Stat 512 – Day 7 Probability Models, Inference (Ch. 4)

Stat 512 – Day 7

Probability Models, Inference (Ch. 4)

Last Time – Random Sampling Need an unbiased method for selecting

samples from a larger population so you believe sample obtained is representative of the population Simple random sample = every sample of size n

has the same chance of being selected Other probability methods: systematic, multistage,

cluster, and stratified sampling Goal: Allows us to generalize sample results

back to the population of interest

Last Time - Sampling

Compareresults

Randomized?

Getting the observational units in the first place!

Population

Random?sample

parameter

statistic

Scope of Conclusions

Subjects randomly

assigned to groups?

Yes No

Subjects randomly

selected from Yes

population? No

Can generalize to population

Can draw cause-and-effect conclusions

But: Nonsampling Errors

Even if sample is properly selected, are other sources of bias and misleading results

Example

Should Cal Poly allow speeches on campus that might incite violence? 13 Yes, 10 No

Should Cal Poly forbid speeches on campus that might incite violence 4 Yes, 18 No

Lost ticket, would you buy another? 8 Yes, 13 No

Lost $20, would you buy another? 20 Yes, 3 No

200 lives saved? 12 Program A, 11 Program B

400 lives lost? 4 Program A, 18 Program B

Prediction: more likely if lost ticket

Prediction: Program A more likely when in terms of lives saved

Nonsampling Errors

March 6-8, 2004 Wall Street Journal/NBC poll of 1,018 adults

GAY MARRIAGE opinions depend on how the question is asked.

To one poll question, a 52%-43% majority opposes a constitutionalamendment "making it illegal for gay couples to marry." A 54%-42%majority responds favorably to a second query that omits the word"illegal" and more benignly asks about an amendment "that defined

marriage as a union only between a man and a woman."

Sources of Nonsampling Errors Sensitive questions

Social acceptability Wording of question

Collided vs. made contact Appearance of interviewer Delivery of questions, instructions Order of choices Unsure response, change mind, faulty

memory

Second Benefit of Random Sampling Sample statistics follow a predictable,

probabilistic pattern Will always be some random sampling error

but we will be able to estimate its size…

Second Benefit of Random Sampling

Population

samplerandom?


Population

sample

pop mean


Population

samplex x

x

xxx

x x xx

xx

xx x

x =sample meanpop mean


Population

samplex x

x

xxx

x x xx

xx

xx x

x xxx x x x x x x xxx x xx xx x x xx xx x x xxxx x xx xxx

xx xx

pop mean

Sampling distribution

Last Time


xx xx

parameter parameter

Sampling method is unbiased

Which is “better?”


xx xx

parameter


xx xx

parameter

Last Time

Population size…

Population

samplex x

x

xxx

x x xx

xx

xx x


xx xx

pop mean

Example 1: Sampling Penny Ages Suppose I take repeated samples of n=5

pennies and calculate the date on the pennies Expected shape of the population?

Behavior of sampling distribution?

Sampling Penny Ages

http://statweb.calpoly.edu/chance/applets/ SampleData/SampleData.html

Sampling Penny Ages

Distribution of sample means will be reasonably symmetric if Sample size is large

Convention: n > 30

OR Population distribution is normal

Standard deviation of the sample means follows the formula

nXSD /)(

Normal Distribution

N()

Example 2: Body Temperatures

The probability of temp < 98.2 is .2919 About 29% of randomly selected adults would

have a temperature at most 98.20F.


There is only a 3% chance that a random person will have a temperature in excess of 1000F.


Less than .0001 probability that a sample would have an average temperature below 98.20F

Example 2: Body Temperature We have strong evidence that a random

sample of 130 adults from a population with =98.60F would give a sample mean of 98.2 or less by chance alone

So after observing a sample mean of 98.2, we have strong evidence that the statement that =98.60F is wrong

Example 3: Reese’s Pieces

Distribution of Sample

Simple Bar Graph

Sampling Distribution

About 6% of random samples (with 25 candies) will have a sample proportion of .60 or higher when =.45.

Sampling Distribution

About .4% of random samples (with 75 candies) will have a sample proportion of .60 or higher when =.45.

When n=3

Can’t use the CLT! Would need other methods to help your

friend

For Thursday

Submit PP 7 (3 pts) Read Ch. 6

Lots of new terminology but same basic reasoning…

HW 4 (by Friday)

Stat 512 – Day 7 Probability Models, Inference (Ch. 4)

Documents