| Moore sec. 4.3 Hand
in Wednesday (these are the same as those postponed Friday)
DIST. OF XBAR(S) Add this (follow up class work. Finish the other parts):Example: "Normal" body temperature 98.6 deg. on average. (Assume this is true.) Assume normal distribution, & s.d.among many people is 0.6. X = temp of one random person. Probability that one (random) healthy individual's normal temperature is above 98.8? X is normal, mean 98.6, s.d.0.6. X is N(98.6, 0.6). Want P(X > 98.8). Standardize both sides of the inequality: P(X > 98.8) = P(Z > (98.8 - 98.6)/0.6) = P(Z > 0.2/0.6)= P(Z>.333) = 1-.6293 = .3707 Probability that the mean of a sample of 4 is above 98.8? Need sampling distribution of sample means: mean is still 98.6. n = 4, so s.d. of x-bars is s.d. of x's, divided by the square root of 4. Square root of 4 is 2. s.d. of x-bars is 0.6/2 = 0.3. X-bar distribution is N(98.6, 0.3) Want P(X > 98.8) = P(Z > (98.8 - 98.6)/0.3) = P(Z > 0.2/0.3)= P(Z>.667) = 1-.6293 = .3707 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? 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? - - - - - - - - - - - - - - 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." - - - - - - - - - - - - A. Get 4 slips from the Birkenstock box (outside my door if you missed class). Record them, return them. HW: Find their mean xbar. Find xbar - .841, xbar +.841, your "interval estimate" for the unknown mean of the box. ("margin of error" is .841) Bring next time to compare. |
Read,
to discuss |
Optional |
Review Law of Large Numbers (day
27),
HW questions on LLN?
Mostly we have a fixed sample size n. How close
will Xbar be to µX?
Sampling distribution of x-bars, and Central Limit Theorem
(day
27) :
Look at (and add to?) Sampling Distribution of (sample)
means from 10 test scores.
Wednesday: Review Homework, continue
from here.
"Fuzzy Central Limit Theorem:"
Data whose variation is due to many small
independent random influences will have an approximately
normal distribution.
Balls and pins, heights of women, etc.
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
Chapter 6, preview:
SAMPLE from an UNKNOWN population.
Each person take 4 slips from the Birkenstock box, write them down,
return slips.
HW: find the mean, and your mean
+
.841.
Record these for yourself (This is your
"Interval Estimate" of the mean of the Birkenstock population. )
Next time: Your "estimate" of the (unknown) population mean
µ of the numbers in the shoebox is your sample mean plus or minus
the "fudge factor/margin of error" .841.
You'll Record
them also on the sheet going around, and draw
the interval on the graph transparency
going around.
If xbar = 8.0
7.159|_____________8.0_____________|8.841
| Sievers home | Math151-Sp04/Dayf28.htm | 11:15am | 11/1/04 |