Introduction to Testing a Hypothesis Testing a treatment Descriptive statistics cannot determine if differences are due to chance. Sampling error means differing by chance alone. Example of Differences due to chance alone.

Examples: We know that the population mean of IQ is 100. We selected 50 people and give them our new IQ boosting programme. This sample when tested after the treatment has a mean of 110. Did we boost IQ? We select a sample of college students and a sample of university students. We find that the mean of the college students is 109 and the mean of the university students is 113. Is there a difference in the IQs of college and university students? Are booth cases simply due to sampling error? Remember, the sample mean is rarely the population mean and rarely do the means of two randomly selected samples end up being the same. Sampling distribution: describes the amount of sample-to-sample variability to expect for a given statistic. Sampling Error of the mean:

Simplifying Hypothesis Testing 1. Develop research hypothesis (experimental) 2. Obtain a sample(s) of observation 3. Construct a null hypothesis 4. Obtain an appropriate sampling distribution 5. Reject or Fail to Reject the null hypothesis

Null Hypothesis Assume: the sample comes from the same population and that the two sample means (even though they may be different) are estimating the same value (population mean). Why? Method of Contradiction: we can only demonstrate that a hypothesis is false. If we thought that the IQ boosting programme worked, what would we actually test? What value of IQ would we test?

Rejection and Non-Rejection of the Null Hypothesis If we reject, we then say that we have evidence for our experimental hypothesis, e.g., that our IQ boosting programme works. If we fail to reject, we do NOT prove the null to be true. Fisher: we choose either to reject or suspend judgment. Neyman and Pearson argued for a pragmatic approach. Do we spend money on our IQ boosting or not? We must accept or reject the null. But still, accepting does not equal proving it to be true.

Type I & Type II Errors Example: the IQ boosting programme We test: or Type I Error: the null hypothesis is true, but we reject it. The probability of a Type I Error is called and is set at.05. Type II Error: the null hypothesis is false, but we fail to reject it. The probability of a Type II Error is called

TrueFalse Reject the Null Type I Error alpha Correct Power 1- beta Fail to Reject Correct 1- alpha Type II Error Beta Null Hypothesis How sure are we of our decisions?

Power & [a ] [ b ][ --- power ----] Note: The figure is based on the null hypothesis being false and represents the sampling distribution of the means.

One-Tailed and Two Tailed Test of Significance Sampling Distribution of the Mean