Apr. 5

Stat 100

To do

• Read Chapter 21, try problems 1-6

• Skim Chapter 22

Thought Question

• Suppose we think mean classes missed per week is higher for males than for females.

• How would we determine whether this is the case?

Basic Strategy

• Collect data on missed classes for males and females

• Calculate mean for each sex and determine the difference.

• Then what? Suppose the mean for 50 males was 0.5 higher than mean for 50 females.

• Could we generalize that for all students, mean is higher for males?

An Issue

• Did the observed difference occur just by chance or does it reflect a true difference?

• What’s the likelihood the observed difference would be 0.5 if there’s really no difference in the populations?

Significance Test

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

• Statements are called null hypothesis and alternative hypothesis

The two hypotheses

• null hypothesis : a “nothing happening” statement– no difference between groups, or no

relationship, or nothing new

• alternative hypothesis : "something's happening”– there is a difference, or there is a relationship,

or something’s new

Notation

• H0 represents null hypothesis

• HA represents alternative hypothesis

Missed classes by men and women

• H0: No difference in mean classes missed for men and women

• HA: There is a difference between mean missed classes by men and women

Deciding between the hypotheses – an example

• Students asked to randomly a number between 1 and 10

• In past, a bias toward picking 7 has been noticed

• With random picking, what proportion would pick 7?

• Answer = 1/10 = 0.1 because there are 10 numbers

Are students picking randomly?

• let p represent proportion of all students would pick 7

• null : random picking , p = 0.10

• alternative: bias toward 7 , p> 0.10

The sample data

• Suppose 24 of 100 students in the class pick 7.

• The proportion picking 7 is 24/100 = 0.24 (much higher than .10)

• Could this have happened through random picking?

• Or, does it reflect bias toward 7?

With true random picking

• Distribution of possible sample proportions would be bell curve

• Centered at 0.10 (chance of randomly picking 7)

• SD = Sqr root[(0.1)(1-0.1)/100] = 0.03• About 99.7% chance that sample proportion

would fall in range 0.10 3(.03), or .01 to .19

Where the observed statistic falls

• Sample value of 0.24 is outside interval of what might normally occur with random picking

• Reasonable to reject the hypothesis that picking is random

Where the observed statistic falls

• Observed = 0.24 picked 7

• Find percentile rank of 0.24 in bell curve with mean 0.10 and SD = 0.03

• z score = (0.24-0.10) / 0.03 = 0.14 / 0.03= 4.67

• Page 137 table, proportion to left of z is around 0.995

Statistically Significant

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

• In our example, the result was statistically significant

Thought Question

• Imagine all the criminal trials in the United States

• In each trial, what is the null hypothesis?

• What is the alternative hypothesis?

• Overall criminal trials what are the two types of mistakes that can b made?

Statistical Errors

• Type 1: rejecting the null hypothesis when you should not (like a convicting innocent person)

• Type 2: not rejecting the null hypothesis when you should (like failing to convict 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 1 error: deciding new treatment is

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

better when it actually is