**Confidence Intervals for Means**

In the preceding considerations
(Confidence Intervals I), the confidence interval for the mean

Suppose that the underlying distribution is normal and that

Select

Thus the observations of a random sample provide a

#### Example 1

Let *X* equals the amount of butterfat in pound produced by a typical cow during a 305-day milk production period between her first and second claves. Assume the distribution of *X* is

481 | 537 | 513 | 583 | 453 | 510 | 570 |

500 | 487 | 555 | 618 | 327 | 350 | 643 |

499 | 421 | 505 | 637 | 599 | 392 | - |

For these data,

Let *T* have a *t* distribution with *n*-1 degrees of freedom. Then, *s*. If the observed *s* is smaller than

If it is not possible to assume that the underlying distribution is normal but *t* distribution.

Generally, this approximation is quite good for many normal distributions, in particular, if the underlying distribution is symmetric, unimodal, and of the continuous type. However, if the distribution is highly skewed, there is a great danger using this approximation. In such a situation, it would be safer to use certain nonparametric method for finding a confidence interval for the median of the distribution.

**Confidence Interval for Variances**

The confidence interval for the variance

In order to find a confidence interval for *a* and *b* should selected from tabularized Chi Squared Distribution with *n*-1 degrees of freedom such that

That is select *a* and *b* so that the probabilities in two tails are equal:

Thus the probability that the random interval

It follows that

#### Example 2

Assume that the time in days required for maturation of seeds of a species of a flowering plant found in Mexico is *n*=13 seeds, both parents having narrow leaves, yielded

A confidence interval for

Although *a* and *b* are generally selected so that the probabilities in the two tails are equal, the resulting *a* and *b* that yield confidence interval of minimum length for the standard deviation.