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in estimating a population mean with a confidence interval based on a random sample x1,...,xn from a normal distribution with an unknown mean and a known variance, compare the length of the 95% and 90% confidence intervals for the unknown mean based on x1,...,xn.

Sagot :

95% confidence interval is wider than 90% confidence interval.

Given as that unknown mean and a known variance.

let's suppose parameter for mean as µ and

for standard deviation σ.

we have sample of size n  as x1,x2,x3,x4.......xn.

Confidence means what degree of assurance do we desire that the interval estimate contains the  populace mean µ.

The likelihood that the interval falls within the level of confidence c

The population parameter is contained in estimate.

If the level of confidence is 90%, this means that we are 90%

confident that the interval contains the population mean, µ

The corresponding z-scores for 90% confidence interval are ± 1.645.

If the level of confidence is 95%, this means that we are 95%

confident that the interval contains the population mean, µ

The corresponding z-scores for 95% confidence interval are ± 1.96.

So In comparison to a 95% Confidence Interval, a 90% Confidence Interval would be narrower. This happens because the reliability of an interval containing the actual mean declines as the confidence interval's precision rises (i.e., CI width decreases) (less of a range to possibly cover the mean). Consider the case of 100% confidence; the interval must contain all possible values in order to guarantee that the mean is captured with 100% certainty.

To know more about normal distribution here,

https://brainly.com/question/13906025

#SPJ4

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