Confidence Intervals

Question 1

How do we derive the formula for a confidence interval?

Answer 1

First, recall the STANDARD NORMAL VARIABLE is

Z = (Xbar - mu) / (sigma /sqrt(n))

Because the AREA under the STANDARD NORMAL CURVE between -1.96 and 1.96 is .95,

P(-1.96 < (Xbar - mu) / (sigma /sqrt(n)) < 1.96) = .95

Now manipulate the inequalities inside the parentheses s.t. they appear in the equivalent form l < mu < u.

1. multiply through by sigma/sqrt(n):
   - 1.96 *sigma/sqrt(n) < Xbar-mu < 1.96 *sigma/sqrt(n)
2. subtract Xbar from each term:
   - Xbar - 1.96 *sigma/sqrt(n) < -mu < -Xbar + 1.96 *sigma/sqrt(n)
3. Multiply through by -1 to eliminate the minus sign which reverses the direction of the inequality:
   Xbar + 1.96 *sigma/sqrt(n) > mu > Xbar - 1.96 *sigma/sqrt(n)

Xbar - 1.96 *sigma/sqrt(n) < mu < Xbar + 1.96 *sigma/sqrt(n)

The equivalence of ea set of inequalities to the original set implies that

P(Xbar - 1.96 *sigma/sqrt(n) < mu < Xbar + 1.96 *sigma/sqrt(n) ) = .95

Think of a RANDOM INTERVAL with endpoints
Xbar +-1.96 *sigma/sqrt(n)

Then in interval notation we get

(Xbar -1.96 *sigma/sqrt(n), Xbar +1.96 *sigma/sqrt(n))

in summary, the interval is a RANDOM INTERVAL that “captures” the true value of mu

Question 2

How should we interpret a confidence interval w.r.t. its confidence value?

Answer 2

A correct interpretation of “95% confidence” relies on the LONG-RUN relative FREQUENCY interpretation of PROBABILITY:

Let A be the event that TRUE mu is within the interval Xbar +/- 1.96*sigma/sqrt(n). To say that event A of being CAPTURED BY AN INTERVAL has probability .95 is to say that if the experiment on which A is defined is performed infinite times (because Xbar is a SAMPLING statistic), in the long run A will occur 95% of the time.

Recall that Xbar will FLUCTUATE around the TRUE MEAN mu. Then the CI will ALSO FLUACTUATE, s.t. any CI constructed around Xbar may NOT always contain the TRUE mu.

Question 3

What is the general formula to compute an 100(1-alpha)% confidence interval?

Answer 3

A 100(1-alpha)% CI for mean mu of a NORMAL POPULATION when the value of sigma is KNOWN is given by
xbar +/- z_alpha/2*sigma/sqrt(n)