Math151 Day30

Math 151 , Day 30, Friday, Nov. 5, 2004After class hit reload..

HW Day30 Sec. 6.1 Read it (again and again).
Memorize the definition of a C.I. p. 302, esp. the "repeated samples" bit (and where z* comes from), or below,
and the formula p.306 for a CI for the mean (or below).
Closed Book Quiz Wednesday(!) on the two Confidence Interval things.
Note on reading: p. 306, table at top: "Tail area" is area in One tail, which is what you look up in the Normal table.

Sec.6.1 :
To be handed in Monday:
p.302, 6.1 poll of women
6.2 95% confidence?
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Using formula p. 306 for C.I.:
A. Find the critical value z* for C = 84% (like example 6.3, pp.304-5)
6.6 potassium again. (a) n=1 (b) n = 3.
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. For b), Find xbar with a calculator. Read pp. 312-13 before doing part c.
= = = = = = = = = = = = = = = = = = = =
Postpone!! 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.

Read,
to discuss

Op-
tional

Chapter 6, If you didn't Wednesday:
SAMPLE from an UNKNOWN population. Each person take 4 slips from the Birkenstock box,
      find the mean, and your mean + .841.
      Record these for yourself . This is your "Interval Estimate" of the mean of the shoebox population.
          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.
      Record them also on the sheet going around, and draw the interval on the graph transparency going around.
         If xbar = 8.1       7.259|_____________8.1_____________|8.941

I have candy for everyone whose interval contains the population mean. Who gets candy?
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Extended Standard Normal Table
z         P(Z < z)                         P(Z > z)   = same in scientific notation: E-03 = 10^-3
3.00   .9986501019683700    .0013498980316301    1.35E-03
4.00   .9999683287581670    .0000316712418331    3.17E-05
5.00   .9999997133484280    .0000002866515718    2.87E-07
6.00   .9999999990134120    .0000000009865877    9.87E-10
7.00   .9999999999987200    .0000000000012799    1.28E-12
8.00   .9999999999999990    .0000000000000007    6.66E-16 Below this, machine can't compute.

If your assumptions lead you to a(n almost) impossible z value, question your assumptions!
(The basis of significance/hypothesis testing)
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Central Limit Theorem again. HW questions?Day 29
How big does n have to be for Xbar to have a normal distribution? (about 25 is always good.)
"Fuzzy Central Limit Theorem:"
# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #
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
(Each is a "point estimate")

"xbar IS µ" Never true exactly
"xbar is close to µ" True for most xbars, depending on "close", and sample size n.
"xbar is probably close to µ" ["probably"?? It is or it isn't. We have "confidence" ours is a close one]
"xbars are usually close to µ" True.

If we have only one sample -- one xbar, we have NO IDEA if ours is one of the "close" ones.
Quantify "usually" and "close."

Fall 2002: 33% (16 of 48) xbars recorded were within 1 of µ. (between 68.4 and 70.4).

83% (40 of 48) xbars recorded were within 4 of µ. (between 65.4 and 73.4).

94% (45 of 48) xbars recorded we

within 5 of µ

Fall 2001: 65% (15 of 23) xbars recorded were within 4 of µ. (between 65.4 and 73.4).

78% (18 of 23) xbars recorded were within 5 of µ. (between 64.4 and 74.4).

Spring 2002: 83% (35 of 42) xbars recorded were within 4 of µ. (between 65.4 and 73.4).

90% (38 of 42) xbars recorded were within 5 of µ. (between 64.4 and 74.4).

(I excluded impossible xbars)

Note tradeoff between "close"(accuracy) and "usually" (confidence)

Interval estimate: xbar + margin of error (fudge factor) estimates population mean µ (69.4)

    69.75 + 1:   "µ is between 68.75 and 70.75" True
    69.75 + 4:   "µ is between 65.75 and 73.75" True
       73.5 + 4:    "µ is between 69.5 and 77.5" False
       73.5 + 5:    "µ is between 68.5 and 78.5" True
        64.25 + 4:   "µ is between 60.25 and 68.25" False
        64.25 + 5:   "µ is between 59.25 and 69.25" False

Confidence interval estimate of a(n unknown) population parameter:

an Interval constructed from the data, +
a Confidence level C: where

by this method

Confidence Interval of the form estimate + margin-of-error for the mean µ with Confidence level C: (p.306)

the estimate is xbar
margin of error m is : z* times Standard deviation of sample mean

Probability C is between -z* and +z*

(Table A, or Table C, t dist. bottom row)

Standard deviation of sample mean: Sigma /sqrt(n)

know

m = z* (sigma)/ sqrt(n),

so CI is xbar + z* (sigma)/ sqrt(n)

z*? Probability C is between -z* and +z* in the standard normal table.
To get the z* for C = 60% from the normal table, note that this is the middle 60%, which leaves 40% to be split between the 2 tails. So 20% above z*, and 80% below. Go into the body of table A, find .8000 is between values .7995 and .8023, closer to .7995. The z value with .7995 below it is .84. z* = .84. (Cf. example 6.3, pp.304-5)
Table C, bottom row, is a restating of table A, normal table, but with probabilities (areas) on the edge, and z values in the body. Bottom row gives the most common C's. Read off, for C =60%, z* = .841.

Example: Sample of size 9 from a Normal population with unknown mean and pop. s.d. sigma = 6, xbar = 12.
Find a 90% CI estimate for the unknown mean µ:
n=9. (sigma)/ sqrt(n) = 6/3=2
z* = 1.645, so m = 3.290;
CI is 12 + 3.290, or 8.710 to 15.290.

The Birkenstock box contains numbers from a normally distributed population, with population standard deviation 2.
You each constructed a 60% confidence interval for the unknown mean:
    n = 4.
    Standard deviation of sample mean = 2/sqrt(4) = 2/2 = 1
    z* for C = 60% is .841, so margin of error m is .841 times 1= .841.
How many people captured the true mean?
( previous classes, 11/20 = 55% , 22/29= 76%.   9/18 = 50% , 11/20 = 55%, 15/22= 68%, 14/22 = 64%
16/18 = 88% Combined 100/151 = 66%This class?
Quite variable for small numbers of samples, but settling down.)

Why does the formula work?

1. If a particular xbar is within m of the population mean, then the interval xbar + m contains the population mean.

xbar is farther than m from the population mean

xbar

m doesn't contain the population mean.

(this is the point of HW # 6.3)

2. We choose z* (and from it m) so that the probability that Xbar is within m of the population mean is C.

Probability C is between -z* and +z*

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