing. 1. Define the research question and formulate the appropriate null and alternative hypotheses.
1 Review: Let X 1, X,..., X n denote n independent random variables sampled from some distribution might not be normal!) with mean µ) and standard deviation σ). Then X µ σ n In other words, X is approximately
More informationSTATISTICS SYLLABUS UNIT I
STATISTICS SYLLABUS UNIT I (Probability Theory) Definition Classical and axiomatic approaches.laws of total and compound probability, conditional probability, Bayes Theorem. Random variable and its distribution
More informationKumaun University Nainital
Kumaun University Nainital Department of Statistics B. Sc. Semester system course structure: 1. The course work shall be divided into six semesters with three papers in each semester. 2. Each paper in
More information7.2 One-Sample Correlation ( = a) Introduction. Correlation analysis measures the strength and direction of association between
7.2 One-Sample Correlation ( = a) Introduction Correlation analysis measures the strength and direction of association between variables. In this chapter we will test whether the population correlation
More informationGlossary. The ISI glossary of statistical terms provides definitions in a number of different languages:
Glossary The ISI glossary of statistical terms provides definitions in a number of different languages: http://isi.cbs.nl/glossary/index.htm Adjusted r 2 Adjusted R squared measures the proportion of the
More informationMathematical Notation Math Introduction to Applied Statistics
Mathematical Notation Math 113 - Introduction to Applied Statistics Name : Use Word or WordPerfect to recreate the following documents. Each article is worth 10 points and should be emailed to the instructor
More informationWe know from STAT.1030 that the relevant test statistic for equality of proportions is:
2. Chi 2 -tests for equality of proportions Introduction: Two Samples Consider comparing the sample proportions p 1 and p 2 in independent random samples of size n 1 and n 2 out of two populations which
More informationMath 10 - Compilation of Sample Exam Questions + Answers
Math 10 - Compilation of Sample Exam Questions + Sample Exam Question 1 We have a population of size N. Let p be the independent probability of a person in the population developing a disease. Answer the
More informationExam 2 (KEY) July 20, 2009
STAT 2300 Business Statistics/Summer 2009, Section 002 Exam 2 (KEY) July 20, 2009 Name: USU A#: Score: /225 Directions: This exam consists of six (6) questions, assessing material learned within Modules
More information375 PU M Sc Statistics
375 PU M Sc Statistics 1 of 100 193 PU_2016_375_E For the following 2x2 contingency table for two attributes the value of chi-square is:- 20/36 10/38 100/21 10/18 2 of 100 120 PU_2016_375_E If the values
More informationEXAM 3 Math 1342 Elementary Statistics 6-7
EXAM 3 Math 1342 Elementary Statistics 6-7 Name Date ********************************************************************************************************************************************** MULTIPLE
More informationPOLI 443 Applied Political Research
POLI 443 Applied Political Research Session 6: Tests of Hypotheses Contingency Analysis Lecturer: Prof. A. Essuman-Johnson, Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College
More informationPurposes of Data Analysis. Variables and Samples. Parameters and Statistics. Part 1: Probability Distributions
Part 1: Probability Distributions Purposes of Data Analysis True Distributions or Relationships in the Earths System Probability Distribution Normal Distribution Student-t Distribution Chi Square Distribution
More informationStatistical Inference: Estimation and Confidence Intervals Hypothesis Testing
Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing 1 In most statistics problems, we assume that the data have been generated from some unknown probability distribution. We desire
More informationAlgebra 2B Review for the Final Exam, 2015
Name:: Period: Grp #: Date: Algebra 2B Review for the Final Exam, 2015 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Tell whether the function y = 2(
More informationM(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1
Math 66/566 - Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)
More informationSummary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)
Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ
More informationSTATS 200: Introduction to Statistical Inference. Lecture 29: Course review
STATS 200: Introduction to Statistical Inference Lecture 29: Course review Course review We started in Lecture 1 with a fundamental assumption: Data is a realization of a random process. The goal throughout
More informationPractical Statistics
Practical Statistics Lecture 1 (Nov. 9): - Correlation - Hypothesis Testing Lecture 2 (Nov. 16): - Error Estimation - Bayesian Analysis - Rejecting Outliers Lecture 3 (Nov. 18) - Monte Carlo Modeling -
More informationy ˆ i = ˆ " T u i ( i th fitted value or i th fit)
1 2 INFERENCE FOR MULTIPLE LINEAR REGRESSION Recall Terminology: p predictors x 1, x 2,, x p Some might be indicator variables for categorical variables) k-1 non-constant terms u 1, u 2,, u k-1 Each u
More informationINTERVAL ESTIMATION AND HYPOTHESES TESTING
INTERVAL ESTIMATION AND HYPOTHESES TESTING 1. IDEA An interval rather than a point estimate is often of interest. Confidence intervals are thus important in empirical work. To construct interval estimates,
More informationEC2001 Econometrics 1 Dr. Jose Olmo Room D309
EC2001 Econometrics 1 Dr. Jose Olmo Room D309 J.Olmo@City.ac.uk 1 Revision of Statistical Inference 1.1 Sample, observations, population A sample is a number of observations drawn from a population. Population:
More informationMAT2377. Ali Karimnezhad. Version December 13, Ali Karimnezhad
MAT2377 Ali Karimnezhad Version December 13, 2016 Ali Karimnezhad Comments These slides cover material from Chapter 4. In class, I may use a blackboard. I recommend reading these slides before you come
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 6 Sampling and Sampling Distributions Ch. 6-1 6.1 Tools of Business Statistics n Descriptive statistics n Collecting, presenting, and describing data n Inferential
More information4.1. Introduction: Comparing Means
4. Analysis of Variance (ANOVA) 4.1. Introduction: Comparing Means Consider the problem of testing H 0 : µ 1 = µ 2 against H 1 : µ 1 µ 2 in two independent samples of two different populations of possibly
More informationThe Components of a Statistical Hypothesis Testing Problem
Statistical Inference: Recall from chapter 5 that statistical inference is the use of a subset of a population (the sample) to draw conclusions about the entire population. In chapter 5 we studied one
More informationSociology 6Z03 Review II
Sociology 6Z03 Review II John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review II Fall 2016 1 / 35 Outline: Review II Probability Part I Sampling Distributions Probability
More informationIntroduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test
Introduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test la Contents The two sample t-test generalizes into Analysis of Variance. In analysis of variance ANOVA the population consists
More informationSTAT 135 Lab 5 Bootstrapping and Hypothesis Testing
STAT 135 Lab 5 Bootstrapping and Hypothesis Testing Rebecca Barter March 2, 2015 The Bootstrap Bootstrap Suppose that we are interested in estimating a parameter θ from some population with members x 1,...,
More informationChapter 5: HYPOTHESIS TESTING
MATH411: Applied Statistics Dr. YU, Chi Wai Chapter 5: HYPOTHESIS TESTING 1 WHAT IS HYPOTHESIS TESTING? As its name indicates, it is about a test of hypothesis. To be more precise, we would first translate
More informationLecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2
Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Fall, 2013 Page 1 Random Variable and Probability Distribution Discrete random variable Y : Finite possible values {y
More informationtheir contents. If the sample mean is 15.2 oz. and the sample standard deviation is 0.50 oz., find the 95% confidence interval of the true mean.
Math 1342 Exam 3-Review Chapters 7-9 HCCS **************************************************************************************** Name Date **********************************************************************************************
More informationThe Chi-Square Distributions
MATH 03 The Chi-Square Distributions Dr. Neal, Spring 009 The chi-square distributions can be used in statistics to analyze the standard deviation of a normally distributed measurement and to test the
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. describes the.
Practice Test 3 Math 1342 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) The term z α/2 σn describes the. 1) A) maximum error of estimate
More informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
More informationQuantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing
Quantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More informationQuantitative Methods for Economics, Finance and Management (A86050 F86050)
Quantitative Methods for Economics, Finance and Management (A86050 F86050) Matteo Manera matteo.manera@unimib.it Marzio Galeotti marzio.galeotti@unimi.it 1 This material is taken and adapted from Guy Judge
More informationIntroduction to Survey Analysis!
Introduction to Survey Analysis! Professor Ron Fricker! Naval Postgraduate School! Monterey, California! Reading Assignment:! 2/22/13 None! 1 Goals for this Lecture! Introduction to analysis for surveys!
More informationGlossary for the Triola Statistics Series
Glossary for the Triola Statistics Series Absolute deviation The measure of variation equal to the sum of the deviations of each value from the mean, divided by the number of values Acceptance sampling
More informationInferential Statistics
Inferential Statistics Eva Riccomagno, Maria Piera Rogantin DIMA Università di Genova riccomagno@dima.unige.it rogantin@dima.unige.it Part G Distribution free hypothesis tests 1. Classical and distribution-free
More informationLecture 10: Generalized likelihood ratio test
Stat 200: Introduction to Statistical Inference Autumn 2018/19 Lecture 10: Generalized likelihood ratio test Lecturer: Art B. Owen October 25 Disclaimer: These notes have not been subjected to the usual
More informationChapter 3. Comparing two populations
Chapter 3. Comparing two populations Contents Hypothesis for the difference between two population means: matched pairs Hypothesis for the difference between two population means: independent samples Two
More informationMath 1040 Final Exam Form A Introduction to Statistics Fall Semester 2010
Math 1040 Final Exam Form A Introduction to Statistics Fall Semester 2010 Instructor Name Time Limit: 120 minutes Any calculator is okay. Necessary tables and formulas are attached to the back of the exam.
More information1; (f) H 0 : = 55 db, H 1 : < 55.
Reference: Chapter 8 of J. L. Devore s 8 th Edition By S. Maghsoodloo TESTING a STATISTICAL HYPOTHESIS A statistical hypothesis is an assumption about the frequency function(s) (i.e., pmf or pdf) of one
More informationStatistics Revision Questions Nov 2016 [175 marks]
Statistics Revision Questions Nov 2016 [175 marks] The distribution of rainfall in a town over 80 days is displayed on the following box-and-whisker diagram. 1a. Write down the median rainfall. 1b. Write
More informationCherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009 there were participants
18.650 Statistics for Applications Chapter 5: Parametric hypothesis testing 1/37 Cherry Blossom run (1) The credit union Cherry Blossom Run is a 10 mile race that takes place every year in D.C. In 2009
More informationChapter 8 of Devore , H 1 :
Chapter 8 of Devore TESTING A STATISTICAL HYPOTHESIS Maghsoodloo A statistical hypothesis is an assumption about the frequency function(s) (i.e., PDF or pdf) of one or more random variables. Stated in
More informationFinal Exam Bus 320 Spring 2000 Russell
Name Final Exam Bus 320 Spring 2000 Russell Do not turn over this page until you are told to do so. You will have 3 hours minutes to complete this exam. The exam has a total of 100 points and is divided
More informationProbability and Statistics Notes
Probability and Statistics Notes Chapter Seven Jesse Crawford Department of Mathematics Tarleton State University Spring 2011 (Tarleton State University) Chapter Seven Notes Spring 2011 1 / 42 Outline
More informationThis gives us an upper and lower bound that capture our population mean.
Confidence Intervals Critical Values Practice Problems 1 Estimation 1.1 Confidence Intervals Definition 1.1 Margin of error. The margin of error of a distribution is the amount of error we predict when
More informationSolution: First note that the power function of the test is given as follows,
Problem 4.5.8: Assume the life of a tire given by X is distributed N(θ, 5000 ) Past experience indicates that θ = 30000. The manufacturere claims the tires made by a new process have mean θ > 30000. Is
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science
UNIVERSITY OF TORONTO Faculty of Arts and Science December 2013 Final Examination STA442H1F/2101HF Methods of Applied Statistics Jerry Brunner Duration - 3 hours Aids: Calculator Model(s): Any calculator
More informationSTAT100 Elementary Statistics and Probability
STAT100 Elementary Statistics and Probability Exam, Monday, August 11, 014 Solution Show all work clearly and in order, and circle your final answers. Justify your answers algebraically whenever possible.
More informationIntroduction to Statistics
MTH4106 Introduction to Statistics Notes 15 Spring 2013 Testing hypotheses about the mean Earlier, we saw how to test hypotheses about a proportion, using properties of the Binomial distribution It is
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests
Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests 1999 Prentice-Hall, Inc. Chap. 8-1 Chapter Topics Hypothesis Testing Methodology Z Test
More informationMath 2000 Practice Final Exam: Homework problems to review. Problem numbers
Math 2000 Practice Final Exam: Homework problems to review Pages: Problem numbers 52 20 65 1 181 14 189 23, 30 245 56 256 13 280 4, 15 301 21 315 18 379 14 388 13 441 13 450 10 461 1 553 13, 16 561 13,
More informationMathematical Notation Math Introduction to Applied Statistics
Mathematical Notation Math 113 - Introduction to Applied Statistics Name : Use Word or WordPerfect to recreate the following documents. Each article is worth 10 points and can be printed and given to the
More information4 Hypothesis testing. 4.1 Types of hypothesis and types of error 4 HYPOTHESIS TESTING 49
4 HYPOTHESIS TESTING 49 4 Hypothesis testing In sections 2 and 3 we considered the problem of estimating a single parameter of interest, θ. In this section we consider the related problem of testing whether
More informationBasic Concepts of Inference
Basic Concepts of Inference Corresponds to Chapter 6 of Tamhane and Dunlop Slides prepared by Elizabeth Newton (MIT) with some slides by Jacqueline Telford (Johns Hopkins University) and Roy Welsch (MIT).
More informationSampling Distributions of Statistics Corresponds to Chapter 5 of Tamhane and Dunlop
Sampling Distributions of Statistics Corresponds to Chapter 5 of Tamhane and Dunlop Slides prepared by Elizabeth Newton (MIT), with some slides by Jacqueline Telford (Johns Hopkins University) 1 Sampling
More informationLecture 14. Analysis of Variance * Correlation and Regression. The McGraw-Hill Companies, Inc., 2000
Lecture 14 Analysis of Variance * Correlation and Regression Outline Analysis of Variance (ANOVA) 11-1 Introduction 11-2 Scatter Plots 11-3 Correlation 11-4 Regression Outline 11-5 Coefficient of Determination
More informationLecture 14. Outline. Outline. Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA)
Outline Lecture 14 Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA) 11-1 Introduction 11- Scatter Plots 11-3 Correlation 11-4 Regression Outline 11-5 Coefficient of Determination
More informationUniversity of Chicago Graduate School of Business. Business 41000: Business Statistics
Name: OUTLINE SOLUTION University of Chicago Graduate School of Business Business 41000: Business Statistics Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper for the formulas.
More information12.10 (STUDENT CD-ROM TOPIC) CHI-SQUARE GOODNESS- OF-FIT TESTS
CDR4_BERE601_11_SE_C1QXD 1//08 1:0 PM Page 1 110: (Student CD-ROM Topic) Chi-Square Goodness-of-Fit Tests CD1-1 110 (STUDENT CD-ROM TOPIC) CHI-SQUARE GOODNESS- OF-FIT TESTS In this section, χ goodness-of-fit
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Chapter 7 Exam A Name 1) How do you determine whether to use the z or t distribution in computing the margin of error, E = z α/2 σn or E = t α/2 s n? 1) Use the given degree of confidence and sample data
More informationInferences About Two Proportions
Inferences About Two Proportions Quantitative Methods II Plan for Today Sampling two populations Confidence intervals for differences of two proportions Testing the difference of proportions Examples 1
More informationThe Purpose of Hypothesis Testing
Section 8 1A:! An Introduction to Hypothesis Testing The Purpose of Hypothesis Testing See s Candy states that a box of it s candy weighs 16 oz. They do not mean that every single box weights exactly 16
More informationInference for Proportions, Variance and Standard Deviation
Inference for Proportions, Variance and Standard Deviation Sections 7.10 & 7.6 Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c Department of Mathematics University of Houston Lecture 12 Cathy
More informationAnalysis of 2x2 Cross-Over Designs using T-Tests
Chapter 234 Analysis of 2x2 Cross-Over Designs using T-Tests Introduction This procedure analyzes data from a two-treatment, two-period (2x2) cross-over design. The response is assumed to be a continuous
More informationChapte The McGraw-Hill Companies, Inc. All rights reserved.
er15 Chapte Chi-Square Tests d Chi-Square Tests for -Fit Uniform Goodness- Poisson Goodness- Goodness- ECDF Tests (Optional) Contingency Tables A contingency table is a cross-tabulation of n paired observations
More informationChapter Fifteen. Frequency Distribution, Cross-Tabulation, and Hypothesis Testing
Chapter Fifteen Frequency Distribution, Cross-Tabulation, and Hypothesis Testing Copyright 2010 Pearson Education, Inc. publishing as Prentice Hall 15-1 Internet Usage Data Table 15.1 Respondent Sex Familiarity
More informationParameter Estimation, Sampling Distributions & Hypothesis Testing
Parameter Estimation, Sampling Distributions & Hypothesis Testing Parameter Estimation & Hypothesis Testing In doing research, we are usually interested in some feature of a population distribution (which
More informationChapter. Hypothesis Testing with Two Samples. Copyright 2015, 2012, and 2009 Pearson Education, Inc. 1
Chapter 8 Hypothesis Testing with Two Samples Copyright 2015, 2012, and 2009 Pearson Education, Inc 1 Two Sample Hypothesis Test Compares two parameters from two populations Sampling methods: Independent
More informationThe t-test Pivots Summary. Pivots and t-tests. Patrick Breheny. October 15. Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/18
and t-tests Patrick Breheny October 15 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/18 Introduction The t-test As we discussed previously, W.S. Gossett derived the t-distribution as a way of
More informationCHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007)
FROM: PAGANO, R. R. (007) I. INTRODUCTION: DISTINCTION BETWEEN PARAMETRIC AND NON-PARAMETRIC TESTS Statistical inference tests are often classified as to whether they are parametric or nonparametric Parameter
More informationStatistics for IT Managers
Statistics for IT Managers 95-796, Fall 2012 Module 2: Hypothesis Testing and Statistical Inference (5 lectures) Reading: Statistics for Business and Economics, Ch. 5-7 Confidence intervals Given the sample
More information1 MA421 Introduction. Ashis Gangopadhyay. Department of Mathematics and Statistics. Boston University. c Ashis Gangopadhyay
1 MA421 Introduction Ashis Gangopadhyay Department of Mathematics and Statistics Boston University c Ashis Gangopadhyay 1.1 Introduction 1.1.1 Some key statistical concepts 1. Statistics: Art of data analysis,
More informationCh. 7. One sample hypothesis tests for µ and σ
Ch. 7. One sample hypothesis tests for µ and σ Prof. Tesler Math 18 Winter 2019 Prof. Tesler Ch. 7: One sample hypoth. tests for µ, σ Math 18 / Winter 2019 1 / 23 Introduction Data Consider the SAT math
More informationHANDBOOK OF APPLICABLE MATHEMATICS
HANDBOOK OF APPLICABLE MATHEMATICS Chief Editor: Walter Ledermann Volume VI: Statistics PART A Edited by Emlyn Lloyd University of Lancaster A Wiley-Interscience Publication JOHN WILEY & SONS Chichester
More information