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