Mathematics
Confidence Interval for Population Mean
A confidence interval for a population mean is a range of values within which we are reasonably confident the true population mean lies. It is calculated using sample data and provides a measure of the uncertainty associated with estimating the population mean. The confidence level indicates the probability that the interval contains the true population mean.
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10 Key excerpts on "Confidence Interval for Population Mean"
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 - PDF- Barbara Illowsky, Susan Dean(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Remember that a confidence interval is created for an unknown population parameter like the population mean, μ. Confidence intervals for some parameters have the form: (point estimate – margin of error, point estimate + margin of error) The margin of error depends on the confidence level or percentage of confidence and the standard error of the mean. When you read newspapers and journals, some reports will use the phrase "margin of error." Other reports will not use that phrase, but include a confidence interval as the point estimate plus or minus the margin of error. These are two ways of expressing the same concept. 440 Chapter 8 | Confidence Intervals This OpenStax book is available for free at http://cnx.org/content/col11562/1.17 NOTE Although the text only covers symmetrical confidence intervals, there are non-symmetrical confidence intervals (for example, a confidence interval for the standard deviation). Have your instructor record the number of meals each student in your class eats out in a week. Assume that the standard deviation is known to be three meals. Construct an approximate 95% confidence interval for the true mean number of meals students eat out each week. 1. Calculate the sample mean. 2. Let σ = 3 and n = the number of students surveyed. 3. Construct the interval ⎛ ⎝ x ¯ − 2 ⋅ σ n , x ¯ + 2 ⋅ σ n ⎞ ⎠ . We say we are approximately 95% confident that the true mean number of meals that students eat out in a week is between __________ and ___________. 8.1 | A Single Population Mean using the Normal Distribution A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of x ¯ = 10 and we have constructed the 90% confidence interval (5, 15) where EBM = 5.
eBook - PDFBiostatistics
Basic Concepts and Methodology for the Health Sciences, 10th Edition International Student Version
- Wayne W. Daniel, Chad L. Cross(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
Probabilistic Interpretation In repeated sampling, from a normally distributed population with a known standard deviation, 100 1 a ð Þ percent of all intervals of the form " x z 1a=2 ð Þ s " x will in the long run include the population mean m. The quantity 1 a , in this case .95, is called the confidence coefficient (or confidence level), and the interval " x z 1a=2 ð Þ s " x is called a confidence interval for m. When 1 a ð Þ ¼ :95, the interval is called the 95 percent confidence interval for m. In the present example we say that we are 95 percent confident that the population mean is between 17.76 and 26.24. This is called the practical interpretation of Expression 6.2.2. In general, it may be expressed as follows. Practical Interpretation When sampling is from a normally distributed population with known standard deviation, we are 100 1 a ð Þ percent confident that the single computed interval, " x z 1a=2 ð Þ s " x , contains the population mean m. In the example given here we might prefer, rather than 2, the more exact value of z, 1.96, corresponding to a confidence coefficient of .95. Researchers may use any confidence coefficient they wish; the most frequently used values are .90, .95, and .99, which have associated reliability factors, respectively, of 1.645, 1.96, and 2.58. 6.2 CONFIDENCE INTERVAL FOR A POPULATION MEAN 167 Precision The quantity obtained by multiplying the reliability factor by the standard error of the mean is called the precision of the estimate. This quantity is also called the margin of error . EXAMPLE 6.2.2 A physical therapist wished to estimate, with 99 percent confidence, the mean maximal strength of a particular muscle in a certain group of individuals. He is willing to assume that strength scores are approximately normally distributed with a variance of 144. A sample of 15 subjects who participated in the experiment yielded a mean of 84.3.
eBook - PDFStatistics Using IBM SPSS
An Integrative Approach
- Sharon Lawner Weinberg, Sarah Knapp Abramowitz(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
INFERENCES INVOLVING THE MEAN OF A SINGLE POPULATION WHEN σ IS KNOWN 266 INTERVAL ESTIMATION Interval estimation involves the estimation of a population parameter by means of a line segment (or interval) on the real-number line within which the value of the parameter is thought to fall. For example, we might estimate the mean height (to the nearest inch) of all American males by using the interval 65–71 inches centered at the sample-mean value X = 68 inches rather than by using the sample-mean value, X = 68 inches itself. We could then say we believe µ to be one of the numbers within the interval 65–71 or, equivalently, that we believe the interval 65–71 contains µ . Of course, this is not as precise a statement as saying that we believe µ = 68 inches exactly. But because both statements are statements of belief, we may ask: “In which of the two statements (the interval statement or the point statement) do we have more confidence that our belief is true?” In general, we can place more confidence in interval estimations than in point estima-tions and, by extension, more confidence in interval estimation using longer intervals than in interval estimation using shorter intervals. There is a trade-off between the precision of an estimate and our confidence that the estimate is true. If we develop a procedure for constructing intervals of estimation that has a prescribed probability of giving an interval that contains µ , then we can use this probability as our measure of confidence that the population mean falls within the interval. Such procedure is based on the Central Limit Theorem. Recall that under the conditions of the Central Limit Theorem, the sampling distribu-tion of means calculated on samples of size N drawn at random from a population with mean µ and standard deviation σ , is either exactly or approximately normally distributed, with mean µ and standard error σ x = σ / N .
eBook - PDFStatistics Using R
An Integrative Approach
- Sharon Lawner Weinberg, Daphna Harel, Sarah Knapp Abramowitz(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
INTERVAL ESTIMATION Interval estimation involves the estimation of a population parameter by means of a line segment (or interval) on the real-number line within which the value of the parameter is thought to fall. For example, we might estimate the mean height (to the nearest inch) of all American males by using the interval 65–71 inches centered at the sample-mean value X = 68 inches rather than by using the sample-mean value, X = 68 inches, itself. We could then say we believe μ to be one of the numbers within the interval 65–71 or, equivalently, that we believe the interval 65–71 contains μ. Of course, this is not as precise a statement as saying that we believe μ = 68 inches exactly. But because both statements are statements of belief, we may ask: “In which of the two statements (the interval statement or the point statement) do we have more confidence that our belief is true?” In general, we can place more confidence in interval estimations than in point estima- tions and, by extension, more confidence in interval estimation using longer intervals than in interval estimation using shorter intervals. There is a trade-off between the precision of an estimate and our confidence that the estimate is true. If we develop a procedure for constructing intervals of estimation that has a prescribed probability of giving an interval that contains μ, then we can use this probability as our measure of confidence that the population mean falls within the interval. Such procedure is based on the central limit theorem. Recall that under the conditions of the central limit theorem, the sampling distribution of means calculated on samples of size N drawn at random from a population with mean μ and standard deviation σ, is either exactly or approximately normally distributed, with mean μ and standard error σ X ¼ σ= ffiffiffiffi N p .
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- Ned Freed, Stacey Jones, Timothy Bergquist(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
By treating the mean ( x ) of any random sample as a value that’s been randomly selected from the sampling distribution of the sample mean, we can construct an interval like x 1 x and argue convincingly that the interval is 68.3% likely to contain , the population mean. Similarly, we can construct the interval x 2 x and be 95.5% confident that this interval contains . To generalize, the three sampling distribution properties we’ve identified give rise to the basic confidence interval x z x or, equivalently, Interval Estimate of a Population Mean x z 2n (7.3) where x is the mean of a randomly selected sample of size n, is the standard deviation of the population that produced the sample, and z represents the number of standard deviations on a normal distribution required for any given level of “confidence” or probability. NOTE: Looking at things in a slightly different way, it can be shown that when samples are selected from a normal population, the quantity z x / 2n , z has a standard normal distribution. This is what allows us to use the standard normal distribution to build confidence intervals in the manner shown here. When sample sizes are large, the requirement that the population is normal can be relaxed. Notice the factors that determine how wide the interval will be: Factors Influencing Interval Width 1. Confidence—that is, the likelihood that the interval will contain . A higher confidence level will mean a larger z, which, in turn, will mean a wider interval. 2. Sample size, n. A larger sample size will produce a narrower interval. 3. Variation in the population, as measured by . The greater the variation in the population of values, the wider the interval. Applying the general interval expression to our social media example is easy enough.
eBook - PDFBiostatistics
A Foundation for Analysis in the Health Sciences
- Wayne W. Daniel, Chad L. Cross(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Interpreting Confidence Intervals How do we interpret the interval given by Expression 6.2.2? In the present example, where the reliability coefficient is equal to 2, we say that in repeated sampling approximately 95 percent of the intervals constructed by Expression 6.2.2 will include the population mean. This interpretation 6.2 Confidence Interval for a Population Mean 149 is based on the probability of occurrence of different values of x. We may generalize this inter- pretation if we designate the total area under the curve of x that is outside the interval ± 2 x as and the area within the interval as 1 − and give the following probabilistic interpretation of Expression 6.2.2. Probabilistic Interpretation In repeated sampling, from a normally distributed population with a known standard deviation, 100(1 − ) percent of all intervals of the form x ± z (1−∕2) x will in the long run include the population mean . The quantity 1 − , in this case .95, is called the confidence coefficient (or confidence level), and the interval x ± z (1−∕2) x is called a confidence interval for . When (1 − ) = .95, the inter- val is called the 95 percent confidence interval for . In the present example, we say that we are 95 percent confident that the population mean is between 17.76 and 26.24. This is called the practical interpretation of Expression 6.2.2. In general, it may be expressed as follows. Practical Interpretation When sampling is from a normally distributed population with known standard deviation, we are 100(1 − ) percent confident that the single computed interval, x ± z (1−∕2) x , contains the population mean . In the example given here we might prefer, rather than 2, the more exact value of z, 1.96, corresponding to a confidence coefficient of .95. Researchers may use any confidence coefficient they wish; the most frequently used values are .90, .95, and .99, which have associated reliability factors, respectively, of 1.645, 1.96, and 2.58.
No longer available |Learn more- Jessica Utts, Robert Heckard(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Our confidence is in the procedure. We don’t know whether an individual confidence in-terval estimate covers the truth. But we do know that in the long run, of all confidence intervals that we compute using a 95% level of confidence, about 19 out of 20 of them (i.e., 95%) will cover the true population value. About one out of 20 of them will not. Therefore, we don’t know whether the mean TV viewing hours for Penn State students Explore further using Applet 11.2: Interpreting the Confidence Level. Unless otherwise noted, all content on this page is © Cengage Learning. CI Module 3: Copyright 2014 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. Estimating Means with Confidence 425 is really between 1.84 and 2.34 hours. The population mean is between these values if we got one of the 95% of possible samples for which the sample mean is close enough to the true population mean for our procedure to work. The population mean isn’t between these values if we happened to get one of the 5% of possible samples for which the sample mean is too far from the population mean for the procedure to work. A Common Misinterpretation of a Confidence Interval for a Mean A common mistake is to think a confidence interval for a mean covers the specified percentage of individual values in the population. For instance, in Example 11.5, we found that a 95% confidence interval for the mean forearm length of men is 24.33 cm to 26.67 cm. We can be fairly certain that the population mean forearm length is cov-ered by this interval.
- Jay Devore, Nicholas Farnum, Jimmy Doi, , Jay Devore, Nicholas Farnum, Jimmy Doi(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 318 CHAPTER 7 Estimation and Statistical Intervals 7.4 SMALL-SAMPLE INTERVALS BASED ON A NORMAL POPULATION DISTRIBUTION Suppose we select a random sample of components of a certain type and determine the lifetime of each one, resulting in data x 1 , x 2 , . . . , x n . This sample can be used as a basis for calculating one of three different kinds of statistical intervals: 1. An interval of plausible values for the population mean lifetime, that is, a confi-dence interval for H9262 2. An interval of plausible values for the lifetime of a single component of this type that you are planning to buy at some time in the near future, that is, a prediction interval for a single x value 3. An interval of values that includes a specified percentage, for example, 90%, of the lifetime values for components in the population, that is, a tolerance interval for a chosen percentage of x values in the population distribution We have already seen how to calculate a z confidence interval for H9262 when n is large. In this section, we assume that the sample has been selected from a normal population distribution, and show how each of the three types of intervals can be obtained. t Distributions and the One-Sample t Confidence Interval When the population distribution is normal, the sampling distribution of x is also normal for any sample size n . This in turn implies that z 5 ( x 2 H9262 ) y ( H9268 y 1 n ) has a standard nor-mal distribution (the z curve).
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