1. Hypotheses and test procedures

2. Introduction
A parameter can be estimated either by a single number (point estimate) or an entire interval (a confidence interval).
However, at times this is not the aim of the investigation.
Frequently the objective is decide which of two contradictory claims about the parameter is correct.

3. Large-sample confidence intervals for a population mean (continued)
If we replace by , we introduce randomness into the denominator (in addition to that which is already in the numerator through )
Thus we would expect more variation than before.
However, for large , replacing by introduces little variability, so this variable has approximately a standard normal distribution.
Algebraic manipulation, as before, yields a confidence interval.

4. Proposition If , has approximately a standard normal distribution
This implies that is a large-sample CI for with confidence level approximately % .

5. Large sample upper and lower confidence bounds
A large sample upper confidence bound for is
A large sample lower confidence bound for is

6. Intervals based on a normal population
The confidence interval for given in Section 7.2 depends on a large value of n. However, what if n is small?
One could proceed to determine confidence intervals for each particular distribution, such as for the uniform, Gamma, etc.
Because the normal distribution is used so frequently, we focus on that one here.

7. Intervals based on a normal population (continued)
When n is small, is no longer likely to be close to , and thus the probability distribution of is much more spread out that the standard normal distribution.
The statistic has what is called a t distribution with n-1 degrees of freedom (df).

8. Properties of t distributions
Let denote a distribution with df.
Each distribution is bell-shaped and centered at 0.
Each curve is more spread out than the standard normal distribution.
As increases, the spread of the corresponding curve decreases.
As , the sequence of curves approaches the standard normal curve.

9. The confidence intervals
To obtain the confidence intervals, simply replace percentiles for with percentiles for and compute
The confidence interval for takes the form , where
An upper confidence bound for is

10. Degrees of freedom and critical values of the distribution
Note that we use n-1 as the degrees of freedom and not n
The number of degrees of freedom for T is because there are only n-1 free bits of information, once we specify
Critical values of the distribution are given in table A-9.
The df correspond to rows, the probabilities of exceeding values correspond to the columns

11. Prediction intervals
Here we predict a single future value of X rather than estimate the mean value.
The setup is that we have a random sample
and wish to predict the value of
Note that , and

12. Prediction intervals (continued)
Thus
has a standard normal distribution. Replacing by ,
has a distribution
with n-1 degrees of freedom.

13. Proposition A prediction interval (PI) at level is .
An upper prediction bound results from replacing by and discarding the “– “ part; a similar modification gives a lower prediction bound.

14. Confidence intervals for the variance of a normal population
In this case the interval is based on the statistic
That statistic has a distribution with df.

15. Form of confidence limit
Since
Replacing by gives the confidence interval for Taking the square root of the limits gives a confidence bound for Also, replace by for one-sided bounds.

16. Critical values of the distribution
Critical value for the chi-square distribution are given in Table A-7.
Table A-11 has a more complete table.