Math151 Day26

Math 151 , Day 26, Wednesday, Oct. 30, 2002 After class

-From # 4.40 p.241 Add your 3 means to the circulating list.
HW Day26 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)

Moore sec. 4.3 Hand in Friday:
LLN: p. 238, 4.39 betting on the numbers (I'm told the numbers pays more on average than the NY Lottery)

Read, try if you like, but don't hand in till Monday:
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
More problems:
p. 243 4.42 unbiasedness, sample size
p. 249 4.52 hypokalemia
p. 249 4.48 dust Note, the dust actually weighs 123mg, but the weighings may not be accurate enough for us to find the actual weight. "Distribution of this mean" = "Distribution of means from all possible sets of 3 weighings from these scales." When I took physics, we did not have digital scales; they were balance beams; and we weighed everything 3 times and found the average. (Have you ever gotten on the scale, said "that can't be right!" gotten off and on again a couple times?)
.
p.250 4.54 (labeled 4.53?)pollutants; backward from value to probability. You might want to know L so that if you tested your 125 cars and found a high value of x-bar, you would be able to compare it with L; if it was greater than L, you would go back to the manufacturer and say "I believe you sold me a batch of bad cars, because the chances of getting an average emission level this high if the exhaust system is working properly is only 1 in 100. It is more reasonable to believe the exhaust system is not working, than that we hit that 1 in 100 possibility."

Read,
to discuss

Optional

- HW questions? Continuous Random Variables. Normal Random Variables.

How does sample mean behave? (4.3)
                         Sample Chosen from a Population
                         (varies)               (fixed, but usually unknown)
Calculate Numerical summary: Statistic (Latin) Parameter(Greek letter)
                                    xbar                 µ
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 X bar.

Law of Large Numbers (p.237, "LLN") Take observations at random from a population with population mean µ. Then as the number of observations n increases, the sample mean xbar gets closer and closer to µ. (Even if the population is infinite!
Note--we don't say how big n needs to be for how close here.)
OR Let the sample size n get bigger. Then the xbars will eventually get very close to the population mean µ.
OR As the sample size increases, the sample mean gets closer to the population mean µ.
OR For a very large sample, the sample mean will (almost certainly) be very close to the population mean.
Activstats p 15-3 activities.

Start here Friday:Now: keep a fixed sample size n:
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? This is the distribution of means of all possible SRS's of size n.
We'll call it the "sampling distribution of the sample mean" (Sec. 4.3)

(sampling dist. of proportion: Spinning penny )

Shape, center, spread, (outliers?)
Look at results from #4.40.
Activstats 16-1-2

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 µ (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.

Example: "Normal" body temperature 98.6 deg. on average. (Assume this is true.)
Assume normal distribution, & s.d.among many people is 0.6.
Probability that one individual's normal temperature is below 98.0 degrees?
       Take SRS of 9 people. Sampling distribution of the mean? Probability that the mean is below 98.0?
   Probability that one (random) healthy individual's normal temperature is above 98.8?
   Probability that the mean of a sample of 4 is above 98.8?
   Probability that the mean of a sample of 36 is above 98.8?
   Probability that the mean of a sample of 100 is above 98.8?

SPSS simulation: average of spinners which can land on any number between 0 and 1.
Population--one spinner. distribution flat between 0 and 1, mean .49 s.d. = .29
n = 2, Average of 2 spinners is Xbar. Distribution triangular between 0 and 1, mean .50, s.d. .21. .29/sqrt(2) =.205
n = 4, Average of 4 spinners is Xbar. Distribution normalish between 0 and 1, mean .50, s.d. .15. .29/sqrt(4) =.145
n = 15, Average of 15 spinners is Xbar. Distribution normal between 0 and 1, mean .50, s.d. .09. .29/sqrt(15) =.076

Xbars from SRS:
   Mean of Xbars is mean of population.
   Standard deviation of Xbars is s.d. of population divided by square root of n.
   As sample size increases, sampling distribution of Xbars gets more and more normal-shaped.
                (Central Limit Theorem)

Central Limit Theorem...
How large is "large"? How approximate is "approximate"?
If the population was close to normal, n doesn't need to be very large.
Even if the population is pretty weird, n=25 gives a pretty good approximation to normal.
Pictures on overhead.

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