CONFIDENCE INTERVALS I


 Gerald Chandler
 1 years ago
 Views:
Transcription
1 CONFIDENCE INTERVALS I ESTIMATION: the sample mean Gx is an estimate of the population mean µ point of sampling is to obtain estimates of population values Example: for 55 students in Section 105, 45 of 55 work: p s = 8%; for those who work, the mean number of hours xg = inference: 8% of ASU students work an average of hours a week; p = 0.8 and µ = Problem: the sampling distribution is a continuous distribution the probability that Gx actually equals µ is zero Gx is not an accurate estimate of µ; in this case we cannot even state the probability that Gx is accurate Gx is called a point estimate of µ statisticians generally prefer to give an interval estimate: "There is a 90% probability that µ is between 1 and 17.5." interval estimate has two features the estimate in interval form a probability statement: taken as an assessment of the reliability or accuracy of the estimate the probability hinges on the probabilities found in the sampling distribution of Gx INTUITION Consider the sampling distribution of sample means for samples of size 100 drawn from a population of salaries in which µ = 33,000 and σ = 5,000 E(Gx ) = 33,000 σ xg = 5, = 500.
2 In the z table find z values that demarcate the middle 95% of a normal distribution: z = ± 1.96 The interval µ 1.96 σ xg to µ σ xg contains 95% of the sampling distribution or contains 95% of all the possible Gx s that could ever be drawn from this population the interval noted is 33,000 ± or 33,000 ± 980. Any sample mean in this interval differs from the actual population mean by no more than % of all possible sample means differ from µ by no more than 980. for any Gx, there is a 95% probability that it differs from µ by no more than 980; that is, by no more than the amount 1.96 σ xg choose a sample and calculate Gx; now consider an interval of the form Gx ± 1.96 σ xg necessarily, the population mean m lies within the limits of 95% of all such intervals there is a 95% probability that the population mean lies within the limits of any interval of the form Gx ± 1.96 σ xg A C% confidence interval: An interval of the form A C% xg ± confidence z C σ interval xg for the population mean is given by where z C is chosen so that C% of the normal distribution Gx ± z C σ lies within the interval xg z C to +z C. CALCULATING CONFIDENCE INTERVALS FOR THE POPULATION MEAN C is the confidence level z C is found as a z value such that z C to +z C incorporates the middle C% of a normal distribution; ±z C demarcates a symmetric interval which has area C
3 Example: Find the appropriate z value for a 9% confidence interval the interval must be symmetric: take out the middle 9% leaves 8% to be split between upper and lower tails. The required z values demarcate the lower 4% and lower 96% of the z distribution. Alternatively, let L = (100 C)/; here L = (100 9)/ = 4% or In the cumulative z table find area The closest seems to be ; reading back to the margins z = 1.75; therefore the required z C = 1.75 Check by finding a z value such that 96% of the distribution is less than that value. Examples: A population of Christmas trees has unknown µ, but it is known that the population is normally distributed with σ = 4. A sample of 5 trees has Gx = Find a 95% confidence interval for the mean height of the population. given n = 5, so that σ xg = σ n = 4/5 = 0.8 L = (100 C)/ = 5/ =.5% or From the z table 0.05 of the z distribution is less than 1.96, so z C = applying the formula above Gx ± z C σ xg 16.6 ± ± 1.568, or the interval to Stating the interval: A 95% confidence interval for the population mean is 16.6 ± We are 95% confident that the population mean is in the interval to There is a 95% probability that µ is at least but no more than Find a 90% confidence interval for the same population, same sample find z 90 by reference to z table. L = (100 C)/ = the nearest entry is z = then we have 16.6 ± = 16.6 ± 1.31, or the interval to For a sample of 64 drawn from this population, we got the same Gx. Find a 90% confidence interval for the population mean. since n = 64, σ xg = 4 64 = 0.5 confidence interval: 16.6 ± = 16.6 ± 0.8, or the interval to 17.4
4 Messages: the width of the confidence interval varies in the same direction as the confidence level in our first example, width = = 3.136, while in the second example, width = =.64 width of the interval is z C σ xg : called the precision of the estimate there is a tradeoff between precision and confidence common sense: for very wide intervals, we can be quite confident that we've captured µ, but as the interval narrows, the probability that it includes µ drops to zero As the sample size increases, precision increases at the same level of confidence the third interval above has width 1.64 with sufficiently large sample, we can achieve whatever combination of confidence and precision we desire as n increases, σ xg decreases FINDING THE RIGHT SAMPLE SIZE The distance e = z C σ xg is the error in the estimate e is onehalf the width of the confidence interval within the limits of our confidence statement, we are sure that the population mean differs from the sample mean by no more than e: we might say we're 90% confident that the true mean differs from the sample mean by no more than e. Hence, e is the maximum error in the estimate Suppose that there is some maximum tolerable value for e, or maximum tolerable error for a given confidence level, the value of n necessary to keep e within tolerable limits σ e = zc, solve for n to find n z n = C σ e for given z, chosen for the appropriate confidence level, this formula gives us the sample size necessary to achieve an error of no more than e in general, the result of this calculation is not an integer, so the rule is to make the sample size equal to the next largest integer. NOTE CAREFULLY: This refers to the maximum tolerable error in the sampling procedure, or in the estimate of µ, NOT to the tolerance in a manufacturing process.
5 Examples: Cigarette filters are supposed to have µ = 15 mm in length; σ = 0.1 mm. Machinery will jam if the length of a filter exceeds 15.3 mm, and the probability of such a filter increases as the mean length increases; must have an accurate estimate of the mean length of filters. Let us require e 0.01 mm. and 90% confidence intervals. How large must n be? z n = C σ = e = the next greatest integer is the required sample size (that is, ALWAYS round n upwards in these problems); here n = 69 ordering Tshirts to give to contestants in a road race; average chest size unknown but for all chests everywhere σ = 4 in. Measure a sample of the participants when they register, and require that the sample be accurate to within ±1.5 in. How large must the sample be to have 99% confidence in the result? first, z 99 =? n = (.58 4 / 1.5) = rounding upwards, we require n = 48
6 CONFIDENCE INTERVALS II: σ UNKNOWN WHEN TO USE A z VALUE IN CONSTRUCTING CONFIDENCE INTERVALS To this point, we have assumed the population standard deviation known. IF NOT Population is normally distributed and σ NOT known the sampling distribution of Gx is NOT normal but rather conforms to Student's t distribution If population is NOT normal and σ is NOT known but the sample is large (that is, n 30), then the sampling distribution of Gx approximates the t distribution In either of these cases, s, the sample standard deviation, estimates σ. RECALL: s = Σ(x xg ) /(n 1) and s = s The standard error of the mean is estimated by s xg = s/ n confidence intervals have the form Gx ± t C s xg the t values used here are numbers of standard deviations in this case, numbers of standard deviations on a t distribution CHARACTERISTICS OF THE t DISTRIBUTION Continuous Symmetric Values near the mean are more probable than values further out so that t distribution looks like a bellshaped curve. How is that any different from a normal distribution? 1. the t distribution has fatter tails and less mass in the center for a given number of standard deviations, probability is higher on the normal distribution than on a t distribution put another way, a given probability level will be further from the center of a t distribution that from the center of the normal distribution or, a given probability level will be more standard deviations (t values) away from the mean than would be the case on a normal distribution Note: t values will always be larger than z values for corresponding confidence level intervals constructed with t will always be wider (less precise) than those constructed with z
7 . there is not one t distribution but a large number, depending on the number of "degrees of freedom" Digression: the concept of degrees of freedom Mechanically df = n k, where n is the sample size and k the number of parameters that must be estimated from the sample before estimating the standard deviation for example: s, the sample standard deviation, is an estimate of σ. To calculate s, we must estimate µ. µ is estimated by xg, and xg is the only statistic we must calculate before we can calculate s. We must thus estimate one parameter, µ, before deriving and estimate of σ, and there are thus n 1 degrees of freedom in our estimate of σ more generally, degrees of freedom represents the number of independent (in the probability sense) random variables in a problem in calculating s we must use Gx. Suppose we are given Gx and n 1 of the values in the sample; then the nth value is already determined and can be derived from what we know The tdistribution: pages E7 and E8 in your textbook how to read the table Upper tail (α) values across the top are the area in one tail of the distribution for a confidence interval use an upper tail value corresponding to the area in one tail of the distribution this will be only half the difference between the confidence level and 1 For example: in preparing a 95% confidence interval, there will be 5% in the tails of the distribution, thus 0.05 in each tail: we should use a t value for uppertail area 0.05 and the appropriate number of degrees of freedom if C is the confidence level, expressed as decimal fraction, use α = (1 C)/ degrees of freedom are in the left hand column as df infinity, the tvalue z value Examples: Find the appropriate t value for 0 degrees of freedom and 90% confidence interval. α = (1 0.9)/ = 0.05 t = for a sample of size 37, find the t value for a 99% confidence interval d.f. = n 1 = 36; α = (1 0.99)/ = t =.7195 CONFIDENCE INTERVAL FOR µ WITH NORMAL POPULATION AND σ UNKNOWN Problem requires use of t with n 1 degrees of freedom. Confidence intervals will have the form ( n 1) d. f. C x ± t s where s xg = s/ n, s being the sample standard deviation note similarity to earlier confidence intervals x
8 Examples: 7 male students are selected at random and an alcoholic beverage is poured down them in tenthounce increments until distinct signs of nonsobriety are observed. The following results were obtained: Individual Amount of Beverage (oz) Researchers feel safe in assuming that the distribution of ounces until nonsobriety is normal in the population. Construct a 95% confidence interval for amount of drink it takes to get the average member of the population drunk. calculate Gx and s: Gx = 3.39, s = Σ(x xg ) /(n 1) = [( ) + + ( ) ] (7 1) = s = = calculate s xg = s/ n = 0.846/ 7 = 0.846/.65 = find appropriate t value, for c =.95 and 6 df =.4469 multiply s xg by t value = Gx ± t s xg = 3.39 ± or the interval.546 to Each of 9 cars in a sample is driven 0,000 miles, the gallons of fuel used recorded, and the fuel mileage calculated. For the sample mean fuel mileage Gx = 34.6 and s = 1.. Assuming that the distribution of fuel mileage is normally distributed, find a 90% confidence interval for the mileage to be expected from all cars of this make. s xg = s/ n = (1.)/3 = 0.4 α = (1.9)/ = 0.05 and d.f. = n 1 = 8 t = Gx ± t s xg = 34.6 ± ± or the interval to In a sample of 41 students who work, xg = and s = Find a 95% confidence interval for the average hours worked by all ASU students who work. s xg = s n = = for 40 degrees of freedom, t 95 =.011 confidence interval: ± ± 1.8
9 We wish to establish the average weight of a population of turkeys; we have chosen a sample of 36, weighed them and have the following results: Construct a 98% confidence interval for the population mean of these turkeys first, find t C =.438 next, find Gx and s: Gx = 14.5, s = 4.90 find s xg = s/ n = 4.90/6 = Gx ± t C s xg 14.5 ± ± 1.99 or 1.6 to 16.4 SAMPLING DISTRIBUTIONS FOR SMALL SAMPLES The t distribution is often thought of as primarily of value with small samples applies whenever population is known to be normal and σ unknown, no matter how small n footnote: who was "Student"? A pseudonym for William Gosset, an Irish brewmaster concerned with controlling biochemical processes in brewing with large samples, if population is not normal, we must rely on Central Limit Theorem And many statisticians and other practitioners will use z procedures with any sample of 30 or more: this is especially prevalent in older practice Another possibility: sample is small, so that CLT does not apply population is not normally distributed or the distribution is unknown Safest course is to take a larger sample and rely on CLT
10 Following schematic may be used to determine proper distribution to use in constructing confidence intervals. Population standard deviation known? Yes No Population normal? Population normal? Yes No Yes No Sample Size Sample Size z value n >= 30 n < 30 n >= 30 n < 30 NOTES: z or t (see note) ERROR t value z or t (see note) ERROR 1. For a nonnormal population and large samples, different practitioners may proceed differently. Some argue that the Central Limit Theorem justifies use of a z value in this case, while others feel that it is more appropriate to use a t value since that gives a less precise estimate (a wider confidence interval). For purposes of this course, use a t in such cases.. For small samples from nonnormal populations: there are techniques which can be used to derive an interval estimate in this case, but they are beyond the scope of this course.
11 CONFIDENCE INTERVALS III CONFIDENCE INTERVAL FOR THE POPULATION PROPORTION Purpose: to use the sample proportion, p s, as the basis of an interval estimate of the population proportion p Reminders: the sample proportion p s = x/n the sampling distribution of p has parameters E(p s ) = p σ ps = p (1 p)/n p s is normally distributed, so that probabilities are found by reference to the z table typically p is unknown, so that we must estimate σ ps by s ps = [p s (1 p s )]/n A confidence interval for p then will have the form p s ± z C s ps Examples: of 55 students in a sample, 45 work. Construct a 95% confidence interval for the proportion in the population who work. p s = 45/55 = 0.8 s ps = [.8 (1.8)]/55 = z C = ±1.96 confidence interval: 0.8 ± ± 0.10 We are 95% confident that in the population somewhere between 7% and 9% work. In a sample of 800 North Carolinians 51% express the intention to vote for Jesse Helms in the next election. Find a 98% confidence interval for the proportion in the population who intend to vote for Helms. p s = 51%; s ps = (51 49)/800 = then we have 51 ± = 51% ± 4.1% or the interval 46.9% to 55.1% from this, we can say, strictly and properly, "We are 95% sure that the proportion in the population who intend to vote Helms is within 4.1% of 51%." or, as we might loosely and a bit improperly put it, "Our survey shows that 51% of the population intend to vote Helms, and this result is accurate to within plus or minus 4%." the election is a tossup or too close to call.
12 Suppose same result with a sample of size n = 1600 s ps = 1.497, and s p z = =.9% confidence interval would be 51% ± 3% POINT: is the minor increase in precision worth the extra cost? FINDING THE NECESSARY SAMPLE SIZE IN PROPORTION PROBLEMS Since we have ± z C s ps, the estimate p s differs from p by at most that amount substituting the definition of s ps, the error is at most z C [p (1 p)]/n notice the use of p in the above expression; the concepts advanced here involve what we know about the sampling distribution before sampling begins for a given confidence level, this error can be reduced by increasing n in the last example above, we noted that doubling the sample size would reduce the error from 4% to 3% suppose we require e < 0.01, that is, accuracy to within ± 1%. How large must n be? the maximum error in the estimate: solve for n, giving n = p (1 e = e z C p) z C p ( 1 p) n
13 A major problem: p, the population proportion is unknown solution 1: assume p = 0.5 this will give largest possible value for n since p (1 p) reaches a maximum when p = 0.5 may result in an unnecessarily large and expensive sample solution : use other information do a pilot study on a small sample and use the resulting p s to estimate p previous experience or knowledge of other populations may give an approximate value for p lacks certainty of solution 1, but may result in somewhat smaller sample Examples: applying the formula above to solution 1, we have. 5 (1.5).33 = 0.01 n = 13,573 this is the sample size necessary to be absolutely sure that a 98% confidence interval is accurate to within ± 1% In the work example above, 95% confidence interval and sample of 55 gave accuracy of ±0.10. What sample size is necessary to hold the error to ±0.015 (1.5%)? solution 1: n = [( ) 1.96 ] = ; taking the next greatest integer, we have 469 solution : for n = 55, we had p s = 0.8. Take that as an estimate of the unknown p. Then n = [( ) 1.96 ] = or 50 using the pilotstudy approach reduces the required sample size by more than 1,749 and might save a considerable amount of money A footnote: in most proportion problems, it doesn t matter whether you use percentages or decimal fractions, as long as you keep them straight. In the samplesize formula above, however, you must use decimal fractions. To use percentages, substitute 100 for 1, so the formula becomes n = [p* (100 p*) z ] e* where p* and e* are defined as percentages. THE z VS. THE t DISTRIBUTION In constructing confidence intervals, use the z distribution whenever the population standard deviation σ is known AND the population is known to normally distributed you wish to calculate a confidence interval for a proportion rule of thumb: n p 5 AND n (1 p) 5 for sufficiently accurate approximation In constructing confidence intervals, use the t distribution if the population is known to be normally distributed AND the population standard deviation σ is UNKNOWN: this holds for any sample size if the population s distribution is NOT normal AND the sample size is at least 30 AND the population standard deviation σ is UNKNOWN
Statistical Inference
Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this
More informationAn interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0.
Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.
More information5.1 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 informationWeek 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 informationSampling and Hypothesis Testing
Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus
More informationConfidence level. Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%)
Confidence Interval A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated
More information4. Introduction to Statistics
Statistics for Engineers 41 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation
More informationWhen σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16
8.3 ESTIMATING A POPULATION MEAN When σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16 Task was to estimate the mean when we know that the situation is Normal
More information4. 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 information6.4 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 informationEstimation and Confidence Intervals
Estimation and Confidence Intervals Fall 2001 Professor Paul Glasserman B6014: Managerial Statistics 403 Uris Hall Properties of Point Estimates 1 We have already encountered two point estimators: th e
More informationPoint 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 informationChapter 7 Review. Confidence Intervals. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Chapter 7 Review Confidence Intervals MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Suppose that you wish to obtain a confidence interval for
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationHypothesis Testing. Bluman Chapter 8
CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 81 Steps in Traditional Method 82 z Test for a Mean 83 t Test for a Mean 84 z Test for a Proportion 85 2 Test for
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationStats for Strategy Exam 1 InClass Practice Questions DIRECTIONS
Stats for Strategy Exam 1 InClass Practice Questions DIRECTIONS Choose the single best answer for each question. Discuss questions with classmates, TAs and Professor Whitten. Raise your hand to check
More informationMargin of Error When Estimating a Population Proportion
Margin of Error When Estimating a Population Proportion Student Outcomes Students use data from a random sample to estimate a population proportion. Students calculate and interpret margin of error in
More informationMath 251, Review Questions for Test 3 Rough Answers
Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,
More informationUnit 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 informationBasic Statistics. Probability and Confidence Intervals
Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the
More informationConfidence 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 informationObjectives. 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 informationConstructing 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 informationEstimation of the Mean and Proportion
1 Excel Manual Estimation of the Mean and Proportion Chapter 8 While the spreadsheet setups described in this guide may seem to be getting more complicated, once they are created (and tested!), they will
More informationHYPOTHESIS TESTING III: POPULATION PROPORTIONS, ETC.
HYPOTHESIS TESTING III: POPULATION PROPORTIONS, ETC. HYPOTHESIS TESTS OF POPULATION PROPORTIONS Purpose: to determine whether the proportion in the population with some characteristic is or is not equal
More informationHypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam
Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests
More informationLecture 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 informationProbability and Statistics Lecture 9: 1 and 2Sample Estimation
Probability and Statistics Lecture 9: 1 and Sample Estimation to accompany Probability and Statistics for Engineers and Scientists Fatih Cavdur Introduction A statistic θ is said to be an unbiased estimator
More informationSampling Distribution of a Sample Proportion
Sampling Distribution of a Sample Proportion From earlier material remember that if X is the count of successes in a sample of n trials of a binomial random variable then the proportion of success is given
More informationStandard Deviation Calculator
CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or
More informationAn Introduction to Sampling
An Introduction to Sampling Sampling is the process of selecting a subset of units from the population. We use sampling formulas to determine how many to select because it is based on the characteristics
More informationModule 5 Hypotheses Tests: Comparing Two Groups
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
More informationChapter 8: Introduction to Hypothesis Testing
Chapter 8: Introduction to Hypothesis Testing We re now at the point where we can discuss the logic of hypothesis testing. This procedure will underlie the statistical analyses that we ll use for the remainder
More informationPractice Exam. 1. What is the median of this data? A) 64 B) 63.5 C) 67.5 D) 59 E) 35
Practice Exam Use the following to answer questions 12: A census is done in a given region. Following are the populations of the towns in that particular region (in thousands): 35, 46, 52, 63, 64, 71,
More informationChapter 3: Data Description Numerical Methods
Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,
More informationLesson 17: Margin of Error When Estimating a Population Proportion
Margin of Error When Estimating a Population Proportion Classwork In this lesson, you will find and interpret the standard deviation of a simulated distribution for a sample proportion and use this information
More informationStatistics 100 Binomial and Normal Random Variables
Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random
More informationConfidence 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 informationNeed 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 informationExpected values, standard errors, Central Limit Theorem. Statistical inference
Expected values, standard errors, Central Limit Theorem FPP 1618 Statistical inference Up to this point we have focused primarily on exploratory statistical analysis We know dive into the realm of statistical
More informationChapter 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 informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationCALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 15 scale to 0100 scores When you look at your report, you will notice that the scores are reported on a 0100 scale, even though respondents
More information3.4 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 informationChapter 7. Section Introduction to Hypothesis Testing
Section 7.1  Introduction to Hypothesis Testing Chapter 7 Objectives: State a null hypothesis and an alternative hypothesis Identify type I and type II errors and interpret the level of significance Determine
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.23, 2.1 Normal distribution Math 283 / October
More informationCentral Tendency and Variation
Contents 5 Central Tendency and Variation 161 5.1 Introduction............................ 161 5.2 The Mode............................. 163 5.2.1 Mode for Ungrouped Data................ 163 5.2.2 Mode
More information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationc. Construct a boxplot for the data. Write a one sentence interpretation of your graph.
MBA/MIB 5315 Sample Test Problems Page 1 of 1 1. An English survey of 3000 medical records showed that smokers are more inclined to get depressed than nonsmokers. Does this imply that smoking causes depression?
More informationChapter 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 informationA Short Guide to Significant Figures
A Short Guide to Significant Figures Quick Reference Section Here are the basic rules for significant figures  read the full text of this guide to gain a complete understanding of what these rules really
More information9.1 (a) The standard deviation of the four sample differences is given as.68. The standard error is SE (ȳ1  ȳ 2 ) = SE d  = s d n d
CHAPTER 9 Comparison of Paired Samples 9.1 (a) The standard deviation of the four sample differences is given as.68. The standard error is SE (ȳ1  ȳ 2 ) = SE d  = s d n d =.68 4 =.34. (b) H 0 : The mean
More informationProb & Stats. Chapter 9 Review
Chapter 9 Review Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More information7 Confidence Intervals
blu49076_ch07.qxd 5/20/2003 3:15 PM Page 325 c h a p t e r 7 7 Confidence Intervals and Sample Size Outline 7 1 Introduction 7 2 Confidence Intervals for the Mean (s Known or n 30) and Sample Size 7 3
More informationPreAlgebra Lecture 6
PreAlgebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals
More informationTwosample inference: Continuous data
Twosample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with twosample inference for continuous data As
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 OneWay 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 informationThe Margin of Error for Differences in Polls
The Margin of Error for Differences in Polls Charles H. Franklin University of Wisconsin, Madison October 27, 2002 (Revised, February 9, 2007) The margin of error for a poll is routinely reported. 1 But
More informationCA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction
CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous
More informationChapter 7  Practice Problems 1
Chapter 7  Practice Problems 1 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Define a point estimate. What is the
More informationTypes of Error in Surveys
2 Types of Error in Surveys Surveys are designed to produce statistics about a target population. The process by which this is done rests on inferring the characteristics of the target population from
More informationStat 5102 Notes: Nonparametric Tests and. confidence interval
Stat 510 Notes: Nonparametric Tests and Confidence Intervals Charles J. Geyer April 13, 003 This handout gives a brief introduction to nonparametrics, which is what you do when you don t believe the assumptions
More information8 6 X 2 Test for a Variance or Standard Deviation
Section 8 6 x 2 Test for a Variance or Standard Deviation 437 This test uses the Pvalue method. Therefore, it is not necessary to enter a significance level. 1. Select MegaStat>Hypothesis Tests>Proportion
More informationStatistics 641  EXAM II  1999 through 2003
Statistics 641  EXAM II  1999 through 2003 December 1, 1999 I. (40 points ) Place the letter of the best answer in the blank to the left of each question. (1) In testing H 0 : µ 5 vs H 1 : µ > 5, the
More informationReview of basic statistics and the simplest forecasting model: the sample mean
Review of basic statistics and the simplest forecasting model: the sample mean Robert Nau Fuqua School of Business, Duke University August 2014 Most of what you need to remember about basic statistics
More informationMath and FUNDRAISING. Ex. 73, p. 111 1.3 0. 7
Standards Preparation Connect 2.7 KEY VOCABULARY leading digit compatible numbers For an interactive example of multiplying decimals go to classzone.com. Multiplying and Dividing Decimals Gr. 5 NS 2.1
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationAP * Statistics Review
AP * Statistics Review Confidence Intervals Teacher Packet AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this
More informationGraphical and Tabular. Summarization of Data OPRE 6301
Graphical and Tabular Summarization of Data OPRE 6301 Introduction and Recap... Descriptive statistics involves arranging, summarizing, and presenting a set of data in such a way that useful information
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
STATISTICS/GRACEY EXAM 3 PRACTICE/CH. 89 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the Pvalue for the indicated hypothesis test. 1) A
More informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 101634501 Probability and Statistics for Engineers Winter 20102011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete
More informationStatistical Inference and ttests
1 Statistical Inference and ttests Objectives Evaluate the difference between a sample mean and a target value using a onesample ttest. Evaluate the difference between a sample mean and a target value
More informationStatistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!
Statistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!) Part A  Multiple Choice Indicate the best choice
More informationConfidence 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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) ±1.88 B) ±1.645 C) ±1.96 D) ±2.
Ch. 6 Confidence Intervals 6.1 Confidence Intervals for the Mean (Large Samples) 1 Find a Critical Value 1) Find the critical value zc that corresponds to a 94% confidence level. A) ±1.88 B) ±1.645 C)
More informationMind on Statistics. Chapter 10
Mind on Statistics Chapter 10 Section 10.1 Questions 1 to 4: Some statistical procedures move from population to sample; some move from sample to population. For each of the following procedures, determine
More informationSampling Distribution of a Normal Variable
Ismor Fischer, 5/9/01 5.1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,
More informationSIMPLE REGRESSION ANALYSIS
SIMPLE REGRESSION ANALYSIS Introduction. Regression analysis is used when two or more variables are thought to be systematically connected by a linear relationship. In simple regression, we have only two
More informationDef: 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 informationStandard 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 informationCarolyn Anderson & Youngshil Paek (Slides created by Shuai Sam Wang) Department of Educational Psychology University of Illinois at UrbanaChampaign
Carolyn Anderson & Youngshil Paek (Slides created by Shuai Sam Wang) Department of Educational Psychology University of Illinois at UrbanaChampaign Key Points 1. Data 2. Variable 3. Types of data 4. Define
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationJointly Distributed Random Variables
Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................
More informationSimple 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 informationHypothesis testing for µ:
University of California, Los Angeles Department of Statistics Statistics 13 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative
More informationChapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The PreTax Position
Chapter 27: Taxation 27.1: Introduction We consider the effect of taxation on some good on the market for that good. We ask the questions: who pays the tax? what effect does it have on the equilibrium
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationRevision Notes Adult Numeracy Level 2
Revision Notes Adult Numeracy Level 2 Place Value The use of place value from earlier levels applies but is extended to all sizes of numbers. The values of columns are: Millions Hundred thousands Ten thousands
More informationExtending Hypothesis Testing. pvalues & confidence intervals
Extending Hypothesis Testing pvalues & confidence intervals So far: how to state a question in the form of two hypotheses (null and alternative), how to assess the data, how to answer the question by
More informationMT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...
MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 20042012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................
More information7. Normal Distributions
7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bellshaped
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationChapter 9, Part A Hypothesis Tests. Learning objectives
Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More information