Compute Population Mean Margin Error 90 Confidence Interval Sigma 4
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April 1 to April 3, 2015, a national poll surveyed 1500 American households to gauge their levels of discretionary spending. The question asked was how much the respondent confidence interval for population mean calculator spent the day before; not counting the purchase of a home, motor margin of error calculator vehicle, or normal household bills. For these sampled households, the average amount spent was \(\bar x\) = \$95
Calculate Margin Of Error
with a standard deviation of s = \$185.How close will the sample average come to the population mean? Let's follow the same reasoning as developed in section 10.2 for proportions.
How To Find Confidence Interval
We have:\[\text{Sample average} = \text{population mean} + \text{random error}\]The Normal Approximation tells us that the distribution of these random errors over all possible samples follows the normal curve with a standard deviation of \(\frac{\sigma}{\sqrt{n}}\). Notice how the formula for the standard deviation of the average depends on the true population standard deviation \(\sigma\). When the population standard deviation is unknown, like how to construct a confidence interval in this example, we can still get a good approximation by plugging in the sample standard deviation (s). We call the resulting estimate the Standard Error of the Mean (SEM).Standard Error of the Mean (SEM) = estimated standard deviation of the sample average =\[\frac{\text{standard deviation of the sample}}{\sqrt{n}} = \frac{s}{\sqrt{n}}\]In the example, we have s = \$185 so the Standard Error of the Mean =\[\frac{\text{\$185}}{\sqrt{1500}} = \$4.78\]Recap: the estimated daily amount of discretionary spending amongst American households at the beginning of April, 2015 was \$95 with a standard error of \$4.78The Normal Approximation tells us, for example, thatfor 95% of all large samples, the sample average will be within two SEM of the true population average. Thus, a 95% confidence interval for the true daily discretionary spending would be \$95 ± 2(\$4.78) or\$95 ± \$9.56.Of course, other levels of confidence are possible. When the sample size is large, s will be a good estimate of \(\sigma\) and you can use multiplier numbers from the normal curve. When the sample size is smaller (say n < 30), then s will be
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Population Standard Deviation
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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 http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_Confidence_Intervals/BS704_Confidence_Intervals_print.html 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. confidence interval 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 margin of error 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 estimates for each populationparameter: the point estimate and confidenc
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