Hardest thing--fundamental understanding of meaning of confidence
interval
and significance test.
Many people don't have the mechanics under control. If
not, please see me Clinic or someone!
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#10) Find z* for 88% CI: Meant to
put into practice the picture you memorized for the
quiz.
C is 88%. To use the z table, you have to figure out what's in
the
tail(s). 1-.88 = .12 Divide by 2, find
6%
in each tail.
88%+6% = 94% to the left of +z*. Use the
table backward, closest is .9394, for z* = 1.55
or .9406 for z*=1.56 (or z*=1.555, halfway
between).
I also gave part credit if you told me it was between the values for
80% and 90%. A few people interpolated in the t-table, finding
the
number 8/10 of the way between the values for 80% and 90% (8/10 of the
way between 1.282 to 1.645). . This is not quite right,
since
it's assuming the normal curve is flat in this region, (equal
intervals
contribute equal area) but numerically it comes out fairly close.
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4a, b) This is patterned on 1 bottle of cola vs. 6, 1 test on
Julie vs. 3 ( 4.51, 4.52) Another of that type is 4.43, p. 247.
Making
the comparison between measuring one individual from the
population
(a sample of size 1, or just the 'distribution of the
population')
and measuring the mean from a sample of size n. The distribution
of the means will have less spread.
4c) The probability that the sum of the 36 people's
baggage
is greater than 36x60 is the same as the probability that the average
is greater than 60, because the average is the sum divided by 36.
(This was the "bonus points" problem; I was pleased
that
as many people got it as did.)
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6b) (answer =) Fish caught must be or act like a Simple Random Sample
from the population. All our formulas depend on that. Part
credit for sample must be representative of the population, no biases
in
the sampling.
Notice, the Sample may well not look normal. If the Population
isn't normally distributed, the Sample won't look normal either.
6c) (answer = ) Central Limit Theorem is the Central
theorem
of classical statistics--it guarantees that for a large(ish) n and a
Simple
Random Sample of size n, the sampling distribution of the (sample) mean
is approximately Normal. This allows us to use the normal
tables (and the t-tables too).
(part credit for) The Law of Large Numbers says
that for large n (and again, a SRS), the sample mean will be close to
the
population mean. But doesn't say anything about the
distribution
of sample means.
6d) You can make a narrower confidence interval
(make
a smaller margin of error) by
i) taking a larger sample size (the sqrt(n)
in the denominator will be bigger, make margin of error smaller)
or iii) require lower confidence level (z* will be smaller,
closer
to the middle).
Higher confidence level <--> lower accuracy
<--> wider interval<--> larger
margin of
error.
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11) Voluntary Response Sample--almost sure to be biased.
Need Simple Random Sample or reasonable facsimile to do confidence
interval, or any sort of inference.
(Or you need some sort of Probability Sample--maybe a more
sophisticated one and a different formula.)
The size of the sample compared to the population of voters
is not a problem. A spoonful of soup gives just as good a
sample of a huge pot of soup as it does of a small pot. A
toothpick full of soup, however, is not a good sample. So the
size of the sample per se is important, but the proportion of the
population it represents is not. Here the sample size is very
large.
(Qualification: If you actually can sample a large part of the
population, more than about 10%, you will get even stronger results
than our formulas give. There is a formula--"the finite
population correction"--for adjusting the standard deviation of the
sample mean when this is true. P. 569, Ch.4 footnote 3.)
If you (correctly) said that the results can only describe the
people who sent in answers, then you should notice that you already
have all of the population you are describing. So
your sample mean can't have any variability; it's just the mean of all
the numbers you are hoping to describe. So putting a margin of
error around it doesn't make any sense.
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