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.
8 Key excerpts on "Confidence Intervals"
eBook - ePub- Lawrence S. Aft(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
...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. Confidence Intervals are generally presented in the following format: Point estimate of the population parameter ± (Confidence factor) (Measure variability) (Adjusting factor) A number of formulas have been developed for particular cases. The remaining sections of this chapter will present, illustrate, and interpret these formulas as they relate to quality control. Confidence Intervals for Means The way a confidence interval for means is developed depends on the sample size. As noted earlier, the larger the sample size, the better the estimate. Statistically, large samples are arbitrarily defined as those with 30 or more members. Small samples have fewer than 30 members. Either a z value or a t value is needed to represent the level of significance in confidence interval formulas when one is dealing with averages. Other probability distributions are used when dealing with Confidence Intervals for other statistics, such as standard deviations. z Values For large samples, the standard normal curve is used to specify the confidence. The value corresponding to the confidence is the z value. Example 5.1 What z value corresponds to an α of.05 or 95 percent confidence? Solution We are looking at a confidence interval for which the area outside the interval is symmetrically distributed...
- Bruce B. Frey(Author)
- 2018(Publication Date)
- SAGE Publications, Inc(Publisher)
...Yi-Fang Wu Yi-Fang Wu Wu, Yi-Fang Confidence Interval Confidence interval 358 362 Confidence 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...
eBook - ePubStatistical Misconceptions
Classic Edition
- Schuyler W. Huck(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...Chapter 7 Estimation Students 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 an X̄ = 50. The 95% confidence interval might range from 45 to 55...
eBook - ePub- Mustapha Abiodun Akinkunmi(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
...9 Confidence Intervals for Single Population Mean and Proportion When a statistic is computed from sample data to estimate a population parameter, we have to reduce our risk of estimation by constructing a range in which the exact value of the population parameter lies. This is due to the variability in the sample collected. This process gives us the assurance within a certain percentage that the estimate is capable of predicting the real value of the population of interest. This concept describes the confidence interval or confidence limit. In this chapter, we focus on constructing the confidence interval for both mean and proportion. 9.1 Point Estimates and Interval Estimates When a sample is drawn from a population, a single attribute obtained or quality computed from the measurement is called a statistic. When this statistic closely describes the characteristics obtained from the population, the estimate or a single value from the sampled data is referred to as a point estimate. For example, a human resource manager may want to assess the productivity of workers in a firm. The number of cases closed per month can be used as a metric to measure productivity. It is discovered that the average number of cases closed per month was 22—thus the value 22 is the point estimate. On the other hand, in order to know how well a sample statistic accurately describes a population, we can compute a range of values within which we are confident enough to say the true value of population parameter lies. The interval estimate offers a measure of exactness beyond that of the point estimate by giving an interval that contains plausible values. In most cases, we use a 95% confident limit for the estimation—this implies that we can say that our estimation about a population plausibly falls within the specified range of values in 95 out of 100 cases. For instance, a researcher may like to know the number of cars traveling a road on a daily basis...
eBook - ePub- Michael Smithson(Author)
- 2002(Publication Date)
- SAGE Publications, Inc(Publisher)
...Provide intervals for correlations and other coefficients of association or variation whenever possible (p. 599). Similar arguments were produced in guidelines for medical researchers about 12 years earlier. Bailar and Mosteller (1988) advise researchers to present quantified findings with appropriate indicators of measurement error or uncertainty (such as Confidence Intervals). They admonish them to avoid sole reliance on statistical hypothesis testing and point out that Confidence Intervals offer a more informative way to deal with the significance test than a simple p value does. Confidence Intervals for a mean or a proportion, for example, provide information about both level and variability. Weaknesses and Limitations Perhaps the most obvious difficulty with Confidence Intervals lies in how we interpret what the confidence statement means. Unfortunately, a 95% confidence interval for a proportion does not mean that the probability that the population proportion value lies inside that particular confidence interval is.95. The 95% confidence refers instead to the expected percentage of such intervals that would contain the population value, if we repeatedly took random samples of the same size from the same population under identical conditions. This interpretive quandary is indeed a problem, if for no other reason than the considerable temptation to apply the confidence level to the specific interval at hand. As Savage (1962, p. 98) remarked under a somewhat different context, “The only use I know for a confidence interval is to have confidence in it.” It is difficult to convey a confidence interval without at least implicitly leading the reader to apply the confidence level to the specific interval at hand. Cox and Hinkley (1974, p. 209) are inclined to regard this distinction as “legalistic.” However, yielding to this temptation may result in incoherent inferences, as several authors have shown...
- Richard Kay(Author)
- 2014(Publication Date)
- Wiley(Publisher)
...Chapter 3 Confidence Intervals and p -values 3.1 Confidence Intervals for a single mean 3.1.1 The 95 per cent confidence interval We have seen in the previous chapter that it is not possible to make a precise statement about the exact value of a population parameter, based on sample data, and that this is a consequence of the inherent variation in the sampling process. The confidence interval (CI) provides us with a compromise, rather than trying to pin down precisely the value of the mean μ or the difference between two means, μ 1 − μ 2, for example, we give a range of values, within which we are fairly certain that the true value lies. We will first look at the way we calculate the confidence interval for a single mean μ and then talk about its interpretation. Later in this chapter, we will extend the methodology to deal with μ 1 − μ 2 and other parameters of interest. In the computer simulation in Chapter 2, the first sample (n = 50) gave summary statistics (to 2 decimal places) as follows: The lower end of the confidence interval, the lower confidence limit, is then given by The upper end of the confidence interval, the upper confidence limit, is given by The interval, (79.02, 81.42), then forms the 95 per cent confidence interval. These data arose from a computer simulation where, of course, we know that the true mean μ is 80 mmHg, so we can see that the method has worked in the sense that μ is contained within the range 79.02 to 81.42 The second sample in the computer simulation gave the following summary statistics: mmHg and s = 4.50 mmHg – and this results in the 95 per cent confidence interval as (79.52, 82.02). Again, we see that the interval has captured the true mean. Now, look at all 100 samples taken from the normal population with μ = 80 mmHg. Figure 3.1 shows the 95 per cent Confidence Intervals plotted for each of the 100 simulations...
eBook - ePubSocial Statistics
Managing Data, Conducting Analyses, Presenting Results
- Thomas J. Linneman(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...Chapter 5 Using a Sample Mean or Proportion to Talk About a Population Confidence Intervals This chapter covers.... . . building a probability distribution of sample means. . . how to find and interpret the standard error of a sampling distribution. . . what the Central Limit Theorem is and why it is important. . . population claims and how to put them to the test. . . how to build and interpret Confidence Intervals. . . how a researcher used Confidence Intervals to study popular films. . . how researchers used Confidence Intervals to study Uber and traffic fatalities Introduction In this chapter we continue our exploration of inference, going through some procedures that you will find strikingly similar to those in the chi-square chapter. Whereas in the chi-square chapter we dealt with variables of the nominal or ordinal variety, here we deal with ratio-level variables. Our attention turns away from sample crosstabs and toward sample means (and, at the end of the chapter, proportions). But keep in mind that the inference goal remains the same: we will use sample means in order to make claims about population means. Just as we talked about the chi-square probability distribution, we’ll start this chapter with a distribution of sample means. Sampling Distributions of Sample Means Imagine a hypothetical class with 100 students in it. These students will serve as our population: it is the entire group of students in which we are interested. They get the following hypothetical grades: ■ Exhibit 5.1: Grades for a Population of 100 Students: Frequency Distribution Grade # of Students Receiving This Grade 1.0 1 1.1 1 1.2 1 1.3 2 1.4 2 1.5 2 1.6 2 1.7 3 1.8 3 1.9 3 2.0 4 2.1 4 2.2 5 2.3 6 2.4 7 2.5 8 2.6 7 2.7 6 2.8 5 2.9 4 3.0 4 3.1 3 3.2 3 3.3 3 3.4 2 3.5 2 3.6 2 3.7 2 3.8 1 3.9 1 4.0 1 Source: Hypothetical data. Here is a bar graph of this frequency...
eBook - ePub- Rafael Perera, Carl Heneghan, Douglas Badenoch(Authors)
- 2011(Publication Date)
- BMJ Books(Publisher)
...Probability and Confidence Intervals Probability is a measure of how likely an event is to occur. Expressed as a number, probability can take on a value between 0 and 1. Thus the probability of a coin landing tails up is 0.5. Probability: Event cannot occur = zero Event must occur = one Probability rules If two events are mutually exclusive then the probability that either occurs is equal to the sum of their probabilities. Mutually exclusive: a set of events in which if one happens the other does not. Tossing a coin: either it can be head or tails, it cannot be both. Independent event: event in which the outcome of one event does not affect the outcome of the other event. If two events are independent the probability that both events occur is equal to the product of the probability of each event. E.g. the probability of two coin tosses coming up heads: Probability distributions A probability distribution is one that shows all the possible values of a random variable. For example, the probability distribution for the possible number of heads from two tosses of a coin having both a head and a tail would be as follows: (head, head) = 0.25 (head, tail) + (tail, head) = 0.50 (tail, tail) = 0.25 Probability distributions are theoretical distributions that enable us to estimate a population parameter such as the mean and the variance. Parameter: summary statistic for the entire population The normal distribution The frequency of data simulates a bell-shaped curve that is symmetrical around the mean and exhibits an equal chance of a data point being above or below the mean. For most types of data, sums and averages of repeated observations will follow the normal distribution. Given that the distribution function is symmetrical about the mean, 68% of its area is within one standard deviation (σ) of the mean (μ) and 95% of the area is within (approximately) two standard deviations of μ...
