Apr. 8

Stat 100

To do

• Read Chapter 21, try problems 1-6

• Skim Chapter 22

Significance Test

• Data used to decide between two competing statements (hypotheses) about the population

• Statements are called null hypothesis and alternative hypothesis

Notation

• H0 represents null hypothesis

• HA represents alternative hypothesis

College performance and SAT relationship

• H0 : No relationship between college performance and SAT

• HA: college performance and SAT scores are related

Memorization skills of men and women

• H0: No difference in memorization skills of men and women

• HA: There is a difference between men and women

Thought question

• Consider coin flipping.

• Null hypothesis: chance of heads = .5

• Which is stronger evidence against this null hypothesis -

• 3 heads in 3 flips, or

• 30 heads in 30 flips?

Somewhat obvious answer =

• 30 heads in 30 flips

• What about this that would make us reject the hypothesis that chance of heads = .5

• It seems almost impossible to get this many heads if the coin flipping is “fair”

• Chance of 30 heads in a row is about 1 in a billion

Statistically Significant

• A result is called statistically significant when a null hypothesis is rejected

• Basis for deciding is a probability called the “p-value”

Finding the p-value

• P-value = probability that the observed data would have occurred if the null hypothesis were true.

• Example: data = 30 heads in 30 flips

• null hyp = flipping is random

• p-value = chance of 30 heads if flipping is random

Using the p-value to make decisions

• The smaller the p-value, the stronger the evidence against the null (and for the alternative)

• Why?

• Small p-value means the observed data not likely to happen if the null really is true.

• Example - it would not be very likely that we’d get 100 straight heads

Usual borderline for decision

• If p-value less than .05 (5%), pick alternative hypothesis

• If p-value larger than .05 (5%), pick null hypothesis

Typical research paper statement• “We found a significant difference between

treatment success rates (p <.05).”

• p-value of a test was less than .05 so the researchers rejected a null hypothesis.

• The null would be that there is no difference between treatments

• observed difference was large enough to be “unlikely” if we believed the null to be true

Another typical statement

• The difference between means for the two treatments was not significant (p = .42).

• This means the researchers could not reject a null hypothesis; the p-value = .42 .

• Null: no difference in treatment means

• p-value=.42 means the observed difference would be quite likely to happen if the null were true

Book Ch. 22 Thought Question 1

• Imagine a jury decision in a murder trial.

• It is a mistake if the jury claims the suspect is guilty when in fact he or she is innocent.

• What is the other type of mistake a jury can make?

• Which type of mistake is more serious?

Statistical Errors

• Type 1: rejecting the null hypothesis when you should not (like convicting a person who’s not guilty)

• Type 2: not rejecting the null hypothesis when you should (like not convicting a guilty person)

Example• Experiment is done to see if new treatment

for depression is better than old treatment• null hyp: new treatment not better• alternative hyp: new treatment is better• Type I error: deciding new treatment is

better when it is not• Type 2 error: deciding new treatment is not

better when it actually is

Biggest cause of Type II error

• Not getting enough evidence

• In statistical problems, small sample size