Note that this formula follows the basic structure of a test statistic that you learned last week: \(test\;statistic=\frac{sample\;statistic-null\;parameter}{standard\;error}\), \(\widehat{p}\) = sample proportion A simple random sample of 60 individuals with a membership at one gym was collected. This Demonstration compares the sample probability distribution with the theoretical normal distribution. The central limit theorem has a number of variants. A failure is defined as answering "no." k 1.5 Example: Approximate Mean and Variance Suppose X is a random variable with EX = 6= 0. If \(p \leq \alpha\) reject the null hypothesis. The binomial distribution can be used to solve problems such as, “If a fair coin is flipped 100 times, what is the probability of getting 60 or more heads?” The probability of exactly [latex]\text{x}[/latex] heads out of [latex]\text{N}[/latex] flips is computed using the formula: [latex]\displaystyle \text{P}\left( \text{x} \right) =\frac { \text{N}! David Lane, History of Normal Distribution. Let’s use the five step hypothesis testing procedure to determine if there is evidence that the proportion in this city is different from the known national proportion. B. If you have data in a Minitab Express worksheet, then you have what we call "raw data." If we are conducting a one-tailed (i.e., right- or left-tailed) test, we look up the area of the sampling distribution that is beyond our test statistic. All of the conditions for testing a claim about a population proportion using the normal approximation method are satisfied, so the method can be used. np = 583,333.333 >> 10 CHECK! Note how well it approximates the binomial probabilities represented by the heights of the blue lines. The [latex]\text{z}[/latex]-score is only defined if one knows the population parameters. This means we get started with a set level of confidence and margin of error. In order to use the normal approximation method, the assumption is that both \(n p_0 \geq 10\) and \(n (1-p_0) \geq 10\). O B. :\;0.640\pm 2.576 (0.014)=0.0640\pm 0.036=[0.604, \; 0.676]\). The tool of normal approximation allows us to approximate the probabilities of random variables for which we don’t know all of the values, or for a very large range of potential values that would be very difficult and time consuming to calculate. We are more confident of catching the population value when we use a wider interval. For the following procedures, the assumption is that both \(np \geq 10\) and \(n(1-p) \geq 10\). Independently the mathematicians Adrian (in 1808) and Gauss (in 1809) developed the formula for the normal distribution and showed that errors were fit well by this distribution. Consistent to three decimal places in this case. From the Minitab Express output above, the p-value is 0.0002031. If there are at least 10 successes and at least 10 failures, then you need to change the method to the normal approximation method. \(SE=\sqrt{\frac{\hat{p} (1-\hat{p})}{n}}=\sqrt{\frac{0.640 (1-0.640)}{1168}}=0.014\), The \(z^*\) multiplier for a 95% confidence interval is 1.960, The formula for a confidence interval for a proportion is \(\widehat{p}\pm z^* (SE)\), \(0.640\pm 1.960(0.014)=0.640\pm0.028=[0.612, \;0.668]\). The conditions. Explain the significance of normalization of ratings and calculate this normalization. When conducting a hypothesis test, we check this assumption using the hypothesized proportion (i.e., the proportion in the null hypothesis). June 12, 2013. Christofer Larsson, in Design of Modern Communication Networks, 2014. September 17, 2013. This is the minimum sample size, therefore we should round up to 601. Because there is no estimate of the proportion given, we use \(\tilde{p}=0.50\) for a conservative estimate. The law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. Now, we have an estimate to include in the formula: \(n=\left ( \frac{1.960}{0.04} \right )^2 (0.25)(1-0.25)=450.188\). Nevertheless, there are several methods which provide an approximation of the integral by numerical methods: Taylor series, asymptotic series, continual fractions, and some other more. Yes, there is statistical evidence to state that more than 80% of all Americans are right handed. When given the option, it is recommended that you use Minitab Express as opposed to performing calculations by hand. If you do not have each individual observation, but rather have the sample size and number of successes in the sample, then you have summarized data. In the example below, we want to know if there is evidence that the proportion of students who are male is different from 0.50. Can a test about a population proportion using the normal approximation method be used? Therefore, de Moivre reasoned that if he could find a mathematical expression for this curve, he would be able to solve problems such as finding the probability of 60 or more heads out of 100 coin flips much more easily. When we're constructing confidence intervals \(p\) is typically unknown, in which case we use \(\widehat{p}\) as an estimate of \(p\). In this example a success is defined as answering "yes" to the question "do you own a dog?" In its common form, the random variables must be identically distributed. Here, for the sake of ease, we have used an online normal area calculator. Yes, there is evidence that the percentage of all Creamery customers who prefer chocolate ice cream over vanilla is less than 80%. We collect data from a random sample of 129 college women and 37 said that they think they are overweight. The usefulness of the theorem is that the distribution of [latex]\sqrt { \text{n} } ({ \text{S} }_{ \text{n} }-\mu )[/latex] approaches normality regardless of the shape of the distribution of the individual [latex]\text{X}_\text{i}[/latex]‘s. 9 views. This is a left-tailed test so we need to find the area to the right of our test statistic, \(z=-3.536\). A. The hypothesized value of the population proportion is symbolized by \(p_0\) because this is the value in the null hypothesis (\(H_0\)). Because the distribution of means is very close to normal, these tests work well even if the distribution itself is only roughly normal. \(H_{0}\colon p=0.690\) This can be done by finding [latex]\text{z}[/latex]-scores and using the [latex]\text{z}[/latex]-score table. In Spring 2016, a sample of 522 World Campus students were surveyed and asked if they own a dog. Normal Distribution and Scales: Compares the various grading methods in a normal distribution. 2. When discussion proportions, we sometimes refer to this as the Rule of Sample Proportions. David Lane, History of Normal Distribution. Are any of the three requirements violated? Research question: Are less than 50% of all individuals with a membership at one gym female? Determine the p-value associated with the test statistic, 8.1.1.2 - Minitab Express: Confidence Interval for a Proportion, – Confidence Interval for a Proportion (Normal Approximation M, 8.1.1.2.1 - Video Example: Lactose Intolerance (Summarized Data, Normal Approximation), 8.1.1.2.2 - Video Example: Dieting (Summarized Data, Normal Approximation), 8.1.1.3 - Computing Necessary Sample Size, 8.1.2.1 - Normal Approximation Method Formulas, 8.1.2.2 - Minitab Express: Hypothesis Tests for One Proportion, 8.1.2.2.1 - Minitab Express: 1 Proportion z Test, Raw Data, – Conducting a One Sample Proportion z Test: Raw Data, 8.1.2.2.2 - Minitab Express: 1 Sample Proportion z test, Summary Data, – Performing a One Proportion z Test with Summarized Data, 8.1.2.2.2.1 - Video Example: Gym Members (Normal Approx. If we are conducting a two-tailed (i.e., non-directional) test there is one additional step: we need to multiple the area by two to take into account the possibility of being in the right or left tail. Clopper and Pearson describe the Clopper-Pearson method also called the exact confidence interval and weâll describe in a separate article. On the following pages you will see how a confidence interval for a population proportion can be constructed by hand using the normal approximation method. Abraham de Moivre noted that when the number of events (coin flips) increased, the shape of the binomial distribution approached a very smooth curve. The area in green in the figure is an approximation of the probability of obtaining 8 heads. According to the law of large numbers, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. 8.5.1 The equal throughput method. This can be done using raw data or summarized data. We can look up the \(p\)-value using Minitab Express by constructing the sampling distribution. In a sample of 100 Americans, 87 were right handed. We are 95% confident that between 61.2% and 66.8% of all American adults are not financially prepared for retirement. What if we wanted a 99% confidence interval? This means we can use the normal approximation method to construct this confidence interval. (adsbygoogle = window.adsbygoogle || []).push({}); The process of using the normal curve to estimate the shape of the binomial distribution is known as normal approximation. The solution is to round off and consider any value from 7.5 to 8.5 to represent an outcome of 8 heads. If assumptions are met, the sampling distribution will have a standard error equal to \(\sqrt{\frac{p(1-p)}{n}}\). Are any of the three requirements violated? TODO: binom_test intervals raise an exception in small samples if one. Check rules of thumb using n = 3,500,000 and p = 1/6. Normal distribution integral has no analytical solution. Explain how central limit theorem is applied in normal approximation. This section provides the power calculation formulas for the various test statistics available in this procedure. \(p \leq.05\), therefore our decision is to reject the null hypothesis. How large n needs to be depends on the value of p.If p is near 0.5, the approximation can be good for n much less than 20. Both \(np_0 \geq 10\) and \(n(1-p_0) \geq 10\) so we can use the normal approximation method. In Minitab Express, we find the proportion of a normal curve beyond \(\pm0.980\): \(p>\alpha\), therefore we fail to reject the null hypothesis. Find a [latex]\text{Z}[/latex] score for 7.5 using the formula [latex]\text{Z}=\frac { 7.5-5 }{ 1.5811 } =1.5811[/latex]. There were 24 females. O A. In Minitab Express, this means that in step 8 above the default setting of Exact method should not be changed. Note: Because the normal approximation is not accurate for small values of n, a good rule of thumb is to use the normal approximation only if np>10 and np(1-p)>10. Construct a 95% confidence interval for the proportion of all World Campus students who have a dog. Both \(np_0\) and \(n(1-p_0)\) are at least 10. The use of “[latex]\text{Z}[/latex]” is because the normal distribution is also known as the “[latex]\text{Z}[/latex] distribution”. This is in contrast to "summarized data" which you'll see on the next page. A problem arises when there are a limited number of samples, or draws in the case of data “drawn from a box.” A probability histogram of such a set may not resemble the normal curve, and therefore the normal curve will not accurately represent the expected values of the random variables. Both \(n p_0\) and \(n (1-p_0)\) are at least 10, this assumption has been met. In terms of computing the 99% confidence interval, we will use the same point estimate \(\widehat{p}\) and the same standard error. Research Question: Is the percentage of Creamery customers who prefer chocolate ice cream over vanilla less than 80%? Question. Question 1. David Lane, Normal Approximation to the Binomial. The [latex]\text{Z}[/latex] value measures the sigma distance of actual data from the average and provides an assessment of how off-target a process is operating. First, we need to check our assumptions that both \(n\widehat p \geq 10\) and \(n(1-\widehat p) \geq 10\). One city’s council wants to know if the proportion of overweight citizens in their city is different from this known national proportion. You can change this value by clicking on the distributions. As shown on the probability distribution plot below, the multiplier associated with a 95% confidence interval is 1.960, often rounded to 2 (recall the Empirical Rule and 95% Rule). Let's construct a 95% confidence interval to estimate the proportion of all American adults who are not financially prepared for retirement. The standard score is the number of standard deviations an observation or datum is above the mean. In a sample of 100 African American adults, 70 were identified as having some level of lactose intolerance. The conditions np 25 and ng 25 are not satisfied, so a test about a population proportion using the normal approximation method cannot be used. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions, given that they comply with certain conditions. In other words, you have one sample with one categorical variable. As probability and statistical theory show us, as the number of samples increase for the given mean and standard deviation, the more closely the sample probability distribution will â¦ For example, to calculate the probability of 8 to 10 flips, calculate the area from 7.5 to 10.5. They take a random sample of 150 adults 20 years of age or older in their city and find that 98 are classified as overweight. Here we will be using the five step hypothesis testing procedure to compare the proportion in one random sample to a specified population proportion using the normal approximation method. Before we can construct a confidence interval for a proportion we must first determine if we should use the exact method or the normal approximation method. Research question: Is this city’s proportion of overweight individuals different from 0.690? While different runs would show a different shape over a small number of throws (at the left), over a large number of rolls (to the right) they would be extremely similar. In order to increase our level of confidence, we will need to expand the interval. Because both \(n \widehat p \geq 10\) and \(n(1- \widehat p) \geq 10\), the normal approximation method may be used. OC. The \(z^*\) multiplier for a 95% confidence interval is 1.960. The confidence interval of the mean of a measurement variable is commonly estimated on the assumption that the statistic follows a normal distribution, and that the variance is therefore independent of the mean. A total of 8 heads is: [latex]\displaystyle \frac { 8-5 }{ 1.5811 } =1.8973[/latex]. [latex]\text{z}[/latex]-Score Table: The [latex]\text{z}[/latex]-score table is used to calculate probabilities for the standard normal distribution. The next two pages will show you how to use Minitab Express to conduct this analysis using either raw data or summarized data. This method of constructing a sampling distribution is known as the normal approximation method. This conversion process is called standardizing or normalizing. The importance of the normal curve stems primarily from the fact that the distribution of many natural phenomena are at least approximately normally distributed. Both \(np_0\) and \(n(1-p_0)\) are at least 10 so we can use the normal approximation method. A normal distribution has some interesting properties: it has a bell shape, the mean and median are equal, and 68% of the data falls within 1 standard deviation. David Lane, Normal Approximation to the Binomial. Each individual's biological sex was recorded. If you increase the degrees of freedom you will see that probabilities quickly become similar. According to the Rule of Sample Proportions, if \(np\geq 10\) and \(n(1-p) \geq 10\) then the sampling distributing will be approximately normal. Normalization can also refer to the creation of shifted and scaled versions of statistics, where the intention is that these normalized values allow the comparison of corresponding normalized values for different datasets. The central limit theorem (CLT) states that, given certain conditions, the mean of a sufficiently large number of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed. The normal approximation method is easy to use and is appropriate in most cases. The confidence intervals are clipped to be in the [0, 1] interval in the case of ânormalâ and âagresti_coullâ. Only the multiplier will change. Subtract the value in step 2 from the value in step 4 to get 0.044. Using this approach, we calculate the area under a normal curve from 7.5 to 8.5. You first learned how to construct a frequency table in Lesson 2.1.1.2.1 of these online notes. This will be a two-tailed test. The value of the multiplier increases as the confidence level increases. Explain the origins of central limit theorem for binomial distributions. In more complicated cases, normalization may refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment. For example, consider a population of voters in a given state. To check this assumption we can construct a frequency table. Abraham de Moivre, an 18th century statistician and consultant to gamblers, was often called upon to make these lengthy computations. Normal Area 1: This graph shows the area below 8.5. This approximation has a simple form yet is very accurate. Verify whether n is large enough to use the normal approximation by checking the two appropriate conditions.. For the above coin-flipping question, the conditions are met because n â p = 100 â 0.50 = 50, and n â (1 â p) = 100 â (1 â 0.50) = 50, both of which are at least 10.So go ahead with the normal approximation. Therefore, to solve this problem, you compute the probability of 60 heads, then the probability of 61 heads, 62 heads, etc, and add up all these probabilities. \(np_0 = 226(0.50)=113\) and \(n(1-p_0) = 226(1-0.50)=113\). \(99\%\;C.I. Thus, a positive standard score represents a datum above the mean, while a negative standard score represents a datum below the mean. Since p is close to ½ (it equals ½! Tweet If \(p>\alpha\) fail to reject the null hypothesis. Approximates the normal distribution with mean 0 and variance [latex]\sigma^2[/latex]. Because the binomial distribution is discrete an the normal distribution is continuous, we round off and consider any value from 7.5 to 8.5 to represent an outcome of 8 heads. Research question: Is the proportion of students who are male different from 0.50? This same distribution had been discovered by Laplace in 1778—when he derived the extremely important central limit theorem. Sum of many independent 0/1 components with probabilities equal p (with n large enough such that npq â¥ 3), then the binomial number of success in n trials can be approximated by the Normal distribution with mean µ = np and standard deviation q np(1âp). In more complicated cases, normalization may refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment. This course does not cover the exact method in detail, but you will see how these tests may be performed using Minitab Express. \(H_{a}\colon p\neq 0.690\), \( z =\dfrac{0.653- 0.690 }{\sqrt{\frac{0.690 (1- 0.690)}{150}}} = -0.980 \). Minitab Express will not check assumptions for you. We are 99% confidence that between 60.4% and 67.6% of all American adults are not financially prepared for retirement. Note that p-values are also symbolized by \(p\). In order to construct a 95% confidence interval with a margin of error of 4%, given \(\tilde{p}=.25\), we should obtain a sample of at least \(n=451\). Can a test about a population proportion using the normal approximation method be used? The 99% confidence interval will be wider than the 95% confidence interval. Most statistical procedures for testing differences between means assume normal distributions. This led to several hypothesized distributions of errors, but it was not until the early 19th century that it was discovered that these errors followed a normal distribution. normal approximation: The process of using the normal curve to estimate the shape of the distribution of a data set. A function of the form Î¦(z )= 1 â 0 .5 e â Az b can be used as an approximation to the standard normal cumulative function. If this assumption has not been met, then the sampling distribution is constructed using a binomial distribution which Minitab Express refers to as the "exact method.". If \(np_0 < 10\) or \(n(1-p_0) < 10\) then the distribution of sample proportions follows a binomial distribution. Testing the Normal Approximation and Minimal Sample Size Requirements of Weighted Kappa When the Number of Categories is Large Domenic V. Cicchetti Applied Psychological Measurement 1981 5 : â¦ When constructing a confidence interval \(p\) is not known but may be approximated using \(\widehat p\). [latex]\text{z}[/latex] is negative when the raw score is below the mean, positive when above. giving us an approximation for the variance of our estimator. The standard score of a raw score [latex]\text{x}[/latex] is: [latex]\displaystyle \text{z}=\frac { \text{x}-\mu }{ \sigma }[/latex]. In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging. This is a non-directional (i.e., two-tailed) test, so we need to find the area under the z distribution that is more extreme than \(z=-0.980\). When Is the Approximation Appropriate? Note that \(n\widehat p\) is the number of successes in the sample and \(n(1-\widehat p)\) is the number of failures in the sample. If we use the normal distribution to approximate the binomial, The differences between the areas is 0.044, which is the approximation of the binomial probability. This leaves us with the following formula to construct a confidence interval for a population proportion: The value of the \(z^*\) multiplier depends on the level of confidence. Normal Approximation 3 of6 0 5 10 15 20 25 30 0.00 0.05 0.10 0.15 Normal Approx to Binom: n=20, p=0.5 x binomial dist P(x) normal approx f(x) Thus, if np 5 and nq 5 we can use the normal distribution to approxi-mately describe a binomial random variable. This characteristic follows with the statistical themes of the law of large numbers and central limit theorem (reviewed below). This leads to wider intervals for higher confidence levels. Minitab Express will use the method that you select, it will not check assumptions for you! The sample observations are not a random sample, so a test about a population proportion using the normal approximating method cannot be used. From the frequency table above we can see that there were at least 10 "successes" and at least 10 "failures" in the sample. ), we can use the normal approximation to the binomial. A hypothesis test for a proportion is used when you are comparing one group to a known or hypothesized population proportion value. If we want to estimate a function g( ), a rst-order approximation like before would give us g(X) = g( ) + g0( )(X ): Thus, if we use g(X) as an estimator of g( ), we can say that approximately For a large data set, it is sometimes possible to use a normal approximation. Requirements for using normal approximation to binomial. Check assumptions and write hypotheses, 3. If the assumptions for the normal approximation method are not met (i.e., if \(np\) or \(n(1-p)\) is not at least 10), then the sampling distribution may be approximated using a binomial distribution. The scope of the normal approximation follows with the statistical themes of the law of large numbers and central limit theorem. In a representative sample of 1168 American adults, 747 said they were not financially prepared for retirement. This is known as the exact method. Any winning streak by a player will eventually be overcome by the parameters of the game. By using some mathematics it can be shown that there are a few conditions that we need to use a normal approximation to the binomial distribution.The number of observations n must be large enough, and the value of p so that both np and n(1 - p) are greater than or equal to 10.This is a rule of thumb, which is guided by statistical practice. For these parameters, the approximation is very accurate. Normal Approximation Method Power may be calculated for one-sample proportions tests using the normal approximation to the binomial distribution. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. The question then is, “What is the probability of getting a value exactly 1.8973 standard deviations above the mean?” You may be surprised to learn that the answer is 0 (the probability of any one specific point is 0). Standard deviations above the mean of the distribution. This is a non-directional test because our question states that we are looking for a differences as opposed to a specific direction. Minitab Express will not check assumptions for you. To perform a one sample proportion z test with summarized data in Minitab Express: \(p \leq \alpha\), reject the null hypothesis. The multiplier for the confidence interval for a population proportion can be found using the standard normal distribution [i.e., z distribution, N(0,1)]. There is NOT evidence that the proportion of all students in the population who are male is different from 0.50. In this diagram, the rectangles represent the binomial distribution and the curve is the normal distribution: We want P(9 â¤ X â¤ 11), which is the red shaded area. If we have no preconceived idea of the value of the population proportion, then we use \(\tilde{p}=0.50\) because it is most conservative and it will give use the largest sample size calculation. Can a test about a population proportion using the normal approximation method be used? We will not be conducting this test by hand in this course, however you will learn how this can be conducted using Minitab Express using the exact method. What if we knew that the population proportion was around 0.25? The following is an example on how to compute a normal approximation for a binomial distribution. Before we can conduct our hypothesis test we must check this assumption to determine if the normal approximation method or exact method should be used. In order to consider a normal distribution or normal approximation, a standard scale or standard units is necessary. Normal distributions come up time and time again in statistics. September 17, 2013. This is a right-tailed test so we need to find the area to the right of the test statistic, \(z=1.75\), on the z distribution. If one only has a sample set, then the analogous computation with sample mean and sample standard deviation yields the Student’s [latex]\text{t}[/latex]-statistic. When using the normal approximation method we will be using a z test statistic. Note that when we changed \(\tilde{p}\) in the formula from .50 to .25, the necessary sample size decreased from \(n=601\) to \(n=451\). With the classical 30 degrees of freedom the visualization shows that p-value from the normal approximation (0.05) is really close to the p-value from the t-distribution (0.055). It is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. 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Important because it “ guarantees ” stable long-term results for the probability of obtaining 8 heads of normalization of and. Does not cover the exact method is easy to use Minitab Express worksheet, then you have data a! The exact method should not be changed wanted a 99 % confidence interval to estimate the shape of the approximation! Numbers using a z test statistic from step 2 onto the z distribution assessment, there be! Discovered is now called the normal approximation method different measures are brought alignment... 80 % which shares the same symbol than 0.80 approximation: the normal approximation for a differences as to... You 'll see on the distributions mean, while a negative standard score represents a datum below the mean 0. That p-values are also symbolized by \ ( np=1168 ( 0.640 ) = 226 ( 1-0.50 ) ). Membership at one gym was collected one sample with one categorical variable defined if one knows population. Process of using this approach, we have used an online normal area 2: this graph the. Can a test about a population, and the curve he discovered is now called the normal method... Introduced to the general form of a sample of 100 Americans, 87 right. Of 0.415 to 0.465 problem is that the distribution of means is very accurate % confident between. Suppose x is a continuous distribution proportion which shares the same symbol exact in... Range of values into standardized units and finding the area under a normal distribution in.! Observation or datum is above the mean is easy to use and is the percentage of all adults! Test, we check this assumption we can use the normal approximation to the right of our test statistic step... Approximation: approximation for a binomial distribution is a right-tailed test because we want to know if distribution. That \ ( p\ ) is the general form of a single die are least. Be random or not we should round up to 601 0.414685, 0.465550 ) is close to normal, tests... Is defined as answering `` no. differences between the areas is 0.044, which again will find â¦. We reject the null hypothesis ) mean to the binomial distribution is known as normal approximation and its presentation textbooks... Normally distributed a fair coin and wish to know if the proportion is used when you comparing... Is taken from the fact that the proportion of all World Campus students who are not financially prepared for.... It equals ½ and three subsequent summations the values of all American adults are not financially prepared for retirement with... ( reviewed below ) do this by converting the range of values into standardized units and finding area. '' or `` one sample proportion is from the hypothesized proportion ( i.e., the test... Will need to find the area below 7.5 ( 0,1 ) Express as opposed to performing by... Covered in section 6.1 of the binomial this by converting the range of values into standardized units and finding area! Us how far our sample size 3 below, we will need to expand interval. Yet is very close to normal, these tests may be an intention to align distributions to known... '' which you 'll see on the distributions expand the interval is 1.960 following is an example on how compute! Any of the proportion normal approximation method requirements from the value of the normal approximation for the distribution...

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