Stat 321 – Day 22

Confidence intervals cont.

Reminders

Exam 2 Average .79

Communication, binomial within binomial Course avg > .80 Final exam 20-25%

HW 7 due Tuesday Quiz 6 Thursday on HW 6 Lab 7 due Friday

Last Time – t-intervals

When don’t know the population standard deviation (pretty much always!), can use the sample standard deviation but then, to compensate for the extra uncertainty, use a critical value from the t distribution instead of the normal distribution t critical value depends on sample size (df =n-1) Widens interval to achieve stated confidence level Approaches z critical value as n increases

Technical conditions: Random sample Normal population but robust

Example

Given: n = 5; Sample mean = 41.8, sample SD = 2.39; 90% confidence

df = 5-1= 4, t = 2.132 41.8 + 2.132(2.39/sqrt(5)) 41.8 + 2.28 (“margin of error”) (39.52, 44.08) I’m 95% confident that the population mean

chest measurement is in this interval (39.8) How many militiamen are in this interval?

Prediction Intervals

To specify range of plausible values of individuals, consider the prediction for the average and then that a typical deviation from the average is …

41.8 + 2.132(2.39)sqrt(1+1/5) 41.8 + 5.58 We are 90% confident that a random Scottish

militiaman’s chest is between 36.22in and 47.38in

Example 2

A Gallup poll conducted Dec 5-8, 2005 by phoning a randomly selected sample of 1,013 adults, found that 66% of Internet users never read blogs

“The margin of error is at most 3 percentage points.”

Example 2

Let p represent the proportion of all adult internet users who never use blogs Sample proportion vs. population proportion?

Apply this method to a population proportion What use for estimate? What use for standard deviation? What use for critical value?

Sample proportion?

= X/n where X is binomial (n, p) Expected value? Standard deviation? Shape?

Confidence interval?

p̂

p̂

Confidence Interval for p (not )

nppzp ˆ1ˆˆ 2

Technical conditions:• • Data are SRS from population of interest

10ˆ1,10ˆ pnpn

Some Precautions

Finding voters Margin-of-error doesn’t measure “non-sampling” errors

Alien visits U.S. Senate, wants to estimate proportion of humans who are female Biased sample Confidence interval not needed if one’s data is from

population, not sample

We are 100% confident that p =.16!

Claimed to vote…

For Tuesday

Back to Ch. 6! HW 7

Note addition to exercise 33 in problem 4