Download Chapter 9 Sampling distributions Parameter – number that

Chapter 9 Sampling distributions
Parameter – number that describes the population
A parameter is a fixed number, but in reality we do not know its value
because we can not examine the entire population.
Statistic – number that describes a sample
Use statistic to estimate an unknown parameter.
μ = mean of population
x = mean of the sample
Sampling variability – the differences in each sample mean
P(p-hat) - the proportion of the sample
160 out of 515 people believe in ghosts
P(hat) =
= .31
.31 is a statistic
Proportion of US adults – parameter
Sampling variability
Take a large number of samples from the same population
Calculate x or p(hat) for each sample
Make a histogram of x(bar) or p(hat)
Examine the distribution displayed in the histogram for shape, center,
and spread as well as outliers and other deviations
Use simulations for multiple samples – much cheaper than using actual
samples
Sampling distribution of a statistic is the distribution of values taken by
the statistics in all possible samples of the same size from the same
population
Describing sample distributions
Ex 9.5 page 494-495, 496
Bias of a statistic
Unbiased statistic- a statistic used to estimate a parameter is unbiased if
the mean of its sampling distribution is equal to the true value of the
parameter being estimated.
Ex 9.6 page 498
Variability of a statistic is described by the spread of its sampling
distribution. The spread is determined by the sampling design and the
size of the sample. Larger samples give smaller spreads!!
Homework read pages 488 – 503 do problems 1-4, 12- 15
9.2 Sampling proportions
P(hat) =
=
Mean of sampling distributions
μp(hat) = p
standard deviation of sampling distribution
σp(hat) =
Must always check these two items!!!!!!!!!!!!
Rule of thumb 1- population must be at least 10x greater than sample
size (used to see if you can use standard deviation formula)
Rule of thumb 2 – we will use normal approximation as long as
np
10
n(1-p)
10
example 9.7
Given μp(hat) = .35
n = 1500
What percent fall with 2% of the true value?
______________________________
.33
σp(hat) =
.35
.37
(check for rule 1)
σp(hat) =
= .0123
check to see if we can use normal distribution
np
10
n(1-p)
1500(.35)
10
10
1500(.65)
Yes
yes
Check is population > 10 times 1500? Yes
Need to find z-scores
.33-.35
.37 - .35
.0123
.0123
= -1.63
= 1.63
.9484
.0516
____________________________
.33
.37
.9484 - .0516 = .8968
89.68% fall within 2% of the true value
P(.33
p(hat)
.37)
Homework read pages 504 – 512 problems 19 – 23
10
9.3 Sampling
μ = mean of population
x = mean of sample
μx = μ
σx =
N(μ,
(when taking a sample and finding the standard deviation)
) normal distribution of a sample mean
Z=
(when sample size is 1)
Z=
(when sample size is greater than 1)
Ex. X = height of randomly selected woman
μ = 64.5 inches
σ = 2.5 inches
take a sample of size 10
find σx =
=
= .79
p(x > 66.5)
z=
.9943
= 2.53
.0057
__________________
0.57 % chance that an SRS of 10 women would have an average height
of 66.5 inches or taller
.7881
.2119
Individual person
Z=
= .8
______________________
21.19 % chance of a woman being taller than 66.5 inches.
Do not forget sample of size n z =
Homework problems 32, 33, 39
Central limit theorem
Draw an SRS of size n from any population what so ever with mean μ
and finite standard deviation σ. When n is large, the sampling
distribution of the sample mean x is close to normal N(μ, ).
Page 522 shows examples of graphs
Homework problems 45, 47 – 53
Take home quiz