When the distribution from which we are sampling is not normal, the procedure of this section is still used to obtain approximate confidence bounds. Select the gamma distribution with shape parameter 1 and scale parameter 1.

The procedure works well as long as the sample size is large and the distribution is not too far from normal. Select two-sided intervals and confidence level 0.90.

It's easy to understand the observed behavior mathematically.

The increases and thus the difference is slight when the sample size is large.

Roughly speaking, variance of an estimator describes, how do estimator value ranges from dataset to dataset.

It's defined as follows: \[ \textrm[ \widehat (x) ]=E[(\widehat (x)-E[\widehat (x)])^ ] \] \[ \textrm[ \widehat (x)]=E[(\widehat (x)^2]-E[\widehat (x)]^2 \] Bias is defined as follows: \[ \textrm[ \widehat (x)]=E[\widehat(x)-f(x)]=E[\widehat(x)]-f(x) \] One could think of a Bias as an ability to approximate function.

One of the most important thing in predictive modelling is how our algorithm will cope with various datasets, both training and testing (previously unseen).

This is strictly connected with the concept of bias-variance tradeoff.

When you run the simulation, the value of the appropriate standard score is recorded in the third table and plotted as a red line on the horizontal axis.

The event that this line falls in the critical interval is equivalent to the event that the confidence interval successfully captured the mean (and thus the success indicator variable .

Select the normal distribution with mean 0 and standard deviation 2, and select two-sided intervals.

This is in contrast with the case in which the variance of the underlying distribution is known, where the length of the confidence interval is fixed. Use a derivation similar to Exercises 1 and 2 to show that a 1 - distribution .

In either case, the density of the chosen distribution is shown in the middle graph.