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 the xbars will eventually
get very close to the population mean mu.
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:
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.
SPSS simulation,
again
# # # # # # # # # # # # # # # # # # # # # # #
# # # # # # # # # # # # # # # # # # # # # #
Chapter 6, Introduction to Inference
Statistical Inference: drawing conclusions about a population from
sample data.
Requires: Random sample or Randomized
experiment. (Simple random sample usually)
First example: Use sample mean xbar
to "estimate" (unknown) population
mean µ
Mean of 4 grades (HW#4.40) estimates population
mean of all 10 ("known"= 69.4) E.g. 69.75, 64.25,
73.5
Confidence interval estimate of a(n unknown) population parameter:
| Activstats: Same as given on Day 26 --work on Confidence Interval stuff, either Activstats or Moore. |
| Hand in Monday:
Moore Sec. 4.3
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 these problems to be assigned Monday: Sec.6.1 . p.302, 6.1 poll of women 6.2 95% confidence? 6.3 density of x-bar, and confidence intervals This problem combines the pictures 6.2 and 6.4 For part d, to draw the confidence interval: just choose any point on the horizontal axis of your graph to be x-bar. Measure off the distance m (half the width of the shaded interval) and extend a bar m wide to the left and the right of your point,below the curve. (Like fig. 6.4, the bars with arrows at the ends. The red dots show what the x-bar is for that confidence interval) Choose another point, and repeat.. If your first x-bar was in the shaded interval, pick your second outside the shaded interval, and vice versa. You should note that if x-bar is in the shaded interval, then the confidence interval bar covers mu (280) and if x-bar isn't, then the bar doesn't. -- - - - - - - - - - - - - - - - - - - - - - - - - - Using formula p. 306 for C.I.: 6.6 potassium again. 6.7 comparing CI's for different confidence levels. Also write down the m (margin of error) for each interval. 6.9 comparing CI's for different sample sizes. 6.5 IQ test scores Read pp. 312-13 before doing this one. |
Read, to discuss | Optional |
| Sievers home | Math151-Sp02/Day27.htm | 1am | 4/5/02 |
