CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE - New York University

20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE

If the population standard deviation is unknown, as it usually will be in practice, we will have to estimate it by the sample standard deviation s.

Since is unknown, we cannot use the confidence intervals described previously. The practical versions presented here use s in place of .

Eg 1: A random sample of 8 "Quarter Pounders" yields a mean weight of x = 0.2 pounds, with a sample standard deviation of s= 0.07 pounds. Construct a 95% CI for the unknown population mean weight for all "Quarter Pounders". (Solution later in this handout).

Remember that s is the square root of the sample variance,

s2

=

n

1 -

1

n

i =1

(

xi

-

x)2

where x1 xn are the data values in the sample.

There are two cases to consider: "Large Sample" and "Small Sample".

? If n before.

30 (and The CI is

is X

?unzkn2owsnn),. just

replace

by

s

and

proceed

as

This works because X - ? is approximately standard normal, sn

regardless of the population distribution, when n 30.

We can think of

s n

as the estimated standard error of X.

? Suppose n < 30 (and is unknown). To get a valid CI for ? in this case, we must assume that the population distribution is normal. This assumption is hard to check, and was not required before.

The CI is

X ? t 2

s n

,

where

t

2

is

defined

below.

? Don't forget what you learned earlier: If is known, we use

X ? z 2

, regardless of the sample size. n

So the only time

when you must remember to use t/2 is when is unknown and

n < 30. Otherwise, use z/2.

SUMMARY OF CONFIDENCE INTERVALS

n 30 n < 30

Sigma Known

X ? z 2

n

X ? z 2

n

Sigma Unknown

X ? z 2

s n

X ? t 2

s n

* Must assume Normal population if n < 30, unknown.

What is t/2, and why must we use it when n ................
................

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