It only takes a minute to sign up. Out of 36 possible combinations of dice outcomes, this is represented as 1/36 on the y-axis. There are many more ways to get sums of 6 and 8, and even more ways to get sums of 7. Is any elementary topos a concretizable category? Adding field to attribute table in QGIS Python script. Flash of Stats concepts for Data science - Part I, How to Find the Value of Sin 15 Degrees (Sin15) Without Using FormulaGraphical Approach, post on probability via a Monty Hall-type problem. This is why I said, earlier, that theres often a confusing lack of transparency surrounding how the Normal distribution is taught, in that why its used has nothing to do with the probability density function (PDF) which happens to describe it. My guess is that its pretty good for a normal distribution and then fails for asymmetric and heavy-tailed distributions. When you work with non-parametric distributions, quantile estimations are essential to get the main distribution properties. The confidence interval is -41.6% to 61.6%. When you are trying to estimate a quantile from data then you can turn the problem into a binomial problem. A. If he wanted control of the company, why didn't Elon Musk buy 51% of Twitter shares instead of 100%? It would also be nice to include the other asymptotic for the other case (when $n$ is large but $p$ is either very close to 0 or 1). The distributional assumption is mostly assessed using quantile-quantile plots. Concealing One's Identity from the Public When Purchasing a Home. Are certain conferences or fields "allocated" to certain universities? Confidence interval for the quantile Besides the point estimate x ^ p we also would like to report a two-sided ( 1 ) 100 % confidence interval ( x p l, x p u) for the desired population quantile. 0.1 Libraries. support@analystprep.com. $$ \begin{align*} \text{Confidence interval at 99%} & = \left\{ 0.24 3 \times 0.05, 0.24 + 3 \times 0.05 \right\} \\& = \left\{ 0.09, 0.39 \right\} \\\end{align*} $$. We are not supposed to say that there is a 95% chance that the true population mean lies between 69.45 inches and 70.55 inches, because an entirely different sample mean at 95% confidence could result in an entirely different interval. Since $\bar{x}$ and $s$ are independent, it is pretty easy to calculate confidence bounds for any linear combination of $\mu$ and $\sigma$. There is a common misunderstanding that a 95% confidence interval is an interval that covers the true parameter value with 95% probability. Dice outcomes are limited to [16], but human height presumably lies on the real number line. Were just going backwards by taking a z-score of a sample and then making a guess about the relative likelihood of whether or not a particular interval on the Normal distribution contains the mean! For instance, for the 7-day low flows the ratio between the estimated confidence interval to the estimated quantile based on ML is about 17% for T 2 while it is about 30% for estimation. The x is the mean of a sample, z is the z-score, the s is the standard deviation of the sample (though we should use if we happen to know the population standard deviation, which we often dont), and n is is the size of the sample. Normally-distributed data forms a bell shape when plotted on a graph, with the sample mean in the middle and the rest of the data distributed fairly evenly on either side of the mean. Your colleagues formula gives just a point estimate, not an interval. What do you call an episode that is not closely related to the main plot? The standard trio is 90%, 95%, and 99%. For each of the samples, find the sample median. The standard Stats 101 strategy at this point is to look up this value in the completely arcane table of High Magic, the dreaded z-table: If = .025, then that means the area under the curve of the Normal distribution for our desired interval will be .975, which we find at the z-score of 1.96 in the table above (the columns represent the hundredths place). It is calculated as: Confidence Interval = x +/- t /2, n-1 *(s/ n) where: x: sample mean; t /2, n-1: t-value that corresponds to /2 with n-1 degrees of freedom; s: sample standard deviation n: sample size The formula above describes how to create a . Finance Train, All right reserverd. stat = calculate_statistic (sample) statistics.append (stat) 2. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. That is, humans may be 6 feet tall exactly, or 6.1 feet tall, or 6.314159 feet tall. Quantiles and percentiles represent useful statistical tools for describing the distribution of results and deriving reference intervals and performance specification in laboratory medicine. Percentile confidence intervals. Thanks for contributing an answer to Cross Validated! B is incorrect. Variance is the result of adding together every squared difference of points in the dataset and the mean. Follow to join The Startups +8 million monthly readers & +760K followers. $$ But so what? Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. This is because t-distribution accounts for bigger uncertainty in samples than normal distribution when sample size is samll, but converges to normal distribution when sample size is bigger than 30. This function provides a confidence interval for any quantile or (per)centile. If one pile of apples has 4 apples, another pile has 5 apples, and a final pile has 9 apples, then the mean = (4 + 5 + 9)/3 = 6. Stack Overflow for Teams is moving to its own domain! Get smarter at building your thing. 2. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Then find the Z value for the corresponding confidence interval given in the table. To find the 95% confidence interval for the 60% percentile, we calculate the "order statistic" as (n+1)p = 36*.6 = 21.6 (as we saw above). x = np.random.normal (size=100) Let's see we want to calculate the 95% confidence interval of the mean value. Two intervals are available: Confidence interval based on the normal distribution; Distribution free confidence interval; Additional available plots So for a 95% CI, we have =1.00 - .95 = .05. Is it enough to verify the hash to ensure file is virus free? Step 3: Finally, substitute all the values in the formula. Instead of the population sigma we use sample sigma and instead of using a normal distribution we use a t distribution. Here the formula is: CI = Sample mean z value Standard error of mean (SEM) Concealing One's Identity from the Public When Purchasing a Home. rev2022.11.7.43011. If = .025, then that means the area under the curve of the Normal distribution for our desired interval will be .975, which we find at the z-score of 1.96 in the table above (the columns . A confidence interval is an interval in which we expect the actual outcome to fall with a given probability (confidence). QUADRATIC-NORMAL DISTRIBUTION Y. L. Goh 1 A. H. Pooi 2 . Wouldnt you want to work with $q_{0.95}$ instead of $0.975$? A two-sided 100 ( 1 ) % confidence interval for x q, the q quantile of the normal distribution, is [ x t ( 1 / 2; n 1, ) s n, x t ( / 2; n 1, ) s n] where t ( ; n 1, ) is the quantile of a noncentral t -distribution with n 1 degrees of freedom and noncentrality parameter = n z ( q) = n z ( 1 p). Assuming a normal distribution, a 99% confidence interval for the expected return isclosest to: For a 99% CI, approximately 99% of all the observations fall in the interval \( 3\). I might run some simulations to check out its MSE compared to other estimates. In fact, dont worry about using the formula, as its sufficient to know that it merely exists to give the shape to the thing we call a bell curve, another name for the Normal distribution. The precision or accuracy of the estimate depends on the width of the interval. Calculate the 99% confidence interval. Start studying for CFA exams right away. What are the best buff spells for a 10th level party to use on a fighter for a 1v1 arena vs a dragon? Use when statistic is unbiased. To find out the confidence interval for the population mean, we will use the following formula: Therefore, the confidence interval is 100,000 3919.928, which is equal to the range 96,080.072 and 103,919.928. KEYWORDS: Confidence interval, Quantile, Hypothesis testing When the distribution of the 1. Generally speaking, statistics is often semantics, and the English (or whatever human language) interpretation of a result often hinges heavily on connotations and assumptions present in the framing of a problem. We then split into two: /2, since our confidence interval will be symmetric around the presumed true mean: .05/2 = .025. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Why am I being blocked from installing Windows 11 2022H2 because of printer driver compatibility, even with no printers installed? Their simulation studies showed four of the methods to behave almost identically. Procedure to find the bootstrap confidence interval for the median 1. Dont worry about where it comes from. What are the correct confidence intervals for a generic $q$, in the three cases: The third case is given by Hahn and Meeker in their handbook Statistical Intervals (2nd ed., Wiley 2017): A two-sided $100(1-\alpha)\%$ confidence interval for $x_q$, the $q$ quantile of the normal distribution, is. Now, all you need to remember is that the 5th percentile of $X$ is, as you note, $\mu-1.64\sigma$. December 22, 2020 Mathematics Statistics Research Quantile Coverage Confidence Interval. As an example we can compute the 0.99 percentile confidence interval for the rate parameter as, alpha <- 0.01 quantile (v_rate_est_bt, probs = c (alpha / 2, 1 - alpha / 2)) ## 0.5% 99.5% ## 4.133315 6.811250. Calculate Confidence Interval. Meanwhile, the correct definition assumes that the true parameter value will be covered by 95% of 95% confidence intervals in the long run. Example 1: constructing confidence interval for with z -quantiles Assume X 1, , X n are i.i.d. Everything about confidence intervals involves this fundamental leap, which is not itself a statistical concept. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode ), while the parameter is its standard deviation. So you can test each possible value using the binomial (how many of your 1,000 values are below the value you are testing). I think theres often a confusing lack of transparency surrounding how the Normal distribution is taught, in that why its used has nothing to do with the probability density function (PDF) which happens to describe it. We construct 100(1-) % confidence. The CIPCTLDF option on the PROC UNIVARIATE statement produces distribution-free confidence intervals for the 1st, 5th, 10th, 25th, 50th, 75th, 90th, 95th, and 99th percentiles as shown in the following example: Here is some R code and a small simulation to check the coverage. The CONFIDENCE.NORM function is used to calculate the confidence interval with a significance of 0.05 (i.e. As always, youre welcome to instead say Oh god, when will I remember to just keep my mouth shut around statisticians, but hopefully after reading this post youll be slightly more capable of making confident claims about, well, your confidence. The calculation assumes a 68% CI: $$\text{Confidence interval at 68%}=0.240.05,0.24+0.05=\{0.19,0.29\}$$, Testing the Variances of a Normally Distributed Population using the Chi-square Test A Read More, Odds for and against an event represent a ratio of the desired outcomes Read More, The time-weighted rate of return (TWRR) measures the compound growth rate of an Read More, All Rights Reserved Standard Normal Distribution. CFA Institute does not endorse, promote or warrant the accuracy or quality of Finance Train. Step 1: Find the number of observations n (sample space), mean X, and the standard deviation . Yes, the idea looks right. Create a normal distribution object by fitting it to the data. Why are standard frequentist hypotheses so uninteresting? @DeltaIV Thanks. Let's calculate all the numbers we need according to the formula of confidence intervals. The confidence interval is a range of values. This is almost halfway between 21 and 22, and so we can use the approach described in Confidence Intervals for Order Statistics, Medians and Percentiles for a median from a sample of even size. It is the value of a standard normal variable . This is nearly identical to finding a confidence interval when sigma is known except that we can't use the population sigma because we don't know the population sigma. Finally, weve reached the titular topic. So what I described above is not exactly what you want. Making statements based on opinion; back them up with references or personal experience. How to construct common classical gates with CNOT circuit? Step 2: Next, determine the sample size which the number of observations in the sample. The usual formula you see for a confidence interval is the estimate plus or minus the 97.5th percentile of the normal or t distribution times the standard error. However, the latter are hardly useful unless we superimpose some confidence intervals to the graph. How can you prove that a certain file was downloaded from a certain website? So, if X is a normal random variable, the 68% confidence interval for X is -1s <= X <= 1s. Normal Approximation Method of the Binomial Confidence Interval The equation for the Normal Approximation for the Binomial CI is shown below. Use MathJax to format equations. Grammar is quite important here, and fine distinctions in this phrasing are often the cause of pedantic scorn, since it is important not to imply that estimations are capable, just from a single sample, of making population parameter-level claims. The four commonly used confidence intervals for a normal distribution are: The confidence interval is generally represented as , where n is the number of standard deviations. So z will be a quantile or z-score of a standard normal distribution, such that. on average the 5th percentile of a standard normal sample will be -1.64 and 95% of the time the sample 5th percentile of a sample of n=1,000 will be below -1.54 (approximate from simulations). Use a normal distribution because the interest rates are normally distributed and \ ( \sigma \) is known. N ( , 2) and is known. The 5 methods that boot package provides for bootstrap confidence intervals are summarized below: Normal bootstrap or Standard confidence limits methods use the standard deviation for calculation of CI. Connect and share knowledge within a single location that is structured and easy to search. In . These variety of ways, or outcomes, is the essence of the Normal distribution. Confidence interval for quantiles. PIMS PDF Hui Huang: On Big Leaps, Dynamical Systems and Partial Differential Equations. INTRODUCTION Let For us to define a \(100(1 )%\) confidence interval for \(\), we must specify two random variables \(_1(X)\) and \(_2(X)\) such that \(P(_1(X) < < _2(X)) = 1 \). Maybe one based on normal approximation and one based on Poisson? Quantile Confidence Interval Menu location: Analysis_Nonparametric_Quantile Confidence Interval. Why are UK Prime Ministers educated at Oxford, not Cambridge? Instead of copy & pasting the formula into my answer, I would encourage you to post it as a separate answer, preferably with a bit more context. . The best answers are voted up and rise to the top, Not the answer you're looking for? Posted on novembro 3, 2022 by - . Step 2: Decide the confidence interval of your choice. So a more general approach would define medianinterval (d,p=0.95) = quantile (d,1- (1+p)/2),quantile (d, (1+p)/2). When the Littlewood-Richardson rule gives only irreducibles? Lets finally look at how to construct a confidence interval. The standard error depends on the sample size and the dispersion in the variable of interest. Allow Line Breaking Without Affecting Kerning. The interval ( x p l, x p u) should, hence, fulfill the following condition: P ( ( x p l, x p u) x p) = 1 , Due to its shape, it is sometimes referred to as "the Bell Curve", but there are other distributions which result in bell-shaped curves, so this may be misleading. Draw N samples ( N will be in the hundreds, and if the software allows, in the thousands) from the original sample with replacement. $$ Further, GARP is not responsible for any fees or costs paid by the user to AnalystPrep, nor is GARP responsible for any fees or costs of any person or entity providing any services to AnalystPrep. The returns are normally distribution. Replace first 7 lines of one file with content of another file. We can only make sample-level claims, and interpret those claims on a sample-to-sample basis. We will then say the Poisson mean is 0.035 with 95% confidence interval of (0.019, 0.059). Calculate the 99% confidence interval. Which . m = x.mean () s = x.std () dof = len (x)-1 confidence = 0.95 We now need the value of t. What sorts of powers would a superhero and supervillain need to (inadvertently) be knocking down skyscrapers? We can also obtain both percentiles in one line of code using: > quantiles<-qdata (c (.025,.975),Tstar) > quantiles quantile p 2.5% 0.1914578 0.025 97.5% 3.4841547 0.975 December 8, 2020 Mathematics Statistics Research Research: Weighted quantile estimators Quantile Confidence Interval. The difference here, and the main intuitive leap, is that a Normal distribution deals with continuous variables, as opposed to discrete variables. The best way to think about a Normal distribution is as a pseudo-histogram of an infinite number of samples of some random phenomenon, like rolling dice. It may also be correct to say that 95% of the time the interval will capture the true mean, but lets get into that below. Limited Time Offer: Save 10% on all 2022 Premium Study Packages with promo code: BLOG10. The true population mean could be hiding at the lower end of this interval, or the higher end, but theres no way to tell without taking another sample. The fact that the infinite sampling of all continuous data sets converges to the Normal distribution is due in part to the Central Limit Theorem, which I will again avoid expositing for the sake of brevity and simplicity. This is a quick tutorial on how to make a 95% confidence interval in R using the normal distribution. 2022. For 1,000 data points you want to know what value will have 5% of values below it (population values, the sample is just the estimate) and 95% above it. Confidence intervals for quantiles are commonly known as. You might want to review your class notes and carefully study the article "Coverage probability of confidence intervals: A simulation approach". Searching for this on CRAN, we found the following functionality: Package::Function Version Description MKmisc::quantileCI Implements an exact but very slow \(O(n^2)\)search as well as an asymptotic Should I avoid attending certain conferences? To learn more, see our tips on writing great answers. PDF | Suppose we have a random sample from a non-normal distribution known as the quadratic-normal distribution. the results. See my post on probability via a Monty Hall-type problem for the probability version of this post. It is denoted by n. The calculation assumes a 95% CI: $$\text{Confidence interval at 95%}=0.2420.05,0.24+20.05=\{0.14,0.34\}$$. Confidence Intervals and the Normal Distribution A confidence interval is a range of values that gives the user a sense of how precisely a statistic estimates a parameter. Sometimes the simplest refreshers are best, and when it comes to statistics, concepts like parameter, statistic, z-score, t-test, Students t-distribution, standard deviation, Chebyshevs rule, and confidence interval can tend to merge into disorienting word salads, much like this sentence itself. C is incorrect. where $t_{(\gamma;\,n-1,\,\delta)}$ is the $\gamma$ quantile of a noncentral $t$-distribution with $n-1$ degrees of freedom and noncentrality parameter $\delta = -\sqrt{n}z_{(q)}=\sqrt{n}z_{(1-p)}$. One of those methods, which they calledthe Lawless method (Lawless, 2003, p. 231), is the method used in this . The t distribution is nearly identical . 95% confidence interval = 10% +/- 2.58*20%. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. 3.6 An approach based on large-sample theory In frequentist statistics, a confidence interval ( CI) is a range of estimates for an unknown parameter. The fact that the Normal distribution is the logical conclusion of infinite sampling, and that its mathematical derivation is extremely involved and unintuitive, explains the bizarre and almost magical appearance of the Normal distribution, as well as the frustration of students trying to understand how it relates to the relatively simple concepts of mean, variance, and standard deviations. Chakraborti and Li (2007) compare several methods of confidence interval estimation of a Normal percentile. How to obtain a confidence interval for a percentile? Exact 95% confidence interval for Poisson mean is: Lower bound = 7.65 / 400 =0.019135 for lower bound and. This is done by first ordering the statistics, then selecting values at the chosen percentile for the confidence interval. For example, if a sample of 50 human heights resulted in a mean of 70 inches, with a sample standard deviation of 2, we use the CI formula: 70 +/- 1.96(2/7.07)= 70 +/- .55 = (69.45, 70.55). However, I don't think you need to link to an external site for the sake of answering point 2 (the asymptotic case).I can copy that formula into your answer if you don't feel like doing it yourself. This is demonstrated in the following diagram. If neither distribution can be used, explain why. In fact, collection of data from every subject in a large population is not only economically unviable but also very time-consuming. Stock Price Movement Using a Binomial Tree, Confidence Intervals for a Normal Distribution, Calculating Probabilities Using Standard Normal Distribution, Option Pricing Using Monte Carlo Simulation, Historical Simulation Vs Monte Carlo Simulation, R Programming - Data Science for Finance Bundle, 68% of values fall within 1 standard deviation of the mean (-1s <= X <= 1s), 90% of values fall within 1.65 standard deviations of the mean (-1.65s <= X <= 1.65s), 95% of values fall within 1.96 standard deviations of the mean (-1.96s <= X <= 1.96s), 99% of values fall within 2.58 standard deviations of the mean (-2.58s <= X <= 2.58s). Does English have an equivalent to the Aramaic idiom "ashes on my head"? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. R removing zeros for pseudomedian and its confidence interval in wilcox.test? So 14 is the variance. Confidence interval for the mean - Normal distribution or Student's t-distribution? It only takes a minute to sign up. The subscript , rather than / 2, implies a one-tailed test. By dividing any given data point by the standard deviation, we end up with whats called a z-score, which is the average number of standard deviations from the mean. All Ill say here, for the sake of brevity and simplicity, is that the Normal distribution fundamentally involves circles and the fact that pi is the same for all circles, and that because the act of creating a z-score involves squaring the difference of each data point from the mean, the value of pi is implicitly involved in the standardization of all data sets through z-score conversion. Is it possible for a gas fired boiler to consume more energy when heating intermitently versus having heating at all times? They are commonly intended as the sample estimate of a population parameter and therefore they need to be presented with a confidence interval (CI). We use the following formula to calculate a confidence interval for a difference in population means: Confidence interval = (x1-x2) +/- t* ( (sp2/n1) + (sp2/n2)) where: x1, x2: sample 1 mean, sample 2 mean t: the t-critical value based on the confidence level and (n1+n2-2) degrees of freedom So you can test each possible value using the binomial (how many of your 1,000 values are below the value you are testing). The confidence interval for data which follows a standard normal distribution is: Where: CI = the confidence interval X = the population mean There is some more detail on Wikipedia or by Googling "quantile confidence interval". My profession is written "Unemployed" on my passport. The Normal distribution, in short, can be described by the function: And it looks like the blue-green-yellow picture at the top of this post. I suspect that the formula wouldn't make much sense without more context from the first case. 95% confidence interval = 10% +/- 2.58*20%. A confidence interval is such that you are 95% sure the true mean lies in the interval, that is why you are getting such a small range, because as the sample size gets larger, the interval is narrowing down to one number - the actual mean of the distribution. Calculation of the CI of the mean is relatively simple. (taking into account the fundamental assumption that our population is, in fact, described by the Normal distribution). As such, \(P(_1(X) < < _2>(X)) = 0.95\) specifies \(_1(X)\) and \(_2(X)\) such that there is a 95% chance of finding the true value of \(\) in the interval. Note: Some statisticians believe that, under conventional inference, P -values and percentile confidence limits should not be estimated by interpolation. I guess there should be at least a couple answers hereone for quantiles which aren't close to 0 or 1, and one for quantiles which are. The concept is described in detail below. A confidence interval is computed at a designated confidence level; the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used. Because a z-score is the conversion of any data point into a format relative to its own standard deviation and mean, this results in all z-scores falling into the same grand, relativized scope of comparison via. In this case, the t -based formula would be: 95% CI = r tdf = 13SEr The original assignment says "compute a 95% CI for mu" [for each sample]. To find out the confidence interval for the population . Sort your bootstrap statistics into rank order. Why was video, audio and picture compression the poorest when storage space was the costliest? 68% CI: approximately 68.3% of all the observations fall in the range . I. How would we compare a variance of 14 apples to a variance of 2 inches in a human height dataset? 6 Confidence interval for mean using normal distribution 7 Conclusion Confidence Interval for Mean Confidence Interval = x (t * standard error) Where : x = mean t = t-multiplier is calculated based on degree of freedom and desired confidence interval standard error = sample standard error/ sample size n = sample size Note:- 1. You could also develop and present your ideas about the Poisson approximation there. A stock portfolio has mean returns of 10% per year and the returns have a standard deviation of 20%. Dont worry about how its derived. The 95% Confidence Interval . It is all based on the idea of the Standard Normal Distribution, where the Z value is the "Z-score" For example the Z for 95% is 1.960, and here we see the range from -1.96 to +1.96 includes 95% of all values: The standard deviation is a measure of the average distance from any particular data point in a set of data from the mean of that data. Assuming a normal distribution, the 50% confidence interval for the expected return is closest to: $$ \begin{align*}\text{Confidence interval at 50%} & = \left\{ 0.24 \cfrac {2}{3} \times 0.05, 0.24 + \cfrac {2}{3} \times 0.05 \right\} \\& = \left\{ 0.207, 0.273 \right\} \\\end{align*} $$. What is the formula (if it exists) for the sample variance / confidence interval of a quantile / percentile of the normal distribution? $$, $\delta = -\sqrt{n}z_{(q)}=\sqrt{n}z_{(1-p)}$, This is nice and I upvoted (btw, made a minor edit, please check it out). I.e. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Alternatively, we could say that 5% of the realizations of such intervals would not contain the true value of \(\). Returns the confidence interval for a population mean, using a normal distribution. We can interpret this as with any confidence interval, that we are 95% confident that the difference in the true means (Unattractive minus Average) is between 0.19 and 3.48 years. Statistics and Probability questions and answers. We can then state with 95% confidence that the interval (69.45 inches, 70.55 inches) captured the true population height mean. For example, let's assume we drew $n=20$ samples from a normal distribution with unknown mean and standard deviation. Justify your decision. Rather than having three answers, each focusing on one point, I think it would be better to have, Confidence interval for quantiles: distribution-free, asymptotic and assuming a normal distribution, Mobile app infrastructure being decommissioned. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. When the Littlewood-Richardson rule gives only irreducibles? However, as you can see, things have gotten quite complex from just a few deceptively simple acts. $$ 4. For example, n=1.65 for 90% confidence interval. To estimate the confidence interval for any other value, simply invoke the Student's t quantile function qt () in conjunction with S E. For example, to generate a 90% confidence interval for the mean hours of TV watched per household: mean.int.90 <- mean.x + qt( c(0.05, 0.95), length(x) - 1) * SE.x mean.int.90.
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