Hypothesis Testing Computations

This video is designed to accompany

Module 3 in

Beyond the Numbers Student-Centered Activities for Learning

Statistical Reasoning a publication of the Van-Griner Publishing Company

Brass Tacks

!  There are countless hypotheses that can be tested with statistical science. Some of these are very complex conceptually and mathematically.

!  Almost all share the same logic with respect to the choice that is being made between a null and alternative hypothesis.

Brass Tacks

!  In this video, we are going to learn the details of just one, a very simple one.

!  Others are addressed in the accompanying workbook.

From Words To Symbols

While we may talk about hypotheses in words:

H0: Flibanserin is no better than a Placebo HA: Flibanserin is better than a Placebo

These words eventually have to be translated to symbols, typically symbols representing unknown parameters:

H0: µFlibanserin = µPlacebo HA: µFlibanserin > µPlacebo

where µFlibanserin means the true average number of sexually satisfying events for women using Flibanserin; similarly for the placebo group.

Proportions

We are ONLY going to address the following hypothesis:

H0: p = p0 HA: p > p0

where p is an unknown population proportion.

For some pretty technical reasons this hypothesis is treated the exact same way whether there is just an “=“ in the null or a “< or =“. Your instructor may choose to explain this subtlety.

Stressed?

Stress affects the quality of college students’ sleep far more than alcohol, caffeine or late-night electronics use, a new study shows.

Stress about school and life keeps 68 percent of them awake at night ….

The study of 1,125 students … appears online in the Journal of Adolescent Health ….

Lund HG, et al. Sleep patterns and predictors of disturbed sleep in a large population of college students. J Adolesc Health online, 2009.

The Challenge

68% of the sample said stress kept them awake at night. Is it safe to say that more than 65% of the population of all college students feel the same way?

We are being challenged to test the following hypothesis and decide based on the data if we can safely accept HA.

H0: p ≤ 0.65 HA: p > 0.65

Recall, this means we have to set a value for the Type I error rate, compute a p-value, and compare that p-value to the Type I error rate.

Step 1 of 4 – Set Type I Error Rate

•  Typical to take α = 0.05. This is what we will assume when we read the work of others, unless otherwise stated.

•  We will compute a p-value and compare it 0.05

•  If the p-value is smaller than 0.05 we will reject H0. Else we want.

To test: H0: p ≤ p0 HA: p > p0

Be Aware

! Deciding to compare a p-value to an alpha level can be dangerous.

! In general, the mixing of p-values and Type I error rates has caused almost irreparable confusion in the application of statistical science.

Be Aware

! Still, it is unwise to disconnect you too much from what “everyone else does,” even if what is done is often incorrectly interpreted.

! It will be fine as long as you don’t start to misinterpret a p-value as an error rate.

Sample size/Number of subjects studied

Step 2 of 4 – Compute the Standard Score

Compute the “standard score”:

Sample Proportion Hypothesized value of the population proportion

To test: H0: p ≤ p0 HA: p > p0

Step 3 of 4 – Look up the p-value

Take the standard score to a standard score table and record the p-value

Step 4 of 4 – Make Your Decision

!  If p-value is less than alpha, reject H0.

!  If p-value is not less than alpha, fail to reject H0.

!  DO NOT interpret the p-value as an error rate. It is not.

Step 1 of Example

Take the Type I error rate to be α = 0.05.

This means that in all the experiments being done all around the world with an alpha level of 0.05, only 1 in 20 will wrongly reject their null hypotheses.

To test: H0: p ≤ 0.65 HA: p > 0.65

Step 2 of Example

Compute the “standard score”:

To test: H0: p ≤ 0.65 HA: p > 0.65

Step 3 of Example

The computed value of z was 2.11. Take 2.11 to the standard score table and come out with a p-value of 0.01743

Step 4 of Example

!  The p-value of 0.01743 is smaller than the preset Type I error rate of α = 0.05 so H0 can be rejected.

!  Of all the tests being done with this alpha level, only 1 in 20 will reject the null incorrectly.

Step 4 of Example

!  The p-value has nothing to do with Type I error rate.

!  We can say that the chances of seeing a statistic that supports the null as much or more than the one we computed, assuming the null is true, are about 17 in 1000.

Decision Revisited

!  The p-value is 0.01743.

!  So we will reject H0 in favor of HA since 0.01743 is less than 0.05. It is a safe bet to say that more than 65% of all college students lose sleep because of stress. The results of the study are statistically significant.

Decision Revisited

!  The risk involved in this decision is that HA is really not true.

!  The fixed alpha level of 0.05 helps us get a handle on that risk.

Extensions

H0: p ≤ p0 HA: p > p0

Extensions

H0: p ≤ p0 HA: p > p0

One-Sentence Reflection

Testing a simple hypothesis involves the computation of a standard score, which leads to a p-value that we can compare to our preset Type I error rate.