Confidence Intervals, 6-Step Method

Size: px
Start display at page:

Download "Confidence Intervals, 6-Step Method"

Transcription

1 All rights reserved, Golde I. Holtzman Confidence Intervals, 6-Step Method General Form of the 6-Step Method for Confidence Interval Estimation 1. Model: a. Verbally identify the underlying random variable of interest (characteristic and population). b. Verbally identify the underlying parameter of interest. c. State the assumptions being made regarding the underlying distribution.. Hypotheses: None 3. Formulate Confidence limits and cite a reference: a. State the formula that is appropriate for the parameter under the assumptions, and b. cite a reference, e.g., Zar (1995) or Baldi and Moore (009). 4. Design: a. Choose the confidence coefficient, 1 α, which ultimately determines our confidence in the accuracy of the estimate, and partially determines the width (or length) of the interval, and, thereby, partially determines the precision, of the estimate. b. Choose the sample size, n, which partially determines the width (or length) of the interval, and, thereby, partially determines the precision, of the estimate. 5. Gather the data and: a. Compute the best point estimate of the parameter of interest. b. Compute the standard error of the estimate. Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

2 c. Determine the percentile or critical value of the appropriate distribution. 1 d. Compute the lower (L) and upper (U)confidence limits. 6. State the conclusion verbally: a. Methods (or Statistical Methods). Confidence intervals are calculated assuming that [1-c, verbally] ([cite reference as in 3-b]). b. Results. "I am % confident that is between and." "I am [confidence coefficient] % confident that [parameter of interest, verbally] is between [L] and [U]." Example 1, estimation of population mean when population SD known Reconsider the 5 Ash trees of Region 1, Monongahela National Forest. Based on that sample, estimate the mean DBH of all ash trees in Region 1, with 95% confidence. Assume that DBH is normally distributed, and (unrealistically) that the standard deviation of ash DBH is known to be 10.0 cm. Use the 6-step method. 1. Model: a. The underlying r.v. of interest (Y i ) is the DBH (cm) of the i th randomly sampled ash tree from Region 1, i = 1,,,n. b. The parameter of interest is the population mean, μ, the mean DBH (cm) of all ash trees in Region 1. c. Assume i. Population SD,σ = 10.0 cm. (i.e., we are assuming population SD is known). ii. Y i distributed Normally. 1 Some authors (e.g., Daniel, 1987) call this number the reliability coefficient. Others: (e.g., Koopmans, Zar) call it a critical value. Copyright Golde I. Holtzman 00, 007, 010. ci6step.docx, 7/18/01

3 . Hypotheses: None 3. CI:Formulate confidence limits a. (sample mean) ± (margin of error) (sample mean) ± [(Z)(Pop SE)] σ Y ± z 1 α (quantile notation) n or Y ± z σ (Baldi and Moore simplified notation) n b. See Zar (010), or Baldi and Moore (009) 4. Design: a. Confidence coefficient = (1 α) = b. Sample size = n = Gather the data: (Recall that for the ash trees of Region 1, the sample mean was 15.5 cm.) a. The best estimate of the pop mean = the sample mean = Y = 15.5 cm. b. Pop SE = (Pop SD) n = σ n = = 10 / 5 =.00. c. Z = Z = [Explanation: Since the parameter of interest is the population mean, and since the pop SD is (assumed to be) known, and since the random variable of interest is distributed Normally (also by assumption), it is appropriate to use a percentile of the standard normal distribution, i.e., the appropriate percentile Z 1 (α/). In Step 4, the design step, we specified the confidence level (1 α) = 0.95, therefore α = 0.05, therefore α/= 0.05, therefore (1 α/) = Hence, the required percentile is the 97.5th percentile of the standard normal distribution, Z = 1.96.] Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

4 d. Confidence limits: = estimator ± (margin of error) = estimator ± [(percentile)(standard error)] = (sample mean) ± [z 1 (α/) (Pop SE)] Y ± z σ n = 1 ( α ) = 15.5 ± [(1.96)(.00)] = 15.5 ± 3.9 = (11.6, 19.4) 6. State the conclusion verbally: a. Methods. Confidence intervals are calculated assuming that (i) the population standard deviation is 10.0 cm and (ii) the diameter at breast height of all trees in Region 1 is normally distributed (Sall et al., 001, pp ). b. Results. I am 95% confident that the mean DBH of all ash trees in Region 1 is between 11.6 and 19.4 cm. Example, estimation of population mean when population SD known Estimate the mean DBH of all ash trees in Region 1, with 90% confidence. Assume that DBH is normally distributed, and (unrealistically) that the standard deviation of ash DBH is known to be 10 cm. Use the 6-step method. 1. Model: Same as earlier example.. Hypotheses: None 3. Formulate CL: a. σ Y ± z or Y z 1 α ± n σ n b. See Zar (1995) or Baldi and Moore (009, 01) 4. Design: a. Confidence coefficient = (1 - α) = 0.90 b. n = 5. Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

5 5. Gather the data: a. Sample mean = 15.5 cm. b. Pop SE = ( pop SD) n = = 10.0 / 5 =.00. c. z 1 (α/) = z 0.95 = (from bottom line of Student's t distribution, i.e., a standard normal z = t, a Student s t with infinite degrees of freedom) d. Confidence limits: = 15.5 ± (1.645)(.00) = 15.5 ± 3.9 = (1.1, 18.79) 6. Conclusion. a. Methods. Confidence intervals are calculated assuming that (i) the population standard deviation is 10.0 cm and (ii) the diameter at breast height of all trees in Region 1 is normally distributed (Sall et al., 001, pp ). b. Results. I am 90% confident that the mean DBH of all ash trees in Region 1 is between 1. and 18.8 cm. Notes on Examples 1 and The standard deviation used in Examples 1 and was the population standard deviation (usually denoted by the Greek letter σ) rather than the sample standard deviation (usually denoted by the English letter s). o That the population standard deviation is known to be σ = 10.0 is an assumption. It is something the investigator assumed to be true before he performed the study, before he obtained the sample, and before he calculated the sample standard deviation. o That the standard deviation we used was assumed to be the population standard deviation is what caused us to use the standard normal critical values (z =1.96 for 95% confidence and z = for 90% confidence). If we had not used that assumed population standard deviation and instead had used the sample standard deviation, then we would have used critical values from a different distribution, Student s T Distribution, rather than Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

6 from the Standard Normal Z Distribution. Use of the T Distribution will be illustrated in the next example. There is a tradeoff between confidence in accuracy and precision. The only difference between Examples 1 and is the confidence limits. Example 1 Example Confidence Level ( 1 α ) = ( α ) 1 = Critical Value z = 1.96 z = Confidence Limits (11.6, 19.4) (1., 18.8) Length = = 6.6 Margin of Error (MoE) 7.8/ = / = 3.3 Tradeoff More confidence in accuracy(95%), Less precision (MoE = 3.9) Less confidence in accuracy (90%), More precision (MoE = 3.3) Example 3, estimation of population mean when population SD unknown Reconsider the 5 Ash trees of Region 1, Monongahela National Forest. Based on that sample, estimate the mean DBH of all ash trees in Region 1, with 95% confidence. As in Example 1 above, assume that DBH is normally distributed, but unlike in Example 1, drop the (unrealistically) that the standard deviation of ash DBH is known to be 10.0 cm. Use the 6-step method. In other words: Reconsider the 5 Ash trees of Region 1, Monongahela National Forest. Based on that sample, estimate the mean DBH of all ash trees in Region 1, with 95% confidence. Assume that DBH is normally distributed. Use the 6-step method. Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

7 1. Model a. Let X i represent the DBH (cm) of the i th randomly sampled ash tree from Region 1,i= 1,,, n. b. Let μ represent the mean DBH (cm) of all ash trees in Region 1. c. Assume X i distributed Normally.. Hypotheses: None 3. Formulate confidence limits: ( sample mean ) ± ( Margin of Error) = ( sample mean) ± ( t)( estimated SE) = Y ± t or Y ± t n 1,1 α s n where t tn 1,1 a s (quantile notation) n (Baldi and Moore simplified notation) α = is the ( 1 ) quantile of Student's t distribution with (n 1) degrees of freedom, (Zar 1995, p. 100, Baldi and Moore 009, Chapter 17). 4. Design: a. Confidence coefficient = (1 α) = b. n = Gather the data and compute: a. Sample mean = 15.5 cm b. Estimated SE = s n = = = 1.7. c. (Since the parameter of interest is the population mean, and since the population standard deviation is not (assumed to be) known, and since the underlying random variable of interest is distributed normally (also by assumption), it is appropriate to use a percentile of Student's t distribution, i.e., the appropriate percentile is) t = t 1,1 4,0.975=.0 n α 64. Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

8 d. Confidence limits: estimator ± MOE =estimator ± [(percentile)(standard error)] = 15.5 ± [(.064)(1.7)] = 15.5 ± 3.55 = (1.0, 19.1) 6. Conclusion Methods. Confidenceintervals are computed assuming that the DBH of all ash trees in Region 1 is normally distributed (Zar 1995, p. 100). Results. I am 95% confident that the mean DBH of all ash trees in Region 1 is between 1.0 and 19.1 cm. Example 3-b, estimation of population mean when population SD unknown Yet again, reconsider the 5 Ash trees of Region 1, Monongahela National Forest. Based on that sample, estimate the mean DBH of all ash trees in Region 1, with 90% (rather than 95%) confidence. Assume that DBH is normally distributed. Use the 6-step method. 1. Model a. Let X i represent the DBH (cm) of the i th randomly sampled ash tree from Region 1,i= 1,,, n. b. Let μ represent the mean DBH (cm) of all ash trees in Region 1. c. Assume X i distributed Normally.. Hypotheses: None 3. Formulate confidence limits: Y ± t s (Baldi and Moore simplified notation) n where t tn 1,1 a = is the ( 1 α ) quantile of Student's t distribution with (n 1) degrees of freedom, (Zar 1995, p. 100, Baldi and Moore 009, Chapter 17). Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

9 4. Design: c. Confidence coefficient = (1 α) = d. n = Gather the data and compute: e. Sample mean = 15.5 cm f. Estimated SE = s n = = = 1.7. g. Percentile = t t t n 4,0.95 α = = = ,1 h. Confidence limits: estimator ± MOE =estimator ± [(percentile)(standard error)] = X ± t s n = 15.5 ± [(1.711)(1.7)] = 15.5 ±.94 = (1.56, 18.44) 6. Conclusion Methods. Confidenceintervals are computed assuming that the DBH of all ash trees in Region 1 is normally distributed (Zar 1995, p. 100, Baldi & Moore 009, 01, Chapter 14). Results. I am 90% confident that the mean DBH of all ash trees in Region 1 is between 1.6 and 18.4 cm. Practice Quiz 1. Does increasing the confidence coefficient increase, decrease, or not affect the width of the confidence interval?. A confidence interval is said to be accurate if it contains the true value of the parameter of interest. Consider 90% and 95% confidence intervals for the same data. Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

10 a. Is it possible for the 90% CI to be accurate if the 95% CI is not? b. Is it possible for the 95% CI to be valid if the 90% CI is not? 3. Does increasing the confidence coefficient increase, decrease, or not affect our confidence in the accuracy of the confidence interval. 4. The more narrow a confidence interval is, the more precise it is said to be. Does increasing the confidence coefficient increase, decrease, or not affect the precision of the estimate? 5. Accuracy versus precision. a. Which is more important, accuracy or precision? b. Then which should be determined first, the confidence coefficient or the sample size? 6. Design. a. In general, does a larger sample size yield more, or less, information about the value of the parameter of interest? b. Does a larger sample size yield a larger, or smaller, standard error. c. For a fixed confidence coefficient, e.g. 95%, does a smaller standard error yield a wider, or narrower, confidence interval? d. Is a narrower confidence interval considered more, or less, precise? e. In general, for a fixed confidence coefficient, would a larger sample yield a more, or less, precise interval estimate? 7. How do we know whether to use Student's t distribution rather than the standard normal (z) distribution to find the required percentile? 8. Does use of the t distribution always require assuming that the underlying distribution is normal? 9. Explain the difference between the widths of the confidence for the population mean when the population standard deviation is known, versus when the population standard deviation is unknown, as in Example 1 versus Example 3. Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

11 Example 4-a, estimation of population proportion With 95% confidence, estimate the proportion of trees in Region 1 of the MNF that are ash. 1. Model a. [The r.v. of interest is a categorical variable.] Let X i denote the genus (ash, birch, spruce, ) or the i th randomly sampled tree from Region 1,i= 1,,..., n. b. [The parameter of interest is the population proportion, φ.] Let φ represent the proportion of all trees in Region 1 that are ash trees. c. Assumptions about the underlying distribution: None [When estimating the population proportion of a categorical variable, it is never necessary to make assumptions about the underlying categorical population distribution, as long as there is a simple random sample.]. Hypotheses: None 3. Formulate Confidence Limits: ( sample proportion ) ± ( Margin of error) = p± ( z1 ( p)( 1 p) n), α where z = z1 α is the 1 α percentile of the standard normal Z distribution, and where p is the sample proportion. See Snedecor and Cochran 1967, pp ; Baldi and Moore 009, Chapter 19, Large-sample confidence intervals for a proportion ). 4. Design: a. Confidence coefficient = (1 α) = b. n = Gather the data and compute: a. (Sample proportion) = 5/75 = 1/3 b. Estimated SE ( )( ) = = = Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

12 c. z z z α = = = 1.96 (from bottom line of Student's t distribution, i.e., a 1 standard normal z = t, a Student s t with infinite degrees of freedom) d. Confidence limits: estimator ± MOE estimator ± [(percentile)(standard error)] = 1/3 ± [(1.96)( )] = ± = (0.7, 0.440) 6. Conclusion Methods. Confidence intervals are computed by the method of Snedecor and Cochran 1967, pp Results. I am 95% confident that the proportion of all trees in Region 1 that are ash is between.7% and 44.0%. Example 4 b, estimation of population proportion, BARE BONES It is not necessary to include all of the explanation once you know what you re doing. Here s the brief version. With 95% confidence, estimate the proportion of trees in Region 1 of the MNF that are ash. 1. Model a. Let X i denote the genus (ash, birch, spruce, ) or the i th randomly sampled tree from Region 1,i= 1,,..., n. b. Let φ represent the proportion of all trees in Region 1 that are ash trees. c. Assumptions about the underlying distribution: None. Hypotheses: None 3. Formulate Confidence Limits: ( ( )( ) ) [ ] CL = p± z p 1 p n, Baldi and Moore notation or ( ( )( 1 ) ), 1 [ quantile notation ] α CL = p ± z p p n Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

13 See Snedecor and Cochran 1967, pp ; Baldi and Moore 009, Chapter 19, Large-sample confidence intervals for a proportion ). 4. Design: a. (1 α) = b. n = Gather the data and compute: a. p = 5/75 = 1/3 b. SE = ( 1 3)( 3) 75 = = c. z z z0.975 z α = = = = d. CL= 1/3 ± [(1.96)( )] = ± = (0.7, 0.440) 6. Conclusion Methods. Confidence intervals are computed by the method of Snedecor and Cochran 1967, pp Results. I am 95% confident that the proportion of all trees in Region 1 that are ash is between.7% and 44.0%. References Cited Snedecor, G.W., and W.G. Cochran (1967). Statistical Methods, 6th ed. IowaStateUniversity Press, Ames, Iowa. Baldi, Brigette, and Moore (009), The Practice of Statistics in the Life Sciences, W.H. Freeman and Company, New York. Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

14 Example 5, estimation of population proportion, BARE-BONES This is the 6-Step Method applied to Example 19.3 of Baldi and Moore (009). The National AIDS Behavioral Surveys found that 170 of a sample of 673 adult heterosexuals had multiple partners. What can we say about the population of all adult heterosexuals? Estimate, with 99% confidence, the proportion of all adult heterosexuals who have multiple partners. 1. Model a. X i = whether or not the i th randomly sampled adult heterosexual had multiple partners (multiple, not multiple),i= 1,,..., n. b. φ= the proportion of all adult heterosexuals that who have multiple partners. c. Assumptions about the underlying distribution: None. Hypotheses: None 3. Formulate Confidence Limits: ( ( )( ) ) [ ] CL = p± z p 1 p n, Baldi and Moore notation or ( ( )( 1 ) ), 1 [ quantile notation ] α CL = p ± z p p n where p denotes the sample proportion. See Snedecor and Cochran 1967, pp ; Baldi and Moore 009, Chapter 19, Large-sample confidence intervals for a proportion ). 4. Design: a. (1 α) = 0.99 b. n = Gather the data and compute: a. p = 170 / 673 = ( ) b. SE = = = c. z z z 1 α = = =.576 Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

15 d. CL= ± [(.576)( )] 6. Conclusion = ± = (0.0514, ) Methods. Confidence intervals are computed by the large-sample method of Baldi and Moore, 009. Results. I am 99% confident that the proportion of all adult heterosexuals who have multiple partnersis between 5.14% and 7.58%. Example 6, estimation of population variance and standard deviation With 95% confidence, estimate the variance and standard deviation of the DBH (cm) of all ash trees in Region 1 of the MNF. 1. Model a. The r.v. of interest is X i, the DBH of of the i th randomly sampled ash tree from Region 1 of the MNF, i = 1,,..., n. b. The parameter of interest is the population variance, the variance of all ash trees in Region 1 of the MNF. c. Assume X i distributed normal. Hypotheses: None 3. Formulation of confidence limits: ( lower limit) = ( n 1) s χ n 1,1 α ( upper limit) = ( n 1) s χ n 1, α where χ 1,1 α n is the (1 α/)quantile of the chi-squared distribution with (n 1) degrees of freedom, and where χ n 1, α is the (α/) percentile of the chi-squared distribution with(n 1) degrees of freedom. Percentiles of the Chi-Squared Distribution are posted. See Zar 1995, pp Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

16 4. Design: a. Confidence coefficient = (1 α) = b. n = Gather the data and compute: a. Sample variance = s = (8.6) = b. χ = χ = , 1,1 n α 4,0.975 χ = χ = 1.401, 1, n α 4,0.05 c. Confidence limits: Lower limit = (4)(73.96) / (39.354) = , Upper limit = (4)(73.96) / (1.401) = Conclusion Methods. Confidence intervals are estimated assuming that the DBH (cm) of all trees ins Region 1 are normally distributed (Zar 1995, pp ). Results. I am 95% confident that the variance of all trees in Region 1 of the MNF is between and Or: I am 95% confident that the standard deviation of all trees in Region 1 of the MNF is between 6.7 and 1.0 cm. Department of Statistics Send Suggestions or Comments to Golde Holtzman Last updated: 7/18/01 URL:../STAT5605/ci6step.pdf Copyright Golde I. Holtzman 00, 007, ci6step.docx, 7/18/01

Confidence Intervals for One Standard Deviation Using Standard Deviation

Confidence Intervals for One Standard Deviation Using Standard Deviation Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from

More information

Confidence Intervals for the Difference Between Two Means

Confidence Intervals for the Difference Between Two Means Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

More information

Social Studies 201 Notes for November 19, 2003

Social Studies 201 Notes for November 19, 2003 1 Social Studies 201 Notes for November 19, 2003 Determining sample size for estimation of a population proportion Section 8.6.2, p. 541. As indicated in the notes for November 17, when sample size is

More information

Confidence Intervals for Cp

Confidence Intervals for Cp Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process

More information

Confidence Intervals for Spearman s Rank Correlation

Confidence Intervals for Spearman s Rank Correlation Chapter 808 Confidence Intervals for Spearman s Rank Correlation Introduction This routine calculates the sample size needed to obtain a specified width of Spearman s rank correlation coefficient confidence

More information

Lecture Notes Module 1

Lecture Notes Module 1 Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

More information

Week 4: Standard Error and Confidence Intervals

Week 4: Standard Error and Confidence Intervals Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

More information

Review. March 21, 2011. 155S7.1 2_3 Estimating a Population Proportion. Chapter 7 Estimates and Sample Sizes. Test 2 (Chapters 4, 5, & 6) Results

Review. March 21, 2011. 155S7.1 2_3 Estimating a Population Proportion. Chapter 7 Estimates and Sample Sizes. Test 2 (Chapters 4, 5, & 6) Results MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 7 Estimates and Sample Sizes 7 1 Review and Preview 7 2 Estimating a Population Proportion 7 3 Estimating a Population

More information

How To Check For Differences In The One Way Anova

How To Check For Differences In The One Way Anova MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

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

Fixed-Effect Versus Random-Effects Models

Fixed-Effect Versus Random-Effects Models CHAPTER 13 Fixed-Effect Versus Random-Effects Models Introduction Definition of a summary effect Estimating the summary effect Extreme effect size in a large study or a small study Confidence interval

More information

Standard Deviation Estimator

Standard Deviation Estimator CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

More information

Confidence Intervals for Cpk

Confidence Intervals for Cpk Chapter 297 Confidence Intervals for Cpk Introduction This routine calculates the sample size needed to obtain a specified width of a Cpk confidence interval at a stated confidence level. Cpk is a process

More information

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

More information

Population Mean (Known Variance)

Population Mean (Known Variance) Confidence Intervals Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Population Mean (Known Variance) 1. A random sample of n measurements was selected from a population with

More information

Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name: Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

More information

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE

CONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE 1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,

More information

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete

More information

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics. Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

More information

Chapter Study Guide. Chapter 11 Confidence Intervals and Hypothesis Testing for Means

Chapter Study Guide. Chapter 11 Confidence Intervals and Hypothesis Testing for Means OPRE504 Chapter Study Guide Chapter 11 Confidence Intervals and Hypothesis Testing for Means I. Calculate Probability for A Sample Mean When Population σ Is Known 1. First of all, we need to find out the

More information

" Y. Notation and Equations for Regression Lecture 11/4. Notation:

 Y. Notation and Equations for Regression Lecture 11/4. Notation: Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

More information

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

More information

6.4 Normal Distribution

6.4 Normal Distribution Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

More information

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

More information

Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013 Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

More information

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so:

Chapter 7 Notes - Inference for Single Samples. You know already for a large sample, you can invoke the CLT so: Chapter 7 Notes - Inference for Single Samples You know already for a large sample, you can invoke the CLT so: X N(µ, ). Also for a large sample, you can replace an unknown σ by s. You know how to do a

More information

Point and Interval Estimates

Point and Interval Estimates Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number

More information

Mean = (sum of the values / the number of the value) if probabilities are equal

Mean = (sum of the values / the number of the value) if probabilities are equal Population Mean Mean = (sum of the values / the number of the value) if probabilities are equal Compute the population mean Population/Sample mean: 1. Collect the data 2. sum all the values in the population/sample.

More information

Chapter 2. Hypothesis testing in one population

Chapter 2. Hypothesis testing in one population Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance

More information

z-scores AND THE NORMAL CURVE MODEL

z-scores AND THE NORMAL CURVE MODEL z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A

More information

Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

More information

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

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

Constructing and Interpreting Confidence Intervals

Constructing and Interpreting Confidence Intervals Constructing and Interpreting Confidence Intervals Confidence Intervals In this power point, you will learn: Why confidence intervals are important in evaluation research How to interpret a confidence

More information

Fairfield Public Schools

Fairfield Public Schools Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

More information

Simple linear regression

Simple linear regression Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

More information

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

More information

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

More information

Lesson 10: Basic Inventory Calculations

Lesson 10: Basic Inventory Calculations Lesson 10: Basic Inventory Calculations Review and Introduction In the preceding lessons, you learned how to establish and take measurements in sample plots. You can use a program like LMS to calculate

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

Objectives. 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) CI)

Objectives. 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) CI) Objectives 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) Statistical confidence (CIS gives a good explanation of a 95% CI) Confidence intervals. Further reading http://onlinestatbook.com/2/estimation/confidence.html

More information

Coefficient of Determination

Coefficient of Determination Coefficient of Determination The coefficient of determination R 2 (or sometimes r 2 ) is another measure of how well the least squares equation ŷ = b 0 + b 1 x performs as a predictor of y. R 2 is computed

More information

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

More information

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1. Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.

More information

CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

More information

WISE Power Tutorial All Exercises

WISE Power Tutorial All Exercises ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

More information

Study Guide for the Final Exam

Study Guide for the Final Exam Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

More information

Means, standard deviations and. and standard errors

Means, standard deviations and. and standard errors CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

More information

Lecture 19: Chapter 8, Section 1 Sampling Distributions: Proportions

Lecture 19: Chapter 8, Section 1 Sampling Distributions: Proportions Lecture 19: Chapter 8, Section 1 Sampling Distributions: Proportions Typical Inference Problem Definition of Sampling Distribution 3 Approaches to Understanding Sampling Dist. Applying 68-95-99.7 Rule

More information

AP STATISTICS 2010 SCORING GUIDELINES

AP STATISTICS 2010 SCORING GUIDELINES 2010 SCORING GUIDELINES Question 4 Intent of Question The primary goals of this question were to (1) assess students ability to calculate an expected value and a standard deviation; (2) recognize the applicability

More information

An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

More information

Elements of statistics (MATH0487-1)

Elements of statistics (MATH0487-1) Elements of statistics (MATH0487-1) Prof. Dr. Dr. K. Van Steen University of Liège, Belgium December 10, 2012 Introduction to Statistics Basic Probability Revisited Sampling Exploratory Data Analysis -

More information

Chapter 7 Section 1 Homework Set A

Chapter 7 Section 1 Homework Set A Chapter 7 Section 1 Homework Set A 7.15 Finding the critical value t *. What critical value t * from Table D (use software, go to the web and type t distribution applet) should be used to calculate the

More information

Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing

Introduction. Hypothesis Testing. Hypothesis Testing. Significance Testing Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters

More information

10.2 Series and Convergence

10.2 Series and Convergence 10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

More information

Chapter 10. Key Ideas Correlation, Correlation Coefficient (r),

Chapter 10. Key Ideas Correlation, Correlation Coefficient (r), Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

More information

Simple Regression Theory II 2010 Samuel L. Baker

Simple Regression Theory II 2010 Samuel L. Baker SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

More information

Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Two-Sample T-Tests Assuming Equal Variance (Enter Means) Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

More information

One-Way Analysis of Variance

One-Way Analysis of Variance One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We

More information

2 ESTIMATION. Objectives. 2.0 Introduction

2 ESTIMATION. Objectives. 2.0 Introduction 2 ESTIMATION Chapter 2 Estimation Objectives After studying this chapter you should be able to calculate confidence intervals for the mean of a normal distribution with unknown variance; be able to calculate

More information

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

More information

Factors affecting online sales

Factors affecting online sales Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4

More information

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

More information

12.5: CHI-SQUARE GOODNESS OF FIT TESTS

12.5: CHI-SQUARE GOODNESS OF FIT TESTS 125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability

More information

Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples

Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The

More information

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference)

Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Chapter 45 Two-Sample T-Tests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when no assumption

More information

Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

More information

Recall this chart that showed how most of our course would be organized:

Recall this chart that showed how most of our course would be organized: Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

More information

Exact Confidence Intervals

Exact Confidence Intervals Math 541: Statistical Theory II Instructor: Songfeng Zheng Exact Confidence Intervals Confidence intervals provide an alternative to using an estimator ˆθ when we wish to estimate an unknown parameter

More information

3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples 3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

More information

Hypothesis testing - Steps

Hypothesis testing - Steps Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

More information

Numerical Methods for Option Pricing

Numerical Methods for Option Pricing Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly

More information

Name: Date: Use the following to answer questions 3-4:

Name: Date: Use the following to answer questions 3-4: Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

More information

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

More information

The correlation coefficient

The correlation coefficient The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

NCC5010: Data Analytics and Modeling Spring 2015 Practice Exemption Exam

NCC5010: Data Analytics and Modeling Spring 2015 Practice Exemption Exam NCC5010: Data Analytics and Modeling Spring 2015 Practice Exemption Exam Do not look at other pages until instructed to do so. The time limit is two hours. This exam consists of 6 problems. Do all of your

More information

Regression Analysis: A Complete Example

Regression Analysis: A Complete Example Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

More information

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

Least Squares Estimation

Least Squares Estimation Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

More information

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

More information

Chapter 6: Point Estimation. Fall 2011. - Probability & Statistics

Chapter 6: Point Estimation. Fall 2011. - Probability & Statistics STAT355 Chapter 6: Point Estimation Fall 2011 Chapter Fall 2011 6: Point1 Estimat / 18 Chap 6 - Point Estimation 1 6.1 Some general Concepts of Point Estimation Point Estimate Unbiasedness Principle of

More information

Module 2 Probability and Statistics

Module 2 Probability and Statistics Module 2 Probability and Statistics BASIC CONCEPTS Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The standard deviation of a standard normal distribution

More information

COMP6053 lecture: Relationship between two variables: correlation, covariance and r-squared. jn2@ecs.soton.ac.uk

COMP6053 lecture: Relationship between two variables: correlation, covariance and r-squared. jn2@ecs.soton.ac.uk COMP6053 lecture: Relationship between two variables: correlation, covariance and r-squared jn2@ecs.soton.ac.uk Relationships between variables So far we have looked at ways of characterizing the distribution

More information

Chapter 23 Inferences About Means

Chapter 23 Inferences About Means Chapter 23 Inferences About Means Chapter 23 - Inferences About Means 391 Chapter 23 Solutions to Class Examples 1. See Class Example 1. 2. We want to know if the mean battery lifespan exceeds the 300-minute

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Suppose following data have been collected (heights of 99 five-year-old boys) 117.9 11.2 112.9 115.9 18. 14.6 17.1 117.9 111.8 16.3 111. 1.4 112.1 19.2 11. 15.4 99.4 11.1 13.3 16.9

More information

STATISTICS FOR PSYCHOLOGISTS

STATISTICS FOR PSYCHOLOGISTS STATISTICS FOR PSYCHOLOGISTS SECTION: STATISTICAL METHODS CHAPTER: REPORTING STATISTICS Abstract: This chapter describes basic rules for presenting statistical results in APA style. All rules come from

More information

Quantitative Methods for Finance

Quantitative Methods for Finance Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain

More information

Elementary Statistics Sample Exam #3

Elementary Statistics Sample Exam #3 Elementary Statistics Sample Exam #3 Instructions. No books or telephones. Only the supplied calculators are allowed. The exam is worth 100 points. 1. A chi square goodness of fit test is considered to

More information

Sample Size Issues for Conjoint Analysis

Sample Size Issues for Conjoint Analysis Chapter 7 Sample Size Issues for Conjoint Analysis I m about to conduct a conjoint analysis study. How large a sample size do I need? What will be the margin of error of my estimates if I use a sample

More information

STA-201-TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance

STA-201-TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance Principles of Statistics STA-201-TE This TECEP is an introduction to descriptive and inferential statistics. Topics include: measures of central tendency, variability, correlation, regression, hypothesis

More information

WHERE DOES THE 10% CONDITION COME FROM?

WHERE DOES THE 10% CONDITION COME FROM? 1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

More information

Experiments in Complex Stands

Experiments in Complex Stands Experiments in Complex Stands Valerie LeMay, Craig Farnden and Peter Marshall March 8 to April 3, 009 LeMay, Farnden, Marshall 1 The Challenge Design an experiment based on an objective Constraints: Must

More information

Farm Business Survey - Statistical information

Farm Business Survey - Statistical information Farm Business Survey - Statistical information Sample representation and design The sample structure of the FBS was re-designed starting from the 2010/11 accounting year. The coverage of the survey is

More information

Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

More information