Confidence Interval: Definition

The other way of looking at or using CI's is to estimate the mean of the population from the mean of a sample. First we conjectured a mean for the population, e.g., here it was 100. Then we went through the rest of the steps of the procedure and obtained a CI that was set up around the sample's mean. We had 95% confidence that the mean of the population would be found in this interval. We backed off this position when we realized that we could draw several different samples and come up with several different CI's. The mean of the population could be declared most any number.

What is the difference between confidence intervals and hypothesis ..

The final table from PROC STDRATE presents the two risk estimates and their confidence intervals.

confidence intervals and hypothesis ..

Historically, Hypothesis Testing was the first tool ofstatistical inference that was created to answer questions like "does thisgrowing method improve the yield of cotton?", or "are babies lesslikely to have a particular disease if we know the health of theirparents?" Confidence Interval, on the other hand, was invented later, andit was used in situations where people don't necessary want a "yes"or "no" answer.

Confidence Interval & Hypothesis Testing of Proportion

In this graph, the REFLINE statement is used to display (in red) the observed value of the statistic for the original data. A second REFLINE statement plots (in blue) an approximate 95% confidence interval for the skewness parameter, which is computed in the next section. The bootstrap confidence interval contains 0, thus you cannot conclude that the skewness parameter is significantly different from 0.


Confidence Interval and Hypothesis Test for a Proportion

As it turns out, there are a couple of ways of looking at confidence intervals. CI's have different meanings depending upon whether they are used in hypothesis testing (as above) or in parameter estimation such as estimating a population mean. Since we already looked at hypothesis testing, the following is an example of parameter estimation.

Comparison Confidence Interval and Hypothesis Testing

Recall that confidence intervals measure uncertainty of an estimate.

The asymptotic normality of the ML estimate, which gives the approximation

Hypothesis tests and confidence intervals for a …

The standard deviation of the bootstrap distribution is an estimate for the standard error of the statistic. If the sampling distribution is approximately normal, you can use this fact to construct the usual Wald confidence interval about the observed value of the statistic. That is, if T is the observed statistic, then the endpoints of the 95% two-sided confidence interval are T ± 1.96 SE. (Or use the so-called bootstrap t interval by replacing 1.96 with tα/2, n-1.) The following call to PROC MEANS produces the standard error (not shown):

Testing of Hypothesis and Confidence Interval

The (NNT) can be estimated in various ways. Since NNT is equal to the reciprocal of the risk difference, one way is to obtain the risk difference estimate and standard error from PROC FREQ and then use the delta method to obtain a standard error and confidence limits for NNT. Alternatively, a modeling approach could be used by fitting a logistic model and estimating the appropriate nonlinear function of the logistic model parameters.

Confidence Interval for a mean Hypothesis ..

Suppose the population mean is unknown but conjectured or assumed to be some number. A simple sample of certain N is drawn from this population that has an unknown mu. The mean is calculated for this sample resulting in Xbar, as well as an error term such as the SE or sXbar. Since the mean for the population is unknown, the mean of the sample from the population is the best one-point estimator of the population mean. The CI is set up around this Xbar using the formula above for a 95% confidence interval. Remember this confidence interval is set up around the sample statistic or a single sample mean and not around the population parameter. In this case the interpretation would be that there is 95% confidence that mu or the population mean resides in this interval around the single sample mean.