Hypotheses and test procedures

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.

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.

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

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

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.

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).

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.

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

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

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

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

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.

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.

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.

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.