Chapter 10 Beginning of Inference

This chapter introducecs important methods under the highly unrealistic conditions where we know the population standard deviation but not the population mean.

Point estimates for the average value of X found through samples are generally good estimates, but they are wrong. You can generate a better estimate by creating a confidence interval.

The confidence interval =
x-bar +/- Z* times sigma of x / (Sqrt n).

We get Z* from the t-table for a specific confidence level, for instance when we want a 95% we use 1.96.

In creating a complete solution we first write down all of the given information. Define your variable. Then we determine whether the central limit theorem has "kicked in" or if the underlying data were already normally distributed. Be sure to address whether the data were from a SRS. Graph them if you have them to make sure there are no gaps or outliers. Is the sample size less than 1/10 of the population size???

Identify what you are trying to produce-- a 95% Z interval for mu and give the formula. Show how the numbers are plugged in and calculate the interval.

Write the interpretation of your interval.

We are 95% confident that the true population mean value of [insert the contextual information here] falls between [lower bound] and [upper bound].

Refer to your notes for all of the baaaaaaaaaaaaad interpretations of a confidence interval and NEVER use them. :)

If a value of mu had been proposed before we collected our sample, we could see if the value falls within our interval. If the proposed value does fall in the interval, then it is a reasonable value, although not necessarily correct. If it does not fall in the interval, then it is not a reasonable value according to our sample values.

Hypothesis tests

For hypothesis tests, you develop a null and alternate hypothesis BEFORE you collect data. Both hypotheses use the parameter (NEVER THE STATISTIC) and they are considered logical opposites. The null hypotheses ALWAYS has an "equals" aspect to it: the alternate hypothesis is always <, >, or not equal to.

For instance: H0: mu = 15
Ha: mu > 15.

Although these are not actually opposites, finding evidence that mu is less than 15 provides no support of the alternative hypothesis. You can think of the null hypothesis in this case as mu<=15, which still has an "equals" in it. This is the way I learned hypothesis-writing back in the day and it is still acceptable, but not as common.

Alpha, Beta, Type I error, Type II error, and Power

Alpha is the likelihood of a Type I error--accidentally rejecting the null hypothesis when it was actually correct. (Like convicting the wrong guy.)

Beta is the likelihood of a Type II error--failing to reject the null when it was wrong. (Kind of an error of omission, or not enough evidence to convict.)

Power is the likelihood that the test would have been sensitive enough to pick up the difference between the hypothesized mu and the actual mu (given some other new value for mu). This is the complement of Beta. Yes, 1 - Beta = Power. 1 - power = beta. Power + beta = 1.

Notice that alpha and beta are NEVER added together. They don't live under the same conditions--one assumes that the null was true and the other that the null was false. DO not fall into the trap of EVER adding alpha and beta together (unless you are TOLD to do it and then only if they offer you a lot of money or a passing grade on a test).

Calculating beta is easier than people think on the calculator.
(1) Figure out what the critical values are for rejection of Ho in terms of x-bar.

(2) Find the area under the curve centered at the NEW mu that falls between these critical values. You can use normalcdf(left_critical_value, right_critical_value, new mean, standard dev or error of x-bar).