credible interval

Examples of credible interval in the following topics:

• Variation and Prediction Intervals

  • A prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter.
  • Prediction intervals predict the distribution of individual future points, whereas confidence intervals and credible intervals of parameters predict the distribution of estimates of the true population mean or other quantity of interest that cannot be observed.
  • Then, confidence intervals and credible intervals may be used to estimate the population mean $\mu$ and population standard deviation $\sigma$ of the underlying population, while prediction intervals may be used to estimate the value of the next sample variable, $X_{n+1}$.
  • Alternatively, in Bayesian terms, a prediction interval can be described as a credible interval for the variable itself, rather than for a parameter of the distribution thereof.
  • Formulate a prediction interval and compare it to other types of statistical intervals.

• Estimating the Target Parameter: Interval Estimation

  • Interval estimation is the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter.
  • Interval estimation is the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter.
  • The most prevalent forms of interval estimation are:
  • How can we construct a confidence interval for an unknown population mean $\mu$ when we don't know the population standard deviation $\sigma$?
  • The method for calculating a confidence interval assumes that individual observations are independent.

• What Is a Confidence Interval?

  • A confidence interval is a type of interval estimate of a population parameter and is used to indicate the reliability of an estimate.
  • A confidence interval can be used to describe how reliable survey results are.
  • A confidence interval is a type of estimate (like a sample average or sample standard deviation), in the form of an interval of numbers, rather than only one number.
  • Bayesian inference provides further answers in the form of credible intervals.
  • The confidence interval approach does not allow this, as in this formulation (and at this same stage) both the bounds of the interval and the true values are fixed values; no randomness is involved.

• Descriptive or Inferential Statistics?

  • a confidence interval (or set estimate); i.e., an interval constructed using a data set drawn from a population so that, under repeated sampling of such data sets, such intervals would contain the true parameter value with the probability at the stated confidence level
  • a credible interval; i.e., a set of values containing, for example, 95% of posterior belief

• Interpreting confidence intervals

  • A careful eye might have observed the somewhat awkward language used to describe confidence intervals.
  • Incorrect language might try to describe the confidence interval as capturing the population parameter with a certain probability.
  • Another especially important consideration of confidence intervals is that they only try to capture the population parameter.
  • Our intervals say nothing about the confidence of capturing individual observations, a proportion of the observations, or about capturing point estimates.
  • Confidence intervals only attempt to capture population parameters.

• Interpreting Non-Significant Results

  • The experimenter should report that there is no credible evidence Mr.
  • This is done by computing a confidence interval.
  • If all effect sizes in the interval are small, then it can be concluded that the effect is small.
  • If the 95% confidence interval ranged from -4 to 8 minutes, then the researcher would be justified in concluding that the benefit is eight minutes or less.

• Introduction to Confidence Intervals

  • State why a confidence interval is not the probability the interval contains the parameter
  • These intervals are referred to as 95% and 99% confidence intervals respectively.
  • An example of a 95% confidence interval is shown below:
  • If repeated samples were taken and the 95% confidence interval computed for each sample, 95% of the intervals would contain the population mean.
  • It is natural to interpret a 95% confidence interval as an interval with a 0.95 probability of containing the population mean.

• Level of Confidence

  • The proportion of confidence intervals that contain the true value of a parameter will match the confidence level.
  • If confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will match the confidence level.
  • This is guaranteed by the reasoning underlying the construction of confidence intervals.
  • Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter .
  • In applied practice, confidence intervals are typically stated at the 95% confidence level.

• Lab 3: Confidence Interval (Womens' Heights)

  • The student will calculate a 90% confidence interval using the given data.
  • Now write your confidence interval on the board.
  • Using the class listing of confidence intervals, count how many of them contain the population mean µ; i.e., for how many intervals does the value of µ lie between the endpoints of the confidence interval?
  • Suppose we had generated 100 confidence intervals.
  • When we construct a 90% confidence interval, we say that we are 90% confident that the true population mean lies within the confidence interval.

• Checking model assumptions using graphs

  • If the diagnostics support the model assumptions, this would improve credibility in the findings.
  • Confidence intervals for coefficients in multiple regression can be computed using the same formula as in the single predictor model: