I'm a student in AP Stats and our teacher just told us that z* is the critical value used for calculating a confidence interval, but I was wondering where it actually came from. Like is there a connection between the z* value and a z distribution - why do we need to multiply the margin of error by z*? Thank you so much and have a nice day!
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$\begingroup$ What definition of a confidence interval are you familiar with? $\endgroup$– whuber ♦Commented Feb 29 at 18:36
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$\begingroup$ does this help, @Takki? statisticshowto.com/probability-and-statistics/… $\endgroup$– Alex JCommented Mar 1 at 0:04
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2 Answers
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There is a one-to-one mapping between values of z and p-values (once you specify one- or two-sided). So you can work in two directions:
- Compute z from data and then determine the corresponding p-value.
- Specify a p-value (often 0.05) and determine the corresponding value of z. This is what your teacher referred to as z*. It is also called a critical value of z.
There is a similar relationship between t and t*.
answered Feb 29 at 23:07
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z* in this question represents a critical value of the standard normal distribution, sometimes known as the z distribution. This distribution has a probability density that is a bell-shaped curve, and a critical value corresponds to a cutoff such that the area under the curve (i.e. the associated probability) that is further away from zero than z* matches the desired significance level of a (two-sided) test. The calculation of these critical values is difficult. It is known that there are no closed formulas for the areas in question or the associated critical values. However the probability associated with a particular cutoff value can be expressed as an integral of a known function, and it is possible to approximate the value of the integral using a series expansion.
I'm not 100% sure I know what the original question author means by margin of error in this setting. However it is customary to construct a confidence interval for a mean by choosing an interval centered on the estimated mean and choosing an interval that is z* times the estimated standard deviation of the estimated mean. This accounts for the difference between the standard deviation of the estimated mean and the standard deviation of the z distribution itself, which is one by definition. So the multiplication is just rescaling the z distribution so that it matches the scale of the estimated mean.
