HW- pgs. 491-492 (7.32-7.34, 7.36) Ch. 7 Test MONDAY 12-16-13 www.westex.org HS, Teacher Website 12-11-13 Warm up—AP Stats
Let X = # of girls a couple has in 3 tries. Find m X (expected # of girls) and the standard deviation of X. # of girls Probability 2 xi pi xipi (xi - m X ) pi
m 2 Σpi = 1 X = s X = This can be done with our calculators:
Enter xi in L1 and pi in L2
Stat/Calc/1-var stats (L1,L2)
Calculator gives mX as x & s X . If you want/need
variance just square s X . Name ______Date ______AP Stats 7 Random Variables 7.2 Day 2 Means and Variances of Random Variables
Objectives Explain, and illustrate with an example, what is meant by the law of large numbers. Explain what is meant by the law of small numbers.
Statistical Estimation and the Law of Large Numbers Let’s say we are interested in knowing the mean height µ of the ______of all women between the ages of 18 and 24 years. This µ is the mean mX of the random variable X. Since it is impossible to find the height of ______women in the population of 18 and 24 year old women, we need to estimate mX .
So we choose an SRS of young women (18-24) and use the sample mean ___ to estimate the unknown population mean mX . Since x is COMPUTED FROM THE SAMPLE it is called a
______. mX is a number that describes the POPULATION, so it is called a ______. **We will revisit this terminology in ch. 9, but it is HUGE for Inference which goes from ch. 10-14.***
Statistics (such as ___) obtained from probability samples are random variables because their values would ______in repeated sampling. The ______of statistics are just the probability distributions of these random variables. (More in ch. 9.)
The purpose of an _____ is to represent the population so it makes sense to think that ___ (the mean of our sample) should do a good job of estimating ___ (the mean of our population). Let’s say Rachel takes an SRS of the population of 18-24 year old women and measures their height. Nick B. does the same with his own SRS of the same size as Rachel’s. Would we expect their sample means (___) to be exactly the same? Why not?
So if x is rarely exactly the same as mX then why is it a reasonable estimate of the population mean ( mX )? On average as our sample size increases the accuracy of x compared to the true population mean mX increases. This is called the law of ______numbers.
Law of Large Numbers Draw independent observations at random from any population with finite mean µ. Decide how accurately you would like to estimate µ. As the number of observations drawn increases, the mean of your sample x eventually approaches the mean µ of the population as closely as you specified.
The behavior of x is similar to the idea of probability. In the ______, the proportion of outcomes taking any value gets close to the probability of that value, and the average outcome gets close to the distribution mean. Below is an example of how ______means, x approach the ______mean µ.
Thinking about the Law of Large Numbers The law of large numbers says that the average results of many independent observations are stable and ______. Individual gamblers in a casino will win or lose but the casino will win in the ______because the law of large numbers says what the ______outcome of many thousands of bets will be. Many companies besides casinos use the law of large numbers. Can you think of some examples? Is there a Law of Small Numbers? The rules of probability and the law of large numbers describe the regular behavior of chance phenomena in the ______. Psychologists (my wife!) have discovered that our ______understanding of randomness is very different from the true laws of chance. For example most people believe in the “law of small numbers” even though there is NO SUCH LAW! People often believe that short sequences of random events will show the kind of average behavior that appears only in the long run.
Ex. Write down a sequence of heads and tails that you think imitates 10 tosses of a fair and balanced (like Fox News!) coin.
Gamblers often follow the hot hand theory, betting that a run will continue. Ex. I’ve won three hands in a row at black jack, so it should keep happening. At other times they draw the opposite conclusion. I’ve won 6 hands in a row, surely I’ll start to lose now to get back to the expected probability which is very close to 50%. No compensation is needed to get back to the average in the ______. Our intuition doesn’t do a good job of distinguishing random behavior (sometimes in the short run there will be a string of several heads or tails in a row) from ______influences (something is actually wrong with the coin, so it’s not a 50-50 proposition. ***When looking at data we need statistical inference (Chs. 10-14) because probability calculations can help verify that what we see in the data is more than a ______pattern.***
How large is a large number? The law of large numbers says that the actual mean outcome of many trials gets close to the ______mean µ as more ______are made. But how many trials are needed to get close to µ? That depends on the ______of the random outcomes. The ______variable the outcomes, the more trials are needed to ensure that the mean outcome ___ is close to the distribution mean µ. Think about a person gambling at a casino. If the casino wants the person to be interested in gambling would they want a lot of variability in the outcomes or not too much? Now that we have decided that they would want a lot of variability (Isn’t it boring to play a game if it’s so easy to predict what will occur?) we also know that they need ______hands to be played to ensure that the law of large numbers comes into play.