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# Standard Error Confidence Interval Formula

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 a normal distribution

## Confidence Interval Example

Compute a confidence interval on the mean when σ is estimated View Multimedia

## 90 Confidence Interval

Version When you compute a confidence interval on the mean, you compute the mean of a sample in order to confidence interval for proportion 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 intervals are constructed, we are http://onlinelibrary.wiley.com/doi/10.1002/9781444311723.oth2/pdf 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 size of 9? Recall from the section on http://onlinestatbook.com/2/estimation/mean.html 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 90. Now consider the probability that a sample mean computed in a random sample is within 23.5

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