Mathematics
Confidence Interval for Slope of Regression Line
A confidence interval for the slope of a regression line is a range of values within which the true slope is likely to fall. It provides a measure of the uncertainty associated with estimating the slope from a sample of data. The interval is calculated using the sample data and accounts for variability and potential error in the estimation process.
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eBook - ePubStatistical Misconceptions
Classic Edition
- Schuyler W. Huck(Author)
- 2015(Publication Date)
- Routledge(Publisher)
EstimationStudents can be left with many misconceptions after a standard introductory discussion [of confidence intervals]. One frequent misconception is that a 99% confidence interval is narrower than a 95% confidence interval. Students also tend to misinterpret the confidence interval by considering it to be a fixed quantity, not recognizing its dependence on the particular sample observed. Another common misconception is that the interval is a statement about the distribution of the data, rather than a set of possible “guesses” for the mean of the distribution generating the dataset.*7.1 Interpreting a Confidence Interval
The Misconception
If a 95% confidence interval (CI) has been created to estimate the numerical value of a population parameter, the probability of the parameter falling somewhere between the endpoints of that interval is equal to .95.Evidence of This Misconception *
The first of the following statements comes from an online document entitled “What Are Confidence Intervals?” The second statement comes from a statistics textbook.- In contrast [to tests of null hypotheses], confidence intervals provide a range about the observed effect size. This range is constructed in such a way that we know how likely it is to capture the true—but unknown—effect size. Thus the formal definition of a confidence interval is: “a range of values for a variable of interest [in our case, the measure of treatment effect] constructed so that this range has a specified probability of including the true value of the variable. The specified probability is called the confidence level, and the end points of the confidence interval are called the confidence limits.”
- A confidence interval is a range of values constructed to have a specific probability (the confidence) of including the population parameter. For example, suppose a random sample from a population produced anX̄ = 50. The 95% confidence interval might range from 45 to 55. That is, the probability that the interval 45–55 includes μ is .95.
eBook - ePub- Lawrence S. Aft(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
It is possible to estimate population parameters, such as the mean or the standard deviation, based on sample values. Naturally, how good the predictions are depends on how accurately the sample values reflect the values for the entire population. If a high level of confidence in the inference is desired, a large proportion of the populations should be observed. In fact, in order to achieve 100 percent confidence, one must sample the entire population. Because of the economic considerations typically involved in inspection, the selection of an acceptable confidence interval is usually seen as a trade-off between cost and confidence. Typically, 90, 95, and 99 percent confidence levels are used, with the 99.73 percent level used in certain quality control applications.If one desired to estimate, for example, the mean of a population, the ideal plan would be to measure every member of that population and then calculate the mean. Since this is not usually practical, a sample is generally measured and a sample mean calculated. This sample mean is called a point estimate of the statistic, because it is a single point, or value. How good is this estimate? It is sometimes difficult to answer this question. Although the point estimate maybe the best single estimate, no definitive statement of confidence can be made about it.In addition to stating the point estimate, it is often desirable to establish an interval within which the true population parameter may be expected with a certain degree of confidence to fall. For example, after measuring the tensile strength of steel rods, one might say that the best estimate of the average tensile strength is 732; the true mean is between 725 and 739. (If more confidence were desired for the same data, a wider interval would be specified. A 95 percent confidence interval would be even wider — for example, between 722.49 and 741.51.)A confidence interval is a range of values that has a specified likelihood of including the true value of a population parameter. It is calculated from sample calculations of the parameters.There are many types of population parameters for which confidence intervals can be established. Those important in quality control applications include means, proportions (percentages), and standard deviations.
eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
The general formula for this type of confidence interval is: y ^ i ± c × (s y ⋅ x × n + 1 n + (x i − x ¯) 2 ∑ (x − x ¯) 2) where y ^ i is the predicted value of Y calculated by inserting x i in the estimated regression equation, and c is the two-tailed critical value for the desired level of significance of the t -distribution with df = n − 2. For example, the 95% confidence interval for the predicted reaction time for a person aged 50.2. is: y ^ i ± c × (s y ⋅ x × n + 1 n + (x i − x ¯) 2 ∑ (x − x ¯) 2) = 261.833 ± 2.07 × 3821.224 22 × 24 + 1 24 + (50.2 − 58.742) 2 3624.278 = 261.833 ± 2.07 × 13.179 × 1.03. Thus, the 95% confidence limits for the predicted reaction time of a person aged 50.2 years are 233.722 and 289.944 milliseconds, and the 95% confidence interval is: CI 0.95 = 233.722 ≤ RT ≤ 289.944. Finally notice that the value of X does not necessarily have to be taken from one of the values being sampled. For example, we could find the 95% confidence interval for the predicted value of Y, for a person aged 70. The predicted value of Y estimated using the regression equation is: y ^ i = 161.433 + 2 × 70 = 301.448 and, thus, the 95% confidence interval for the predicted reaction time of a person aged 70 is: 301.448 ± 2.07 × 3821.224 22 × 24 + 1 24 + (70 − 58.742) 2 3624.278 = 301.448 ± 2.07 × 13.179 × 1.038. Hence, the limits of the 95% confidence interval for the reaction times are 273.141 and 329.755. 11.8 Why the term regression? Historically, the term regression is due to Frances Galton (1888). Studying the relationships between the height of parents and offspring, he noted that offspring of extreme parents also tended to be extreme, but not as extreme as the parents
- Bruce B. Frey(Author)
- 2018(Publication Date)
- SAGE Publications, Inc(Publisher)
Yi-Fang Wu Yi-Fang Wu Wu, Yi-FangConfidence Interval Confidence interval358 362Confidence Interval
The term confidence interval refers to an interval estimate that provides information about the uncertainty or the precision of estimation for some population parameter of interest. In statistical inference, confidence intervals are one method of interval estimation, and they are widely used in frequentist statistics. There are several ways to calculate confidence intervals. This entry first emphasizes the importance of confidence intervals by distinguishing interval estimation from point estimation. It then introduces a brief history of confidence intervals. The essentials of constructing confidence intervals are discussed, followed by a brief introduction to other types of intervals in the literature. Confidence intervals have been emphasized in the social and behavioral sciences, but they are often misinterpreted in statistical practice. Thus, the entry concludes with a discussion of common misunderstandings and misinterpretations of confidence intervals.Interval Estimation Versus Point Estimation
The purpose of inferential statistics is to infer properties about an unknown population parameter using data collected from samples. This is usually done by point estimation, one of the most common forms of statistical inference. Using sample data, point estimation involves the calculation of a single value, which serves as a best guess or best estimate of the unknown population parameter that is of interest.Instead of a single value, an interval estimate specifies a range within which the parameter is likely to lie. It provides a measure of accuracy of that single value. In frequentist statistics, confidence intervals are the most widely used method for providing information on location and precision of the population parameter, and they can be directly used to infer significance levels. Confidence intervals can have a one-sided or two-sided confidence bound. They are numerical intervals constructed around the estimate of the unknown population parameter. Such an interval does not directly infer a property of the parameter; instead, it indicates a property of the procedure, as is typical for a frequentist statistical procedure.
eBook - ePub- Michael Smithson(Author)
- 2002(Publication Date)
- SAGE Publications, Inc(Publisher)
2. CONFIDENCE STATEMENTS AND INTERVAL ESTIMATESLet 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
- Richard N. Landers(Author)
- 2018(Publication Date)
- SAGE Publications Ltd(Publisher)
In our case study, Chaitra is currently considering only point estimates – means. While this will give the executives her ‘best guess’ as to her production numbers, this does not capture the day-to-day or month-to-month variation in production, nor does it capture her confidence in that estimate. If she reports only means, the executives might expect her plants to consistently produce at that rate, in which case her estimate may be inaccurate.To address this problem, interval estimates contain information about both the sample mean and the sampling distribution from which it was drawn. They provide a range of realistic values for a particular parameter, given the sample that was used to estimate it. This provides more information about the parameter than a point estimate alone.Different methods are used to compute interval estimates depending on what information you have available about the sample and population. In this chapter, we’ll focus on two ways to compute the most common type of interval estimate for means: the confidence interval.6.2 Confidence Intervals (CI)
A confidence interval is a range of values within which we would expect sample statistics to fall, given a particular sample size, a particular parameter and a particular level of confidence. We often abbreviate them as ‘CI’. Confidence refers to a chosen level of probability that defines the width of the range; for example, with 95% confidence, we determine the range of values in which 95% of sample statistics should fall.The most typical levels of confidence that we see when computing confidence intervals are 95% (CI.95 ) and 99% (CI.99 ). So what do these levels imply?- 95% confidence interval: The means of 95% of samples of the same size drawn from a population with the given mean will fall between these two values.
- 99% confidence interval: The means of 99% of samples of the same size drawn from a population with the given mean will fall between these two values.
How do the sizes of these intervals compare with each other? If we include 99% of samples, we’ll be including the first 95% of samples plus an additional 4%. So the 99% interval will always be bigger.The choice of these values (95% vs 99%) is traditional
eBook - ePub- Robert H. Riffenburgh, Daniel L. Gillen(Authors)
- 2020(Publication Date)
- Academic Press(Publisher)
This leads to the term tolerance interval, because our tolerance in specifying the certainty around this range leads up to a 5% (1−0.95) error. Now suppose that we have constructed a model for Hct values on the basis of patient characteristics (e.g., height, weight, sex, and age) for a sampled population. Such a model might be constructed using linear regression as presented in Chapter 16, Multiple linear and curvilinear regression and multifactor analysis of variance. Given this model, we might want to predict the Hct value for a new randomly sampled patient with a given set of characteristics. However, since our prediction is based upon estimates from sampled data, we also wish to quantify how good our prediction is. Thus we wish to construct an interval around our prediction that quantifies the uncertainty in our “guess.” This interval is termed a prediction interval. A prediction interval is a range of likely values for a new observation that has a specified set of characteristics used to aid in the prediction. The precision of a prediction interval is defined by the probability that the new observation will fall in the specified range. For example, if we produce a 95% prediction interval for the Hct value of a new patient we are treating, the actual Hct of the patient should lie within the interval with probability 0.95. Another type of goal we might have is to characterize the range of plausible values for the true mean Hct value in a population of healthy individuals. Thus we desire a range of values that characterize the population mean Hct. This type of interval is termed a confidence interval, often denoted by CI. What do we mean by confidence? Given that the true population mean Hct in a specified population is a fixed quantity, we wish to control the probability that our interval will contain the population mean
eBook - ePubSocial Statistics
Managing Data, Conducting Analyses, Presenting Results
- Thomas J. Linneman(Author)
- 2017(Publication Date)
- Routledge(Publisher)
Chapter 8 Using Sample Slopes to Talk About Populations Inference and Regression This chapter covers.... . . one last probability distribution, this time using sample slopes. . . what the standard error of the slope really means. . . how to test a slope for statistical significance. . . the role of sample size in testing a slope for significance. . . how a researcher used regression to study the relationship between housing appreciation and support for social security. . . how a researcher used regression to study how family size and grades are related Introduction The goal of this chapter is to take the inferential techniques we’re already covered and apply them to the regression slopes you learned about in the previous chapter. Given that you are a seasoned pro at inference by this point, you will undoubtedly experience some déjà vu (or because you’re reading this, déjà lu). Inferential techniques have a lot in common with one another, so once you learn the overall goal of inference, it’s really just a variation on a theme. The goal of inference with slopes is to be able to claim that, in the population, the independent variable has an effect on the dependent variable. One More Sampling Distribution Chapter 4 included a distribution of hundreds of chi-square values. Chapter 5 had a distribution of hundreds of sample means. Chapter 6 involved a sampling distribution of sample mean differences. Here, I do roughly the same thing, but instead of chi-square values, or sample means, or differences between sample means, my building blocks will be sample slopes. I went back to my original hypothetical dataset of 100 grades I used in previous chapters. To each student’s grade, I added another piece of information: the percentage of classes that student attended during the semester
