B RIDGE TO INFERENCE Sec 9.1

G OING FROM KNOWN TO UNKNOWN What good are our tools when we don’t know about the population. What do we know about sampling distributions: We can be 95% sure that our sample should be within about 2 standard errors of the true proportion (or mean) We can be 99.7% sure that our sample should be within about 3 standard errors of the true proportion (or mean) This is true whatever our population looks like as long as n (sample size) is large enough. For sample means n > 30. For proportions np and nq >15.

G OING FROM KNOWN TO UNKNOWN So we will need values for the standard error. = standard error for a sample proportion. = standard error for a sample mean. We will not know the values for the population proportion and the population standard deviation. What can we use that should be “close” to the population values? We will often use the sample standard deviation and proportion.

B RING IT TOGETHER. A preliminary study suggests that the proportion of college age students that complete their degree is around 25%. To support this another study is done with 1000 students selected from all over the country. These results find that 221 of the 1000 selected students ended up completing their degree. Does the second study support the first or not?

P OINT AND INTERVAL ESTIMATES Sec 8.1

P OINT E STIMATES, There are two ways to approximate a parameter. The first is with a Point Estimate. Point Estimates are single values that form a “best guess” for the parameter. Normally these are found from samples They do not give us information on how trustworthy the estimation is.

P OINT ESTIMATE E XAMPLE ; A business owner wants to find out how much people spend at his store. He samples random patrons and finds they spend an average (mean) of $ This point estimate does not give him an idea of spread, or any idea of how good this estimation might be. In general, point estimates are looked down upon in statistics. They are, however, prevalent in the media.

I NTERVAL ESTIMATES, We can also find Interval estimations for population parameters. An interval estimation is an interval of numbers within which the parameter is thought to be. These intervals are found with inferential methods that we will introduce in the coming sections. Interval estimations give not only possible values for the parameter but also indications of the relative strength of that estimation.

I NTERVAL ESTIMATE E XAMPLE ; A researcher is trying to find the proportion of pygmy tarsiers living in the wild are female. There sample showed a proportion of 35% plus or minus 3.2%. This estimate give a sense of spread. (32.8, 38.2) The tight error (3.2%) give a sense that the approximation is a good estimate. Interval estimates are the standard in most sophisticated applications.

C ONFIDENCE I NTERVALS, A confidence interval is an interval where the population parameter is believed to land and a probability that the parameter is indeed within that interval. These probabilities should be close to one. (95%) We say, “We are 95% confident the parameter is between A and B.”

C ONFIDENCE I NTERVALS, Recall: The sampling distributions for a population’s proportion and mean are normally distributed as long as the sample size is large. We know the parameter we want is in the center of that distribution. We know “about” 95% of those samples lie within two standard deviations of the parameter being measured.

M ARGIN OF E RROR We have said that “about” 95% is within 2 standard deviations (Errors) of the mean. To be more precise lets find the z-score interval for which a proportion of.95 is contained symmetrically about the mean. μ z- score: 1.96z- score: -1.96

M ARGIN OF E RROR So, to be within a 95% confidence interval we will want to be within 1.96(se) of the sampling mean. Which we still don’t know! μ z- score: 1.96z- score: -1.96

L AST PIECE, Let think about this. If Jim is within 10 feet of Pam, Then Pam are within ten feet of ______. If there is a 10% chance Bill is in the same room with Jan. Then there is a 10% chance Jan is in the same room with Bill. If our sample is within 1.96 standard errors of the parameter, Then the parameter should be within 1.96(se) of our sample.

O UR INTERVAL,

E XAMPLE, A business surveys 1000 randomly selected customers and finds that 23% of those sampled get most of their coupons from mailers. Their statistician tells them the standard error for this data is Construct a 95% confidence interval for the population proportion that gets their coupons from the mailers.

E XAMPLE