Suppose a group of researchers is studying the heights of high school basketball players. The researchers take a random sample from the population and establish a mean height of 74 inches. The mean of 74 inches is a point estimate of the population mean.
A point estimate by itself is of limited usefulness because it does not reveal the uncertainty associated with the estimate; you do not have a good sense of how far away this inch sample mean might be from the population mean.
What's missing is the degree of uncertainty in this single sample. Confidence intervals provide more information than point estimates. Assume the interval is between 72 inches and 76 inches. If the researchers take random samples from the population of high school basketball players as a whole, the mean should fall between 72 and 76 inches in 95 of those samples.
Doing so invariably creates a broader range, as it makes room for a greater number of sample means. A confidence interval is a range of values, bounded above and below the statistic's mean, that likely would contain an unknown population parameter.
The resulting datasets are all different where some intervals include the true population parameter and others do not. A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups, which may be related to certain features.
Calculating a t-test requires three key data values. They include the difference between the mean values from each data set called the mean difference , the standard deviation of each group, and the number of data values of each group.
Tools for Fundamental Analysis. Trading Basic Education. Advanced Technical Analysis Concepts. What are confidence intervals? The purpose of taking a random sample from a lot or population and computing a statistic, such as the mean from the data, is to approximate the mean of the population. How well the sample statistic estimates the underlying population value is always an issue. A confidence interval of the prediction is a range that is likely to contain the mean response given specified settings of the predictors in your model.
Just like the regular confidence intervals, the confidence interval of the prediction presents a range for the mean rather than the distribution of individual data points. Going back to our light bulb example, suppose we design an experiment to test how different production methods Slow or Quick and filament materials A or B affect the burn time. After we fit a model, statistical software like Minitab can predict the response for specific settings.
We want to predict the mean burn time for bulbs that are produced with the Quick method and filament type A. Minitab calculates a confidence interval of the prediction of — hours. A prediction interval is a range that is likely to contain the response value of a single new observation given specified settings of the predictors in your model.
The prediction interval is always wider than the corresponding confidence interval of the prediction because of the added uncertainty involved in predicting a single response versus the mean response. A tolerance interval is a range that is likely to contain a specified proportion of the population. To generate tolerance intervals, you must specify both the proportion of the population and a confidence level.
The confidence level is the likelihood that the interval actually covers the proportion. The light bulb manufacturer is interested in how long their light bulbs burn. The analysts randomly sample bulbs and record the burn time in this worksheet. Under Data , choose Samples in columns. In the textbox, enter Hours. To calculate the confidence interval, start by computing the mean and standard error of the sample. Remember, you must calculate an upper and low score for the confidence interval using the z-score for the chosen confidence level see table below.
For the lower interval score divide the standard error by the square root on n, and then multiply the sum of this calculation by the z-score 1. Finally, subtract the value of this calculation from the sample mean.
Therefore, with large samples, you can estimate the population mean with more precision than you can with smaller samples, so the confidence interval is quite narrow when computed from a large sample.
McLeod, S. What are confidence intervals in statistics?
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