Download this is the mean, for sure this is the standard deviation, for sure but

8.1 What about when you take a sample?
when you take a sample size 2 or 3 or whatever, you expect a lower standard deviation
(the mean of the population is still the same)
we call that the standard deviation of x (also called the standard error)
notation: σ
there is also the mean of x
notation: µ
Central Limit Theorem:
ex) if σ=8 and you take a sample of size n=16, what standard deviation do you get? (i.e.
what is σ_ ?)
ex) If σ=1.9 and sample size n=20, find your standard deviation σ_
note: dont round too much, that error will get bigger in each step
this is the mean, for sure
this is the standard deviation, for sure
but...is the distribution of x *normal* (is it a bell curve) ?
we want this because then we can figure out all the probabilities
.......lots of experimenting.....
population is normal -> distribution for x is normal
population is not normal -> take sample of size 30 (to be safe) then
distribution for x is normal
therefore, x is normal if:
population is normal OR n≥30 (and we know σ)
to describe a distribution, you state:
the mean, the standard deviation, the shape
for adult salmon length, µ=42" and σ=6" (normal distribution), find:
a) the probability that one salmon is longer than 46"...
b) the probability that four salmon average longer than 46"
formula for z-score: value - mean
st.dev
note: the text writes the formula for z differently
(with n in it). we will learn it this way - it will make
life easier for the rest of the course
you do:
ex) µ=78, σ=6, sample size n=10
population has normal distribution
find P(x<70)
ex) for all cars, the average lifespan is µ=12.6 yrs (σ=2.1, normal distribution). what is the
probability that nine cars on average last longer than 14 years?
8.2 distribution of sample related to proportions
ex) 62% of mothers want increased athletic programs in the schools
we write: .62
p represents proportion of the population
represents the proportion of the sample
ex) from a census, 51% of US residents are women
notation: p = .51
what proportion are men?
ex) from a survey of 800 teenagers, 673 like Justin Timberlake's music
notation:
note: you wouldnt say "what is the average number of people who
like his music"
what proportion of teenagers do NOT like his music?
notation and formulas ... for proportions:
what is
it comes from the population
note that this is different than with the sample mean x
17% of Americans have high cholesterol. suppose we will take a survey of 80
people. what is the probability that less than 12% of your sample has high cholesterol?
what is µ^ ?
what is σ^ ?
is the distribution of p normal?
so, what is the probability that less than 12% of your sample has high cholesterol?
there are lots of decimal numbers here
there are values the proportion
there are values for probability
be careful as you are doing problems not to confuse them
in this problem,
proportion is .12
probability is .1170
ex) p = .47 n = 60
what is P(p < .43) ?
P(p<.43) = .2676
ex) if p=.28, find P( ^ > .32) with a sample size 180
ex) 62% of mothers want increased athletic funding
what is the probability that more than 40 out of 60 mothers will vote for funding?
ex) find P(x>32.7) ... µ=29.6 σ=8.2 n=34
ex) a 2003 study found that medical
residents work an average of 81.7 hours
per week. suppose the number of hours
worked is normally distributed with a
standard deviation of 6.9
what is the probability that the mean
number of hours worked by a team of 8
residents is less than 75 hours a week?
ex) find P(p< .6) ... p=.55 n=80