Math 151 , Day 25, Monday, April 2, 2001 final version

We only got to here; will continue here Wednesday
-From # 4.40 p.241  Add your 3 means to the circulating list, if you didn't Monday..
Sec. 4.3: Sampling distributions. We take a simple random sample of size n, find the sample mean xbar.  It will be different depending on the sample, so we have a random phenomenon.  We measure the outcome as a number, the sample mean, so we have a random variable Xbar.
Law of large numbers  Let the sample size n get bigger.  Then xbar will eventually get very close to the population mean mu.
OR As the sample size increases, the sample mean gets closer to the population mean mu.
OR For a very large sample, the sample mean will (almost certainly) be very close to the population mean.  Day 22

Usually we have a fixed sample size n.  Assume that's true from now on.
What probability distribution describes the random phenomenon of finding Xbar from a SRS?
That is, what is the distribution of the random variable Xbar, when the experiment is to take a simple random sample of size n?
We'll call it the "sampling distribution of the sample mean" = distribution of means of all possible SRS's of size n.
  Shape, center, spread, (outliers?)

Look at results from #4.40.

Things we know:

• Whatever the population distribution of X, that we draw the sample from, (see p. 242)

The mean of the x-bars = the mean of the population
The standard deviation of the x-bars = the s. d. of the population divided by the square root of n.
• Consequences:

* X-bar "hits" the population mean on average--is unbiased estimator of mu (doesn't systematically go too high or too low.
* Averages are less variable than individual observations.  Averages from large samples are less variable than averages from smaller samples (because of dividing by the square root of n)
• IF the population is Normal, the sampling distribution of Xbar is Normal.
• "The Central Limit Theorem"

In any case, for "large" n, the sampling distribution of Xbar is Approximately Normal.

HW Day25, Monday, April 2  Read sec. 4.3. (Skip ch. 5) Sec. 6.1 next.  Read ahead...
Memorize the tan box on p. 242 (mean and s.d. of sampling dist. of x-bar)

Hand in FRIDAY LLN: p. 234, 4.39 betting on the numbers These problems use only the mean and standard deviation.    p. 243, 4.41 (lab measurements)   p. 250, 4.50 These problems use either the Central Limit theorem, or the "sample mean of n independent observations from a normal distribution has a normal distribution." theorem (both on p. 244)   p. 249, 4.51 cola (you did a, now do b)    p. 247, 4.44 carpet flaws.  Also draw some square yards and mark some flaws.   p. 250, 4.53 auto accidents

Read, to discuss

Optional