## confidence interval for one population mean

Rather, it reflects the amount of random error in the sample and provides a range of values that are likely to include the unknown parameter. In practice, however, we select one random sample and generate one confidence interval, which may or may not contain the true mean. To illustrate let's first take a small subset of the adult respondents to the Weymouth Health Survey and create a small data set consisting of their ages. To get the 95% confidence interval ,then use the prop.test()function. In health-related publications a 95% confidence interval is most often used, but this is an arbitrary value, and other confidence levels can be selected. Instead of "Z" values, there are "t" values for confidence intervals which are larger for smaller samples, producing larger margins of error, because small samples are less precise. The sample is large, so the confidence interval can be computed using the formula: So, the 95% confidence interval is (0.329, 0.361). I can can compute both the point estimate (mean) and the 95% confidence interval using the t.test() command. In public health this is most commonly done by computing a confidence interval. How do I gauge the precision of an estimated mean or an estimated proportion in a single sample? Substituting the sample statistics and the t value for 95% confidence, we have the following expression: Interpretation: Based on this sample of size n=10, our best estimate of the true mean systolic blood pressure in the population is 121.2. Using the subsample in the table above, what is the 90% confidence interval for BMI? We can use the Weymouth health survey data to get the counts of those with or without a history of diabetes using the table() function: Then find the denominator (sum of those with or without diabetes). in which one is conducting hypothesis testing. Consequently, one can always use a t score, even with large sample. We can substitute the equation for Z from the central limit theorem into this equation in order to derive an expression for computing the 95% confidence interval for the population mean, as follows: Link to the step-by-step derivation of this equation. For both continuous variables (e.g., population mean) and dichotomous variables (e.g., population proportion) one first computes the point estimate from a sample. The formulas for confidence intervals for the population mean depend on the sample size and are given below. In other words. If we call treatment a "success", then x=1219 and n=3532. So, the 95% confidence interval is (0.120, 0.152). The sample proportion is p̂ (called "p-hat"), and it is computed by taking the ratio of the number of successes in the sample to the sample size, that is: If there are more than 5 successes and more than 5 failures, then the confidence interval can be computed with this formula: The point estimate for the population proportion is the sample proportion, and the margin of error is the product of the Z value for the desired confidence level (e.g., Z=1.96 for 95% confidence) and the standard error of the point estimate. If there are fewer than 5 successes or failures then alternative procedures, called exact methods, must be used to estimate the population proportion. When the outcome of interest is dichotomous like this, the record for each member of the sample indicates having the condition or characteristic of interest or not. Another way of thinking about a confidence interval is that it is the range of likely values of the parameter with a specified level of confidence (which is similar to a probability). Looking down to the row for 9 degrees of freedom, you get a t-value of 1.833. Table - Z-Scores for Commonly Used Confidence Intervals. [Note: Both the table of Z-scores and the table of t-scores can also be accessed from the "Other Resources" on the right side of the page. Scroll down to the bottom of the table and note that as the sample size becomes larger, the t values become closer to the z value listed at the bottom of the table. Because you want a 95% confidence interval, your z*-value is 1.96. We select a sample and compute descriptive statistics including the sample size (n), the sample mean, and the sample standard deviation (s). The sample size is large and satisfies the requirement that the number of successes is greater than 5 and the number of failures is greater than 5. and the sampling variability or the standard error of the point estimate. One can compute confidence intervals all types of estimates, but this short module will provide the conceptual background for computing confidence intervals and will then focus on the computation and interpretation of confidence intervals for a mean or a proportion in a single group. For Z? Import this data file into R, and compute the mean and 95% confidence interval for the variable "weight," which is the weight of the adult household respondent in pounds, and interpret the result in a sentence. Example: During the 7th examination of the Offspring cohort in the Framingham Heart Study there were 1219 participants being treated for hypertension and 2,313 who were not on treatment. Suppose we compute a 95% confidence interval for the true systolic blood pressure using data in the subsample. Just as with large samples, the t distribution assumes that the outcome of interest is approximately normally distributed. If we are interested in a confidence interval for the mean, we can ignore the t-value and p-value, and focus on the 95% confidence interval. The prevalence of cardiovascular disease (CVD) among men is 244/1792=0.1362. The precision of a confidence interval is defined by the margin of error (or the width of the interval). A town wide health survey was conducted in Weymouth, MA in 2002. With 95% confidence, the true mean is in the range of 167.5 to 170.8 pounds. Standard errors represent variability in estimates of a mean or proportion; i.e., if one had taken many samples to estimate a mean or proportion, the standard error is the estimated standard deviation of the sampling means or sampling proportions.

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