Hypothesis TestingPart II – Computations

This video is designed to accompany

pages 95-116

in

Making Sense of UncertaintyActivities for Teaching Statistical

ReasoningVan-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.

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 PlaceboHA: Flibanserin is better than a PlaceboThese words eventually have to be translated to

symbols, typically symbols representing unknown parameters:

H0: mFlibanserin = mPlacebo

HA: mFlibanserin > mPlacebo

where mFlibanserin 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, form a rejection region and then see if our data fall into that region.

Step 1 of 3 – Determine Rejection Criterion

• Set a level for Type I errors, typically α = 0.05

• Find the cutoff value for rejection region (called the “critical value”). Call this value “k.”

• If the alternative is a simple “>” you will reject H0 only if your statistic is bigger than k.

• For α = 0.05, this k would be 1.645.

z=p̂− p0

√ p0(1− p0)nSample size/Number of subjects studied

Step 2 of 3 – Compute the Appropriate Statistic

Compute the “standard score”:

Sample Proportion

Hypothesized value of the population proportion

To test: H0: p p0

HA: p > p0

Step 3 of 3

Compare the standard score to the cutoff k.

If z > k, then reject H0.

Otherwise, fail to reject H0.

z=0.68−0.65

√ 0.65(1−0.65)1125

=2.11

Step 1 of Example

Compute the “standard score”:

To test:H0: p 0.65HA: p > 0.65

Step 2 of Example

The computed value of z was 2.11. Take 2.11 to the FPR table and come out with an FPR of 0.01743

Practical Upshot

The estimated FPR 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.

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

The estimated FPR helps us get a numerical handle on the risk of this false positive.

Extensions

What would happen if

Then the standard score, z, would be negative.

To find the FPR in this case, look up the positive value of z in the FPR table and add 0.5 to that. That’s 0.5 not 0.05.

In any case, it is clear that the FPR is going to be bigger than 0.5 which is an entire order of magnitude bigger than 0.05.

So if then there is no way HA can be accepted.

H0: p p0

HA: p > p0

One-Sentence Reflection

Testing a simple hypothesis about a proportion is a two-step process involving the computation of a standard score, which is then taken to a table to identify the false positive rate associated with rejecting the null hypothesis.