Sampling• We have a known

population. We ask “what would happen if

I drew lots and lots of random samples from this population?”

Inference• We have a known sample. We ask “what kind of

population might this sample have been drawn from?”

The Central Limit Theorem• If you draw simple random samples of size n

• from a population with mean and variance

• then

the expected mean of x-bar is

the expected variance of x-bar is / n

the expected histogram of x-bar is approximately normal

Estimating mu from sample data

• Is this true?

• amu = sample mean

• Why not? Because the Central Limit Theorem tells us that, if we

drew lots and lots of sample, the sample means vary. Some are bigger than mu and others are smaller than mu.

Estimating mu from sample data

• What about this?

• mu = somewhere in the neighborhood

• of the sample mean

• But how do we define neighborhood?

Example 6.1 We have a sample of 500 high-school seniors, selected at

random from the population of all high-school seniors in California. For the 500 kids in the sample, their average score on the math section of the SAT is 461.

Known: sample mean is 461 Unknown: population mean Assumed: population sigma is 100

The Central Limit Theorem• If you draw simple random samples of size 500

from a population with mean and standard deviation of 100, then

the expected mean of x-bar is

the expected st dev of x-bar is about 4.5

the expected histogram of x-bar is approximately normal

Table A tells us...

• ...about 68% of sample means should fall within 4.5 points

of mu

• ...about 95% of sample means should fall within 9 points

of mu

• ...about 99.75% of sample means should fall within 13.5

points of mu

mu-9 mu mu+9

Sample Means

About 95% of sample means should fall within 9 points of mu

mu is 452

435 440 445 450 455 460 465 470 475 480 485

Sample Means

mu is 470

435 440 445 450 455 460 465 470 475 480 485

Sample Means

mu is 452

435 440 445 450 455 460 465 470 475 480 485

Sample Means

mu is 470

435 440 445 450 455 460 465 470 475 480 485

Sample Means

The 95% Confidence Interval If mu is any number less than 452, then our

sample mean would be surprisingly large. If mu is any number greater than 470, then our

sample mean would be surprisingly small. Therefore, the 95% confidence interval for mu

is the range from 452 to 470. If mu is inside this range, then our sample is

not unusual (according to the 95% rule).

Other confidence intervals• If we suppose that the sample mean is within 1.645

standard deviations of mu, then we get a 90% confidence interval.

• If we suppose that the sample mean is within 2.576 standard deviations of mu, then we get a 99% confidence interval.

Effect of sample size on the confidence interval

• As n gets larger, the expected variability of the sample means gets smaller.

Larger sample sizes produce narrower confidence intervals (other things equal).

Smaller sample sizes produce wider confidence intervals (other things equal).

Some cautions The data must be a simple random sample from the

population The sample mean, and therefore the confidence interval,

may be too heavily influenced by one or more outliers If the sample size is small and population is not

approximately normal, then the CLT doesn’t promise the approximately normal distribution for the sample means