Confidence Intervals and Tests of Hypotheses

IT233: Applied Statistics TIHE 2005

Confidence Intervals and Tests of Hypotheses

Confidence intervals and tests for
Confidence intervals and tests for
Confidence intervals and tests for
Confidence intervals and tests for

Confidence Intervals for

Theorem: If is the variance of a random sample of size from a normal population, a confidence interval for is

where and are - values with degrees of freedom, leaving areas and , respectively, to the right.

Proof:

The confidence interval for can be derived by using the statistic

Consider the following figure:

To derive the confidence interval, we begin by giving a probability statement based on the above figure that

Substituting for , we write

To isolate in the centre of the inequality, we divide each term by and then inverting each term, we obtain

The desired confidence interval for can be read as:

Example 1: A manufacturer of batteries claims that his batteries will last, on average, 3 years with a variance of 1 year. If 5 of these batteries have lifetimes years construct a 95% C.I. for and decide if the claim that is valid.

Solution:First we find the summary data:

= 5, = 15, = 48.26

= 0.815

NOTE: You could use your Calculator to find:

We have,

To answer the last part of the problem, that is, “Is the claim that valid?”, we use the confidence interval method.

Answer:Since above interval reveals that the value lies in the internal, we can accept the claim with a 95% confidence.

Notes: (1) Do not use confidence interval method for testing any claim, if the test is one-tailed.
(2) You could also use the following traditional method.

Test of Hypothesis for

The traditional method of hypothesis testing for the above example is as follows:

Null hypothesis:

Alternative hypothesis: (Two-tailed test)

Critical Region:

=0.025

= 0.025

Acceptance

Region

0 0.484 11.1434

Test Statistic

Conclusion: Since the value of test statistic falls in the acceptance region, we accept .