Confidence Level Standard Error Of The Mean
Contents |
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 standard error confidence interval distribution or a normal distribution Compute a confidence interval on the mean standard error confidence interval calculator when σ is estimated View Multimedia Version When you compute a confidence interval on the mean, you compute the standard error of measurement confidence interval mean of a 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.
Standard Error Confidence Interval Linear Regression
However, to explain how confidence 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 standard error confidence interval proportion distribution of the mean for a sample 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
estimated range being calculated from a given set of sample data. (Definition taken from Valerie J. Easton and John H. McColl's Statistics Glossary v1.1) The common notation for the parameter in question confidence level standard deviation is . Often, this parameter is the population mean , which is estimated
Equation For Standard Error Of The Mean
through the
Margin Of Error Confidence Interval
by the method employed includes the true value of the parameter . Example Suppose a student measuring the boiling temperature of a certain liquid observes the readings (in degrees Celsius) 102.5, 101.7, http://onlinestatbook.com/2/estimation/mean.html 103.1, 100.9, 100.5, and 102.2 on 6 different samples of the liquid. He calculates the sample mean to be 101.82. If he knows that the standard deviation for this procedure is 1.2 degrees, what is the confidence interval for the population mean at a 95% confidence level? In other words, the student wishes to estimate the true mean boiling temperature of the liquid using the http://www.stat.yale.edu/Courses/1997-98/101/confint.htm results of his measurements. If the measurements follow a normal distribution, then the sample mean will have the distribution N(,). Since the sample size is 6, the standard deviation of the sample mean is equal to 1.2/sqrt(6) = 0.49. The selection of a confidence level for an interval determines the probability that the confidence interval produced will contain the true parameter value. Common choices for the confidence level C are 0.90, 0.95, and 0.99. These levels correspond to percentages of the area of the normal density curve. For example, a 95% confidence interval covers 95% of the normal curve -- the probability of observing a value outside of this area is less than 0.05. Because the normal curve is symmetric, half of the area is in the left tail of the curve, and the other half of the area is in the right tail of the curve. As shown in the diagram to the right, for a confidence interval with level C, the area in each tail of the curve is equal to (1-C)/2. For a 95% confidence interval, the area in each tail is equal to 0.05/2 = 0.025. The value z* representin
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 https://onlinecourses.science.psu.edu/stat200/node/49 proportion confidence intervals are not 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 (\(\mu\)) 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 standard error distribution. Similar to the z values that you used as the 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. error confidence interval Using either method, the degrees of freedom will be based on the sample size, n. Since 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=5
be down. Please try the request again. Your cache administrator is webmaster. Generated Wed, 05 Oct 2016 07:37:37 GMT by s_hv1002 (squid/3.5.20)