Unlocking the Mysteries of Hypothesis Testing Brent Griffin Revised Fall 2006

What’s this all about? Hypothesis An educated guess A claim or statement about a property of a population The goal in Hypothesis Testing is to analyze a sample in an attempt to distinguish between population characteristics that are likely to occur and population characteristics that are unlikely to occur.

The Basics Null Hypothesis vs. Alternative Hypothesis Type I vs. Type II Error  vs. 

Null Hypothesis vs. Alternative Hypothesis Statement about the value of a population parameter Represented by H0 Always stated as an Equality Alternative Hypothesis Statement about the value of a population parameter that must be true if the null hypothesis is false Represented by H1 Stated in on of three forms > < 

Type I vs. Type II Error

Alpha vs. Beta a is the probability of Type I error b is the probability of Type II error The experimenters (you and I) have the freedom to set the -level for a particular hypothesis test. That level is called the level of significance for the test. Changing a can (and often does) affect the results of the test—whether you reject or fail to reject H0.

Alpha vs. Beta, Part II It would be wonderful if we could force both  and  to equal zero. Unfortunately, these quantities have an inverse relationship. As  increases,  decreases and vice versa. The only way to decrease both  and  is to increase the sample size. To make both quantities equal zero, the sample size would have to be infinite—you would have to sample the entire population.

Type I and Type II Errors True State of Nature The null hypothesis is true The null hypothesis is false Type I error (rejecting a true null hypothesis)  We decide to reject the null hypothesis Correct decision Decision Type II error (rejecting a false null hypothesis)  page 376 of text We fail to reject the null hypothesis Correct decision

Forming Conclusions Every hypothesis test ends with the experimenters (you and I) either Rejecting the Null Hypothesis, or Failing to Reject the Null Hypothesis As strange as it may seem, you never accept the Null Hypothesis. The best you can ever say about the Null Hypothesis is that you don’t have enough evidence, based on a sample, to reject it!

Seven Steps to Hypothesis Testing Happiness (Traditional or Classical Method)

The Seven Steps… Describe in words the population characteristic about which hypotheses are to be tested State the null hypothesis, Ho State the alternative hypothesis, H1 or Ha Display the test statistic to be used

The Seven Steps… Identify the rejection region Is it an upper, lower, or two-tailed test? Determine the critical value associated with , the level of significance of the test Compute all the quantities in the test statistic, and compute the test statistic itself

The Seven Steps… State the conclusion. That is, decide whether to reject the null hypothesis, Ho, or fail to reject the null hypothesis. The conclusion depends on the level of significance of the test. Also, remember to state your result in the context of the specific problem.

Types of Hypothesis Tests Large Sample Tests, Population Mean (known population standard deviation) Large Sample Tests, Population Proportion (unknown population standard deviation) Small Sample Tests, Mean of a Normal Population

Actually, it’s just the beginning... The End Actually, it’s just the beginning...