Mathematics
Confidence Intervals
Confidence intervals are a statistical tool used to estimate the range within which a population parameter, such as a mean or proportion, is likely to lie. They provide a measure of the uncertainty associated with the estimate and are typically expressed as a range of values with an associated level of confidence, often 95%. Confidence intervals are widely used in hypothesis testing and decision making in research and data analysis.
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10 Key excerpts on "Confidence Intervals"
eBook - PDF- Timothy DelSole, Michael Tippett(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
3 Confidence Intervals When possible, quantify findings and present them with appropriate indicators of measurement error or uncertainty (such as Confidence Intervals). Avoid relying solely on statistical hypothesis testing, such as P values, which fail to convey important information about effect size and precision of estimates. 1 International Committee of Medical Journal Editors (2019) A major goal in statistics is to make inferences about a population. Typically, such inferences are in the form of estimates of population parameters; for instance, the mean and variance of a normal distribution. Estimates of population parameters are imperfect because they are based on a finite amount of data. Therefore, when reporting the estimated value of a population parameter, it is helpful to report its uncertainty. In fact, the estimate itself is almost meaningless without an indication about how far the estimate may be from the population value. A confidence interval provides a way to quantify uncertainty in parameter estimates. A confidence interval is a random interval that encloses the population value with a specified probability. Confidence Intervals are related to hypothesis tests about population parameters. Specifically, for a given hypothesis about the value of a parameter, a test at the 5% significance level would accept the hypothesis if the 95% confidence interval contained the hypothesized value. While hypothesis tests merely give a binary deci- sion – accept or reject – Confidence Intervals give a sense of whether the decision was a “close call” or a “miss by a mile.” This chapter constructs a confidence interval for a difference in means, a ratio of variances, and a correlation coefficient. These Confidence Intervals assume the population has a normal distribution. If the popu- lation is not Gaussian, or the quantity being inferred is complicated, then bootstrap methods offer an important alternative approach, as discussed at the end of this chapter.
eBook - PDF- Sally Caldwell(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
Learning to navigate your way through various approaches to the same type of question, different systems of symbolic notation, or encounters with personal preferences can provide an added boost to your overall level of statistical understanding. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 150 CHAPTER 6 Confidence Intervals Key Terms confidence interval for the mean family of t distributions confidence interval for a proportion level of confidence estimate of the standard error of margin of error the mean CourseMate brings course concepts to life with interac-tive learning, quizzing, flashcards, extra problem sets, and exam prepara-tion tools that support Statistics Unplugged. Logon and learn more at www.cengagebrain.com , which is your gateway to all of this text’s compli-mentary and premium resources. Chapter Problems Fill in the blanks, calculate the requested values, or otherwise supply the correct answer. General Thought Questions 1. A confidence interval for the mean is calculated by adding and subtracting a value to and from the sample . 2. The purpose of constructing a confidence interval for the mean is to the true value of the population mean, based upon the mean of a . 3. A confidence interval for the mean is an interval within which you believe the of the population is located. 4. As the level of confidence increases, the precision of your estimate . 5. There is a(n) relationship between level of confidence and pre-cision of the estimate.
eBook - PDFProbability and Statistics for STEM
A Course in One Semester
- E.N. Barron, J.G. Del Greco, E. N. Barron, J. G. Del Greco(Authors)
- 2022(Publication Date)
- Springer(Publisher)
77 C H A P T E R 4 Confidence and Prediction Intervals This chapter introduces the concept of a statistical interval. The principal types of statistical intervals are Confidence Intervals and prediction intervals. Instead of using a single number (a point estimate) to estimate quantities like the mean, a statistical interval includes the point estimate and an interval around it (an interval estimate) quantifying the errors involved in the estimate. 4.1 Confidence Intervals FOR A SINGLE SAMPLE We will use # to denote a parameter of interest in the pmf or pdf which we are trying to estimate. Normally this is the population mean or SD , or the population binomial proportion p: The following is the precise definition of what it means to be a confidence interval for the parameter # . We give the definition for a continuous random variable X with pdf f X .xI #/. The definition if X is discrete is similar. Definition 4.1 Let X 1 ; X 2 ; : : : ; X n be a random sample from a random variable X with pdf f X .xI #/. Given 0 < ˛ < 1; a 100.1 ˛/% confidence interval for # is an open interval with random endpoints of the form l.X 1 ; X 2 ; : : : ; X n / and u.X 1 ; X 2 ; : : : ; X n / such that P .l .X 1 ; X 2 ; : : : ; X n / < # < u .X 1 ; X 2 ; : : : ; X n // D 1 ˛: The percentage 100.1 ˛/% is called the confidence level of the interval. In the remainder of the book we will use CI to denote Confidence Interval. Remark 4.2 The value of ˛ sets the confidence level, and ˛ represents the probability the interval .l; u/ does not contain the parameter #: The probability that the value of the unknown parameter # lies in the interval can be adjusted depending on our choice of ˛. For example, if we would like the probability to be 0:99 (a confidence level of 99%), then we would choose ˛ D 0:01. If it is acceptable that the probability be only 0:90 (a confidence level of 90%), then we can instead choose ˛ D 0:10.
eBook - PDF- Maria Dolores Ugarte, Ana F. Militino, Alan T. Arnholt(Authors)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 8 Confidence Intervals 8.1 Introduction In Chapter 7, techniques to find point estimators, such as the method of moments and maximum likelihood, were introduced as well as were criteria to evaluate the “goodness” of an estimator; however, even the most e ffi cient unbiased estimator is not likely to estimate the population parameter exactly. Further, a point estimate provides no information about the precision or reliability of the estimate. Consequently, the construction of an interval estimate or confidence interval ( CI ), where the user can control the precision (width) of the interval as well as the reliability (confidence) that the true parameter will be found in the confidence interval, is desirable. A (1 -↵ ) confidence interval for a parameter ✓ , denoted CI 1 -↵ ( ✓ ), is constructed by first selecting a confidence level, denoted by (1 -↵ ) and typically expressed as a percentage, (1 -↵ ) · 100%. The confidence level is simply a measure of the degree of reliability in the procedure used to construct the confidence interval. Typical confidence levels are 90%, 95%, or 99%. A confidence level of 99% implies that 99% of all samples would provide Confidence Intervals that would contain ✓ . Clearly, it is desirable to have a high degree of reliability. Unfortunately, with increased reliability, the width of the confidence interval increases. So, the goal is to construct a confidence interval with a width the practitioner finds useful while maintaining a degree of reliability that is as high as possible. The relationship between the width and confidence level in a confidence interval will become clearer once some actual confidence interval formulas are examined. The confidence interval has two limits, a lower limit denoted by L ( X ) and an upper limit denoted by U ( X ). The confidence level is defined as P ( ✓ 2 ⇥ L ( X ) , U ( X ) ⇤) . That is, an interval should be constructed such that P ( L ( X ) ✓ U ( X ) ) = 1 -↵ .
eBook - PDF- 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.
eBook - PDF- Ken Black, Ignacio Castillo, Amy Goldlist, Timothy Edmunds(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
326 CHAPTER 10 Confidence Intervals deviation). In the case of proportions, identifying the continuous range of population propor- tions that contain a particular sample proportion within their (for example) 95% probability interval is complicated by two factors: • The range of possible sample proportions that can be generated is discrete. • The standard deviation of the sampling distribution of the proportion depends on the population proportion—this means that different population proportions have different interval widths for the same confidence level. A number of different statistical techniques are used to address these complications (all with the goal of producing Confidence Intervals that achieve the nominal rate of success; for example, 95% of Confidence Intervals constructed at the 95% confidence level contain the true population proportion). In this text, we present only the simplest commonly used approach. We use a fixed interval width determined by using the confidence level and the sample pro- portion with the normal approximation to the binomial distribution. In other words, we use the same method as we used for Confidence Intervals of the mean. To make explicit the approximation that we are making, we arrive at the formula for the confidence interval by rearranging the z score formula above: z p n ˆ (1 ) / 2 π π π ± = − − α p z n ˆ (1 ) / 2 π π π = ± × − α Then we eliminate the unknown population proportion ( π ) from the right-hand side of the equation by approximating it with the sample proportion ( p ˆ ): p z p p n ˆ ˆ (1 ˆ ) / 2 π = ± × − α Assumptions The method presented in this text for constructing Confidence Intervals of proportions makes use of some simplifying approximations (approximating the binomial distribution with the normal distribution, and approximating the standard error of the proportion across the range of possible population proportions with the standard error of the proportion at exactly the sample proportion).
- Robert M. Leekley(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Third, a 95% confidence interval does not imply that we have done something wrong 5% of the time. Assuming that we do everything right, we will still be unlucky 5% of the time. Rather, a 95% confidence interval is simply an estimate of an unknown population parameter generated in a way that will be correct 95% of the time and incorrect 5% of the time. As a philosophical aside, some authors (and instructors) object to saying that a particular 95% confidence interval has a 0.95 probability of being right on the grounds that, once it is taken, it is either right or it is not. However, I will not avoid such language. An analogy may help explain the issue. Suppose we were gamblers, and I offered you a bet on the outcome from the flip of a fair coin; to ensure fairness a third person will flip the coin. A fair coin lands heads 50% of the time. Hence, 0.50 is the probability we would each use in deciding what bets we were willing to make. Now, suppose the third person flips the coin, but hides the result. What is the probability that the result is heads now? In one sense, either one or zero; the result is either heads or it is not. Still, since we have not seen the coin, we do not know what came up. In this sense, nothing has changed. And 0.50 is still the probability we would have to use in deciding what bets we were willing to make. Now, suppose that, instead of flipping a coin, the third person is going to estimate a confidence interval in a way that will be right 95% of the time. Then 0.95 is the probability we would each use in deciding what bets we were willing to make. Now, suppose the third person actually estimates the interval. What is the probability that it is right now? In one sense, either one or zero; the result is either right or it is not. Still, we do not know what came up. In this sense, nothing has changed. And 0.95 is still the probability we would have to use in deciding what bets we were willing to make.
eBook - PDFStatistics
Unlocking the Power of Data
- Robin H. Lock, Patti Frazer Lock, Kari Lock Morgan, Eric F. Lock, Dennis F. Lock(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
214 C H A P T E R 3 Confidence Intervals 3.1 SAMPLING DISTRIBUTIONS In Chapter 1 we discuss data collection: methods for obtaining sample data from a population of interest. In this chapter we begin the process of going in the other direction: using the information in the sample to understand what might be true about the entire population. If all we see are the data in the sample, what conclusions can we draw about the population? How sure are we about the accuracy of those conclusions? Recall from Chapter 1 that this process is known as statistical inference. Statistical Inference Statistical inference is the process of drawing conclusions about the entire population based on the information in a sample. Data Collection Statistical Inference Population Sample Statistical inference uses sample data to understand a population Population Parameters and Sample Statistics To help identify whether we are working with the entire population or just a sample, we use the term parameter to identify a quantity measured for the population and statistic for a quantity measured for a sample. Parameters vs Statistics A parameter is a number that describes some aspect of a population. A statistic is a number that is computed from the data in a sample. As we saw in Chapter 2, although the name (such as “mean” or “proportion”) for a statistic and parameter is generally the same, we often use different notation to distinguish the two. For example, we use (mu) as a parameter to denote the mean for a population and x as a statistic for the mean of a sample. Table 3.1 summarizes Table 3.1 Notation for common parameters and statistics Population Parameter Sample Statistic Mean x Standard deviation s Proportion p p Correlation r Slope (regression) 1 b 1 3.1 Sampling Distributions 215 common notation for some population parameters and corresponding sample statis- tics. The notation for each should look familiar from Chapter 2.
- Jay Devore, Nicholas Farnum, Jimmy Doi, , Jay Devore, Nicholas Farnum, Jimmy Doi(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 316 CHAPTER 7 Estimation and Statistical Intervals of a 95% confidence level: Add 2 to both the number of successes and the number of fail-ures and then use the traditional formula. Do this for the data described in this exercise, and compare the resulting interval to the one you calculated in part (a). 27. Let H9266 1 and H9266 2 denote the proportion of successes in population 1 and population 2, respectively. An investigator sometimes wishes to calculate a confi-dence interval for the difference H9266 1 2 H9266 2 between these two population proportions. Suppose random samples of size n 1 and n 2 , respectively, are indepen-dently selected from the two populations, and let p 1 and p 2 denote the resulting sample proportions of successes. If the sample sizes are sufficiently large (apply the rule of thumb appropriate for a single proportion to each sample separately), the statistic p 1 2 p 2 has approximately a normal sampling dis-tribution with mean value H9266 1 2 H9266 2 and standard deviation 1 H9266 1 (1 2 H9266 1 ) y n 1 1 H9266 2 (1 2 H9266 2 ) y n 2 . The estimated standard deviation of this statistic results from replacing each π under the square root by the corresponding p . a. Use the foregoing facts to obtain a large-sample two-sided 95% confidence interval formula for estimating H9266 1 2 H9266 2 .
eBook - PDFStatistics, Student Solutions Manual
Unlocking the Power of Data
- Robin H. Lock, Patti Frazer Lock, Kari Lock Morgan, Eric F. Lock, Dennis F. Lock(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
This confidence interval represents female offspring, where the association is inconclusive. The confidence interval −8.38 to −0.60 has all negative values as plausible values for the slope of the population regression line, so this confidence interval indicates that we are 95% confident that there is a negative association between the two variables. This confidence interval represents male offspring. 3.79 (a) Interval is for the mean, not all students. (b) Interval is for the population mean, not the sample mean. (c) The interval is not uncertain, only whether or not it captures the population mean. (d) Interval is trying to capture the mean, not 95% of individual student pulse rates. (e) Scope of inference could apply to the mean pulse rate for all students at this college, but sample was not taken from all US college students. (f) The population mean pulse rate is a single fixed value. (g) Interval is for the population mean, not other sample means. Section 3.3 Solutions 3.81 (a) Yes. (b) Yes. (c) No. A bootstrap sample has the same sample size as the original sample. (d) No. The value 78 is not in the original sample. (e) Yes. (f) Yes. 3.83 The distribution appears to be centered near 25 so the point estimate is about 25. Using the 95% rule, we estimate that the standard error is about 3 (since about 95% of the values appear to be within 6 of the center). Thus our interval estimate is Statistic ± 2 · SE 25 ± 2(3) 25 ± 6 19 to 31 The parameter being estimated is a mean μ, and the interval 19 to 31 gives plausible values for the population mean μ. Answers may vary. 68 CHAPTER 3 3.85 The distribution appears to be centered near 6 so the point estimate is about 6. Using the 95% rule, we estimate that the standard error is about 4 (since about 95% of the values appear to be within 8 of the center).
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