Large Sample Confidence Interval

8 Key excerpts on "Large Sample Confidence Interval"

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  eBook - ePub

  Introductory Statistics

  • Alandra Kahl(Author)
  • 2008(Publication Date)
  • Bentham Science Publishers

    (Publisher)

  Estimation Alandra Kahl

  1 Department of Environmental Engineering, Penn State Greater Allegheny, PA 15132, USA

  Abstract

  It is critically important for researchers to select the correct sample size to determine the population means. By understanding the difference between small and large sample sets, researchers can then construct intervals of confidence that assist in determining the population means. Confidence intervals are the margins of the error present within the dataset and are included to show the confidence of the researcher in the integrity of their dataset. Common confidence intervals are 90%, 95%, and 99%. The Z confidence level is calculated to show where the mean is likely to fall, and the T confidence level is used only when the sample size is smaller than 30 samples.

  Keywords: Interval, Population mean, Sample estimation.

  INTRODUCTION

  By including sample sizes in their data analysis, researchers can simply describe the uncertainty associated with their dataset as well as to show the error margins present within the set. The distinctions between large and small sample sizes are typically set to encompass both the upper and lower ends of the dataset as well as the margins of error present within the data set. Confidence levels are also included with confidence intervals to show agreement with the assumptions of the dataset. Common confidence levels are 90%, 95%, and 99%. These levels are used to show the surety of the researchers in their measurements as well as to assist in predicting the characteristics of future datasets based on current understandings to ensure that the correct size sample is being used to determine the population means.

  Construction of Confidence Intervals

  The statistical inference technique refers to obtaining conclusions from data using statistical methods. As a result, the most crucial things to consider are testing hypotheses and concluding. As a branch of statistics, estimate theory is responsible for extracting parameters from data that have been contaminated by noise [46

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  Applied Biostatistics for the Health Sciences

  • Richard J. Rossi(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Confidence Interval A confidence interval is an interval estimator that produces an inter-val of estimates that captures the true value of an unknown parameter with a prespecified probability. Confidence Level The confidence level is the prespecified probability for which a con-fidence interval procedure will capture the true value of the parameter being estimated. Interval Estimator An interval estimator is a rule that produces an interval of estimates of an unknown parameter. Large Sample Confidence Interval for 𝝁 A large sample (1 − ? ) × 100% confidence interval for a mean ? is ̄ ? ± ? crit × ? √ ? Large Sample Confidence Interval for p A large sample (1 − ? ) × 100% confidence interval for a proportion ? is ˆ ? ± ? crit × √ ˆ ? (1 − ˆ ? ) ? Large Sample Confidence Interval for 𝒑 𝑿 − 𝒑 𝒀 A large sample (1 − ? ) × 100% confidence interval for the difference of two proportions, ? ? − ? ? , is ˆ ? ? − ˆ ? ? ± ? crit × √ ˆ ? ? (1 − ˆ ? ? ) ? ? + ˆ ? ? (1 − ˆ ? ? ) ? ? Large Sample Confidence Interval for the Odds Ratio A large sample (1 − ? ) × 100% confidence interval for the odds ratio is ? = ? ln( ˆ ?? )− ? crit × ?? to ? = ? ln( ˆ ?? )+ ? crit × ?? . Large Sample Confidence Interval for the Relative Risk A large sample (1− ? )×100% confidence interval for the relative risk is ? = ? ln( ˆ ?? )− ? crit × ?? to ? = ? ln( ˆ ?? )+ ? crit × ?? . Relative Risk The relative risk of contracting a disease or a having a particular condition for a risk factor is RR = ? ( disease | exposure to the risk factor ) ? ( disease | no exposure to the risk factor ) Small Sample Confidence Interval for 𝝁 A small sample (1 − ? ) × 100% confidence interval for a mean ? is ̄ ? ± ? crit × ? √ ? provided the underlying distribution is approximately normal.

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  Statistics

  Principles and Methods

  • Richard A. Johnson, Gouri K. Bhattacharyya(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Of course, this is contingent on the validity of the assumptions underlying the techniques—independent normal observations here. 3.2 Large Sample Confidence IntervalS FOR  Having established the basic concepts underlying confidence interval statements, we now turn to the more realistic situation for which the population standard deviation  is unknown. We require the sample size n to be large in order to dispense with the assumption of a normal population. The central limit theorem then tells us that X is nearly normal whatever the form of the population. Referring to the normal distribution of X in Figure 5 and the discussion accompanying Figure 2, 1 – α μ /2 – z n / μ σ α n / σ /2 α /2 α x /2 + z μ α Figure 5 Normal distribution of X. We again have the probability statement P [ X − z  ∕ 2  √ n <  < X + z  ∕ 2  √ n ] = 1 −  (Strictly speaking, this probability is approximately 1 −  for a nonnormal population.) Even though the interval ( X − z  ∕ 2  √ n , X + z  ∕ 2  √ n ) will include  with the probability 1 − , it does not serve as a confidence interval because it involves the unknown quantity . However, because n is large, replacing  ∕ √ n with its estimator S ∕ √ n does not appreciably affect the probability statement. Summarizing, we find that the Large Sample Confidence Interval for  has the form Estimate ± ( z value ) ( Estimated standard error ). 246 CHAPTER 8/DRAWING INFERENCES FROM LARGE SAMPLES Large Sample Confidence Interval for  When n is large, a 100( 1 −  )% confidence interval for  is given by ( X − z  ∕ 2 S √ n , X + z  ∕ 2 S √ n ) where S is the sample standard deviation. Example 6 Confidence Interval for the Daily Mean Time at Social Network Sites Most college students are enamored with social network sites.

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  eBook - ePub

  Statistics Essentials For Dummies

  • Deborah J. Rumsey(Author)
  • 2019(Publication Date)
  • For Dummies

    (Publisher)

  n whose confidence intervals contain the population parameter. When taking many random samples from a population, you know that some samples (in this case, 95% of them) will represent the population, and some won’t (in this case, 5% of them) just by random chance. Random samples that represent the population will result in confidence intervals that contain the population parameter (that is, they are correct); and those that do not represent the population will result in confidence intervals that are not correct.

  For example, if you randomly sample 100 exam scores from a large population, you might get more low scores than you should in your sample just by chance, and your confidence interval will be too low; or you might get more high scores than you should in your sample just by chance, and your confidence interval will be too high. These two confidence intervals won’t contain the population parameter, but with a 95% confidence level, this type of error (called sampling error ) should only happen 5% of the time.

  Confidence level (such as 95%) represents the percentage of all possible random samples of size n that typify the population and hence result in correct confidence intervals. It isn’t the probability of a single confidence interval being correct.

  Another way of thinking about the confidence level is to say that if the organization took a sample of 1,000 people over and over again and made a confidence interval from its results each time, 95 percent of those confidence intervals would be right. (You just have to hope that yours is one of those right results.)

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  Available until 7 Feb |Learn more

  Probability, Statistics, and Reliability for Engineers and Scientists

  • Bilal M. Ayyub, Richard H. McCuen(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 11 Confidence Intervals and Sample Size Determination

  In this chapter, we discuss the sampling variation of statistics, introduce confidence intervals as measures of the accuracy of statistics, demonstrate the computation of confidence intervals on the mean and variance, calculate the size of samples needed to estimate a mean with a stated level of accuracy, and discuss the fundamentals of quality control.

  CONTENTS

  • 11.1 Introduction
  • 11.2 General Procedure
  • 11.3 Confidence Intervals on Sample Statistics
  • 11.3.1 Confidence Interval for the Mean
  • 11.3.2 Factors Affecting a Confidence Interval and Sampling Variation
  • 11.3.3 Confidence Interval for Variance
  • 11.4 Sample Size Determination
  • 11.5 Relationship between Decision Parameters and Type I and II Errors
  • 11.6 Quality Control
  • 11.7 Applications
  • 11.7.1 Accuracy of Principal Stress
  • 11.7.2 Compression of Steel
  • 11.7.3 Sample Size of Organic Carbon
  • 11.7.4 Resampling for Greater Accuracy
  • 11.8 Simulation Projects
  • 11.9 Problems

  11.1 INTRODUCTION

  From a sample we obtain single-valued estimates such as the mean, the variance, a correlation coefficient, or a regression coefficient. These single-valued estimates represent our best estimate of the population values, but they are only estimates of random variables, and we know that they probably do not equal the corresponding true values. Thus, we should be interested in the accuracy of these sample estimates.

  If we are only interested in whether or not an estimate of a random variable is significantly different from a standard of comparison, we can use a hypothesis test. However, the hypothesis test only gives us a “yes” or “no” answer and not a statement of the accuracy of an estimate of a random variable that may be the object of our attention. A measure of the accuracy of a statistic may be of value as part of a risk analysis.

  In Example 9.3, a water-quality standard of 3 ppm was introduced for illustration purposes. The hypothesis test showed that the sample mean of 2.8 ppm was not significantly different from 3 ppm. The question arises: Just what is the true mean? Although the best estimate (i.e., expected value) is 2.8 ppm, values of 2.75 or 3.25 ppm could not be ruled out. Is the true value between 2 and 4 ppm, or is it within the range 2.75 to 3.25 ppm? The smaller range would suggest that we are more sure of the population value; that is, the smaller range indicates a higher level of accuracy. The higher level of accuracy makes for better decision making, and this is the reason for examining confidence intervals as a statistical tool.

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  eBook - PDF

  Essential Statistics

  • D.G. Rees(Author)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  The sample size, n — the larger the sample size, the smaller the error term and the smaller the width of the interval (other things being equal). 2. The variability of height (as measured by the standard deviation) — the more variable the height, the larger the standard deviation, the greater the error term, and the greater the width of the interval. 3. The level of confidence we wish to have that the population mean height does in fact lie within the specified interval — the greater the confidence, the greater the error term and the greater the width of the interval. This third factor, confidence, is such an important concept that we will devote the next section to it. Note The three statements above are quoted without proof. I hope that you can at least accept them as being 'intuitively reasonable’. 9.2 95% Confidence Intervals For the student height example, the population mean height, jjl , has a fixed numerical value at any given time. This value is unknown to us, but by taking a random sample of 27 heights, we want to specify an interval within which we are reasonably confident that this fixed unknown value lies. Suppose we decide that nothing less than 100% confidence will suffice, implying absolute certainty. Unfortunately, theory indicates that the 100% confidence interval is so wide that it is useless for all practical purposes. Instead, statisticians conventionally choose a 95% confidence level and calculate a 95% confidence interval for the population mean, jjl , Confidence Interval Estimation ■ 117 using formulae we shall introduce in the next sections. For the moment, it is important for you to understand what a confidence level of 95% means. It means that on 95% of occasions when such intervals are calculated the population mean will actually fall inside the interval we have calculated from the sample data. On the other 5% of occasions, it will fall outside the interval.

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  eBook - ePub

  Confidence Intervals

  • Michael Smithson(Author)
  • 2002(Publication Date)
  • SAGE Publications, Inc

    (Publisher)

  2. CONFIDENCE STATEMENTS AND INTERVAL ESTIMATES

  Let us return to the example confidence statement by the pollster, namely that she is 95% confident that the true percentage vote for a political candidate lies somewhere between 38% and 44%, on the basis of a sample survey from the voting population. Her requirements to make this statement are identical to those for estimating a population parameter with a sample statistic, namely a statistical model of how the sample statistic is expected to behave under random sampling error. In this example, the population parameter is the percentage of the voters who will vote for the candidate, but we could be estimating any statistic (e.g., a mean or the correlation between two variables).

  Let us denote the population parameter by θ, whose value is unknown. We may define confidence intervals for values of θ given a confidence level of 100(1 – α)%, where α lies between 0 and 1, and a sample size of N. Confidence intervals may have an upper limit or a lower limit, or both. A 100(1 – α)% upper confidence limit (U) is a value that, under repeated random samples of size N, may be expected to exceed θ’s true value 100(1 – α)% of the time. A 100(1 – α)% lower confidence limit (L) is a value that, under repeated random samples of size N, may be expected to fall below θ’s true value 100(1 – α)% of the time. The traditional two-sided confidence interval uses lower and upper limits that each contain θ’s true value 100(1 – α/2)% of the time, so that together they contain θ’s true value 100(1 – α)% of the time. The interval often is written as [L, U], and sometimes writers will express the interval and its confidence level by writing Pr(L < θ < U) = 1 – α.

  The limits L and U are derived from a sample statistic (often this statistic is the sample estimate of θ) and a sampling distribution that specifies the probability of getting each possible value that the sample statistic can take. This means that L and U also are sample statistics, and they will vary from one sample to another. To illustrate this derivation, we will turn to the pollster example and use the proportion of votes instead of the percentage. This conversion will enable us to use the normal distribution as the sampling distribution of the observed proportion, P. Following traditional notation that uses Roman letters for sample statistics and Greek letters for population parameters, we will denote the sample proportion by P and the population proportion by Π. It is customary for statistics textbooks to state that for a sufficiently large sample and for values of Π not too close to 0 or 1, the sampling distribution of a proportion may be adequately approximated by a normal distribution with a mean of Π and an approximate estimate of the standard deviation sp

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  eBook - ePub

  Social and Behavioral Statistics

  A User-Friendly Approach

  • Steven P. Schacht, Jeffery E. Aspelmeier(Authors)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  t distribution), and alpha levels; taken all together, these terms equal a confidence interval. This chapter also explores a statistical technique that tells the sample size required for a stated margin of error. While all these new terms may seem somewhat overwhelming, in application they really are quite easy to both calculate and understand. Quite simply, what confidence intervals do is estimate population parameters. While an array of different confidence intervals exists to estimate nearly every conceivable population parameter, to keep things simple only estimates of population means are discussed in this chapter.

  Samples

  Before undertaking any actual calculations for confidence intervals, we should review briefly what samples and statistics enable us to do. To assist in this discussion, below is the figure that initially appeared in Chapter 1 (Figure 8.1 ). This figure is important to this chapter’s material because it points out the two things that samples and corresponding statistics do with regard to population parameters: (1) estimate, and (2) hypothesis test. The first, estimates of population parameters, is what this chapter is all about, while hypothesis testing is largely addressed by chapters 9 through 13 .

  Figure 8.1 Population Parameters/Sample Statistics

  To this point, the discussion primarily has been concerned with two different types of sample statistics: means and standard deviations. Until now, however, we have had no way to assess how accurate these and other statistics are in terms of the population parameters they estimate. That is, while sample means, standard deviations, and other descriptive statistics are the best estimates we have for each corresponding parameter, these figures by themselves tell us nothing about how accurate they are. Accuracy, in this context, means how much the sample statistic potentially deviates from the parameter it is estimating.

  This is exactly what confidence intervals do; they enable us to determine the accuracy of our initial estimates. To accomplish this, information from the population and the sample (or, more typically, just from the sample) is used to calculate the given estimate’s accuracy. Moreover, and building upon the material discussed in the previous two chapters, confidence intervals also make estimates of accuracy in terms of probability values. In sum, confidence intervals are probability estimates of the true parameter value in terms of its occurrence between constructed boundaries.

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