Confidence Interval Sample Mean Standard Error
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test AP formulas FAQ AP study guides AP calculators Binomial Chi-square f Dist Hypergeometric Multinomial Negative binomial Normal Poisson t Dist Random numbers Probability Bayes rule Combinations/permutations Factorial Event counter Wizard Graphing Scientific Financial Calculator books AP calculator review confidence interval for sample mean calculator Statistics AP study guides Probability Survey sampling Excel Graphing calculators Book reviews Glossary AP confidence interval for sample mean formula practice exam Problems and solutions Formulas Notation Share with Friends Confidence Interval: Sample Mean This lesson describes how to construct a
Confidence Interval For Sample Mean Difference
confidence interval around a sample mean, x. Estimation Requirements The approach described in this lesson is valid whenever the following conditions are met: The sampling method is simple random sampling. The sampling distribution is approximately normally
Confidence Interval Sample Standard Deviation
distributed. Generally, the sampling distribution will be approximately normally distributed when the sample size is greater than or equal to 30. The Variability of the Sample Mean To construct a confidence interval for a sample mean, we need to know the variability of the sample mean. This means we need to know how to compute the standard deviation or the standard error of the sampling distribution. Suppose k possible samples of size confidence interval sample variance n can be selected from a population of size N. The standard deviation of the sampling distribution is the "average" deviation between the k sample means and the true population mean, μ. The standard deviation of the sample mean σx is: σx = σ * sqrt{ ( 1/n ) * ( 1 - n/N ) * [ N / ( N - 1 ) ] } where σ is the standard deviation of the population, N is the population size, and n is the sample size. When the population size is much larger (at least 20 times larger) than the sample size, the standard deviation can be approximated by: σx = σ / sqrt( n ) When the standard deviation of the population σ is unknown, the standard deviation of the sampling distribution cannot be calculated. Under these circumstances, use the standard error. The standard error (SE) can be calculated from the equation below. SEx = s * sqrt{ ( 1/n ) * ( 1 - n/N ) * [ N / ( N - 1 ) ] } where s is the standard deviation of the sample, N is the population size, and n is the sample size. When the population size is much larger (at least 20 times larger) than the sample s
normal distribution calculator to find the value of z to use for a confidence interval Compute a confidence interval on the mean when σ is known Determine whether to use a t distribution or
Confidence Interval Sample Proportion
a normal distribution Compute a confidence interval on the mean when σ is confidence interval population mean estimated View Multimedia Version When you compute a confidence interval on the mean, you compute the mean of a confidence interval median sample in order to estimate the mean of the population. Clearly, if you already knew the population mean, there would be no need for a confidence interval. However, to explain how confidence http://stattrek.com/estimation/confidence-interval-mean.aspx?Tutorial=AP intervals are constructed, we are going to work backwards and begin by assuming characteristics of the population. Then we will show how sample data can be used to construct a confidence interval. Assume that the weights of 10-year-old children are normally distributed with a mean of 90 and a standard deviation of 36. What is the sampling distribution of the mean for a sample http://onlinestatbook.com/2/estimation/mean.html size of 9? Recall from the section on the sampling distribution of the mean that the mean of the sampling distribution is μ and the standard error of the mean is For the present example, the sampling distribution of the mean has a mean of 90 and a standard deviation of 36/3 = 12. Note that the standard deviation of a sampling distribution is its standard error. Figure 1 shows this distribution. The shaded area represents the middle 95% of the distribution and stretches from 66.48 to 113.52. These limits were computed by adding and subtracting 1.96 standard deviations to/from the mean of 90 as follows: 90 - (1.96)(12) = 66.48 90 + (1.96)(12) = 113.52 The value of 1.96 is based on the fact that 95% of the area of a normal distribution is within 1.96 standard deviations of the mean; 12 is the standard error of the mean. Figure 1. The sampling distribution of the mean for N=9. The middle 95% of the distribution is shaded. Figure 1 shows that 95% of the means are no more than 23.52 units (1.96 standard deviations) from the mean of
our multiplier in our interval used a z-value. But what if our variable of interest is a quantitative variable (e.g. GPA, Age, Height) and we want to estimate the population mean? In such a situation proportion confidence intervals are not https://onlinecourses.science.psu.edu/stat200/node/49 appropriate since our interest is in a mean amount and not a proportion. We apply similar techniques when constructing a confidence interval for a mean, but now we are interested in estimating the population mean (\(\mu\)) http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_Confidence_Intervals/BS704_Confidence_Intervals_print.html by using the sample statistic (\(\overline{x}\)) and the multiplier is a t value. At the end of Lesson 6 you were introduced to this t distribution. Similar to the z values that you used as the confidence interval multiplier for constructing confidence intervals for population proportions, here you will use t values as the multipliers. Because t values vary depending on the number of degrees of freedom (df), you will need to use either the t table or statistical software to look up the appropriate t value for each confidence interval that you construct. Using either method, the degrees of freedom will be based on the sample size, n. Since confidence interval sample we are working with one sample here, \(df=n-1\).Finding the t* MultiplierReading the t table is slightly more complicated than reading the z table because for each different degree of freedom there is a different distribution. In order to locate the correct multipler on the t table you will need two pieces of information: (1) the degrees of freedom and (2) the confidence level. The columns of the t table are for different confidence levels (80%, 90%, 95%, 98%, 99%, 99.8%). The rows of the t table are for different degrees of freedom. The multiplier is at the intersection of the two. ExamplesCups of CoffeeA research team wants to estimate the number of cups of coffee the average Penn State student consumes each week with 95% confidence. They take a random sample of 20 students and ask how many cups of coffee they drink each week. Average HeightSports analysts are studying the heights of college quarterbacks. They take a random sample of 55 college quarterbacks and measure the height of each. They want to construct a 98% confidence interval.Our confidence level is 98%. \(df=55-1=54\)Our t table does not provide us with multipliers for 54 degrees of freedom. To be more conservative, we will use 50 degrees of freedom because that will give us the larger multiplier.Usin
parameters based on sample statistics. There are two broad areas of statistical inference, estimation and hypothesis testing. Estimation is the process of determining a likely value for a population parameter (e.g., the true population mean or population proportion) based on a random sample. In practice, we select a sample from the target population and use sample statistics (e.g., the sample mean or sample proportion) as estimates of the unknown parameter. The sample should be representative of the population, with participants selected at random from the population. In generating estimates, it is also important to quantify the precision of estimates from different samples. Learning Objectives After completing this module, the student will be able to: Define point estimate, standard error, confidence level and margin of error Compare and contrast standard error and margin of error Compute and interpret confidence intervals for means and proportions Differentiate independent and matched or paired samples Compute confidence intervals for the difference in means and proportions in independent samples and for the mean difference in paired samples Identify the appropriate confidence interval formula based on type of outcome variable and number of samples Parameter Estimation There are a number of population parameters of potential interest when one is estimating health outcomes (or "endpoints"). Many of the outcomes we are interested in estimating are either continuous or dichotomous variables, although there are other types which are discussed in a later module. The parameters to be estimateddepend not only on whether the endpoint is continuous or dichotomous, but also on the number of groups being studied. Moreover, when two groups are being compared, it is important to establish whether the groups are independent (e.g., men versus women) or dependent (i.e., matched or paired, such as a before and after comparison). The table below summarizes parameters that may be important to estimate in health-related studies. Parameters Being Estimated Continuous Variable Dichotomous Variable One Sample mean proportion or rate, e.g., prevalence, cumulative incidence, incidence rate Two Independent Samples difference in means difference in proportions or rates, e.g., risk difference, rate difference, risk ratio, odds ratio, attributable proportion Two Dependent, Matched Samples mean difference Confidence Intervals There are two types of