Range Confidence Interval Measure Expected Sampling Error
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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 spent the day before; not counting the purchase of a home, motor margin of error formula vehicle, or normal household bills. For these sampled households, the average amount spent was margin of error calculator \(\bar x\) = \$95 with a standard deviation of s = \$185.How close will the sample average come to the population mean?
Confidence Level Definition
Let's follow the same reasoning as developed in section 10.2 for proportions. 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
Confidence Interval Formula
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 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) = confidence interval for population mean 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 fairly different from \(\sigma\) for some samples - and that means that we we need a bigger multiplier number to account for that. (see the optional material on "t-multipliers" in chapter 21). Confidence Intervals for a population mean (n > 30):For large random samples a confidence interval for a population mean is given by\[\text{sample mean} \pm z^* \frac{s}{\sqrt{n}}\]where z* is a multiplier number that comes form the normal curve and determine
intervals to characterize the results. The most well-known of these are confidence intervals. However, confidence intervals are not always appropriate. In this post, we’ll take a look at the different types of intervals that
Prediction Interval
are available in Minitab, their characteristics, and when you should use them. I’ll cover confidence interval calculator confidence intervals, prediction intervals, and tolerance intervals. Because tolerance intervals are the least-known, I’ll devote extra time to explaining how they confidence interval example work and when you’d want to use them. What are Confidence Intervals? A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population https://onlinecourses.science.psu.edu/stat100/node/58 parameter. Because of their random nature, it is unlikely that two samples from a given population will yield identical confidence intervals. But if you repeated your sample many times, a certain percentage of the resulting confidence intervals would contain the unknown population parameter. The percentage of these confidence intervals that contain the parameter is the confidence level of the interval. Most frequently, you’ll use confidence intervals to bound the mean http://blog.minitab.com/blog/adventures-in-statistics/when-should-i-use-confidence-intervals-prediction-intervals-and-tolerance-intervals or standard deviation, but you can also obtain them for regression coefficients, proportions, rates of occurrence (Poisson), and for the differences between populations. Suppose that you randomly sample light bulbs and measure the burn time. Minitab calculates that the 95% confidence interval is 1230 – 1265 hours. The confidence interval indicates that you can be 95% confident that the mean for the entire population of light bulbs falls within this range. Confidence intervals only assess sampling error in relation to the parameter of interest. (Sampling error is simply the error inherent when trying to estimate the characteristic of an entire population from a sample.) Consequently, you should be aware of these important considerations: As you increase the sample size, the sampling error decreases and the intervals become narrower. If you could increase the sample size to equal the population, there would be no sampling error. In this case, the confidence interval would have a width of zero and be equal to the true population parameter. Confidence intervals only tell you about the parameter of interest and nothing about the distribution of individual values. In the light bulb example, we know that the mean is likely to fall within the range, but the 95% confidence interval does not predict
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 http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_Confidence_Intervals/BS704_Confidence_Intervals_print.html 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 confidence interval 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 margin of error 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 estimat