Question

Describe hypothesis testing. What are the important elements to running a good test? What types of errors could occur? How do I know if I should or should not accept the hypothesis?

Answer 1

Answer :

      Hypothesis testing is used to infer the result of a hypothesisperformed on sample data from a larger population. The test tells the analyst whether or not his primary hypothesis is true. Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed

Types of errors :

           Types of Errors in Hypothesis Testing

Key Terms

o       Type I error

o       Type II error

Objectives

o       Recognize the types of errors that can occur in hypothesis testing

o       Reinforce skills and understanding by way of practice problems

Error and the Significance Level

We accepted the hypothesis that a particular coin was fair because the test statistic for a large data set did not exceed a certain critical value (that is, the number of heads did not vary in a statistically significant sense from the expected value for a fair coin). Nevertheless, we can conceivably imagine a case where it just so happens that a loaded coin "looks" fair in a given set of trials. Although such an occurrence might have a very small probability, it is nonetheless possible. (Alternatively, consider a fair coin: there is a non-zero probability that a series of 1,000 flips comes up all heads! Of course, this is a small probability, but the situation indicates that hypothesis testing does not always produce correct results.)

We can assess the probability of two different types of error for a given significance level. These errors are typically termed Type I and Type II errors. Type I error involves cases where a hypothesis is true, but it is rejected because the test statistic exceeds the critical value for the significance level α. This might be considered a "false negative" result. (The above-mentioned case of a fair coin that so happens by chance to turn up heads an abnormally large number of times, resulting in the rejection of the hypothesis that the coin is fair, is an example of Type I error.) The probability of a Type I error occurring is the same as the significance level--in other words, the probability of a Type I error decreases with decreasing α. Recall that for a random variable (or test statistic) X,

Type II error occurs when the null hypothesis is false, but the data does not indicate that it should be rejected. This situation could be considered a "false positive" result. Such a case might involve a loaded coin that happens to have a fair distribution of heads and tails in a certain series of flips. The probability that a Type II error occurs is the same as the probability that the random variable (or test statistic) does not exceed the critical value c if the alternative hypothesis is assumed to be true:

The probability β is therefore the probability that a Type II error will not occur.

In light of these errors, we can see that the choice of α (and therefore β) is not purely arbitrary--this choice has a significant effect on whether the probability that the analysis yields a correct result. Although we will not go into further depth on appropriate selection of α (or, perhaps more appropriately, c) such that the probability of errors is minimized, it is important that you remain aware that your choice of α is not without consequence.

What Are the Components of a Good Hypothesis?  

A strong hypothesis should meet three fundamental criteria. It needs to state the hypothesis in the proper conditional phrasing. It needs to clearly establish the relationship between the variables included. Finally, it needs to establish that said relationship is scientifically provable.

Before anything else, the student should make certain that the hypothesis being tested is theoretically and contextually relevant to the assignment at hand. In other words, it should relate to the topic. Once the variables have been chosen, the hypothesis itself should suggest direct causality existing between them. For example, "If temperature affects leaf color, then lowering the temperature will affect that color." The emphasis here is on the "if ... then ... will" construction. It is important to remember that a hypothesis is a statement, not a question, and it is also necessary to avoid hedge words such as may or might in its formulation.

Once a viably stated hypothesis has been obtained, the ability to prove its validity should be testable through empirical methods. In short, this means collectible data. For instance, if the hypothesis relates to the color of leaves and temperature, subsequent testing should create an environment where actual leaf samples are subjected to controlled atmospheric fluctuations. If a hypothesis deals with voters' reactions to certain policy decisions, perhaps the tester could conduct surveys intended to monitor shifts in opinion or voting patterns on specific issues. Most importantly, the tests the original student runs on her hypothesis should be capable of being replicated by others if and when they choose. Others should be able to create the same testable conditions with the same materials and achieve precisely the same results.