Discuss 2 General methods with examples to find a family of Confidence Intervals in the Continuous case.?

Question

example:- Pivotal

Answer 1

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.

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If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage ( confidence level) of the intervals will include the unknown population parameter . Confidence intervals are usually calculated so that this percentage is 95%, but we can produce 90%, 99%, 99.9%, confidence intervals for the unknown parameter.

The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter (see precision). A very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter.

Confidence intervals are more informative than the simple results of hypothesis tests (where we decide 'reject H0' or 'don't reject H0') since they provide a range of plausible values for the unknown parameter.

Instructions

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Starting the Applet and setting the conditions
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This applet simulates sampling from a population with a mean of 50 and a standard deviation of 10. For each sample, the 95% and 99% confidence intervals on the mean are computed based on the sample mean and sample standard deviation. The intervals for the various samples are displayed by horizontal lines as shown below. The first two lines represent samples for which the 95% confidence interval contains the population mean of 50. The 95% confidence interval is orange and the 99% confidence interval is blue. In the third line, the 95% confidence interval does not contain the population mean; it is shown in red. In the seventh and last line shown below, the 99% interval does not contain the population mean; it is shown in white.