Introduction to Testing a Hypothesis

Testing a treatment

Descriptive statistics cannot determine if differences are due to chance.A sampling error occurs when apparent differences are by chance alone.

Example of Differences due to chance alone.

2 1 0 0

y 1 1 0 7 y 2 11 7

Examples:

We know that the mean IQ of the population is 100. We selected 50 people and gave them our new IQ boosting program. This sample, when tested after the treatment, has a mean of 110. Did we boost IQ?

We selected a sample of college students and a sample of university students. We found that the mean of the college students was 109 and the mean of the university students was 113. Was there a difference in the IQs of college and university students?

Are both 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 exactly the same number.

Sampling distribution:

Sampling Error of the mean: SS

ny

describes the amount of sample-to-sample variability to expect for a given statistic.

Simplifying Hypothesis Testing

1. Develop research hypothesis (experimental)

2. Obtain a sample(s) of observations

3. Construct a null hypothesis

y y y1 2 0

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 program 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, acceptingdoes not equal proving it to be true.

failing to reject the null hypothesis proving the null hypothesis true

Type I & Type II Errors

Example: the IQ boosting program

We test: or

Type I Error:

Type II Error:

amounts to the same things

the null hypothesis is true, but we reject it. The probability of a Type I error is set at 0.05 and is called alpha

the null hypothesis is false, but we fail to reject it. The probability of a Type II Error is called

True False

Reject the null Type I Error Correct

power = (1- )

Fail to

Reject

Correct

1-

Type II Error

Null Hypothesis as compared to the real world

How sure are we of our decisions?N

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Power &

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Note: The figure is based on the null hypothesis being false and representsthe sampling distribution of the means.

One-Tailed and Two Tailed Test of Significance

Sampling Distribution of the Mean