Sampling Distribution
Using the Z-table
Review: on using TI-84
Syntax: 2nd>vars>normalcdf>min,max,0,1)
+infinity=1EE99
-infinity=-1EE99
1. z > 2.85
0.0022
2. z < 2.85
0.9978
3. z > -1.66
0.9515
4. -1.66< z <2.85
0.9493
Review: on using TI-84
Syntax: 2nd>vars>normalcdf>min,max,mean,sd)
+infinity=1EE99
-infinity=-1EE99
1. x > 0.40, mean: 0.37, sd: 0.04
0.2266
2. 0.40<x< 0.50, mean: 0.37, sd: 0.04
0.2260
3. 0.40<x< 0.50, mean: 0.41, sd: 0.02
0.6915
Sampling distribution of
sample proportion
p
=
Count of success in sample
Size of the sample
=
X
n
The mean of the sampling distributionp is exactly p
The standard deviation of the sampling distribution
p
is
p(1-p)
n
√
One way of checking the effect of under-coverage, non
response, and other sources of error in a sample survey is
to compare the sample with the known facts about the
population.
About 11% of American adults are black, the proportion of
blacks in an SRS of 1500 adults should therefore be close
to 0.11. It is unlikely to be exactly 0.11 because of
sampling variability. If a national sample contains only
9.2% blacks should we suspect the sampling procedure is
somehow under-representing blacks?
ƥ=0.11
ℳ=0.11
σ=0.00808
σ= √
X=.092
p(1-p)
n
P(ƥ ≤ 0.092)= 0.0129
Only 1.29% of all samples would have so few blacks. Therefore
we have a good reason to suspect that the sampling procedure
underrepresented blacks.
Rule of Thumb
1. You can only use the formula for the standard deviation of p-hat
only when the population is at least 10 times as large as the sample
N≧10n
2. Use the normal approximation to the sampling distribution of
p-hat for values of n and p that satisfy np≧10 and n(1-p)≧10
Practice: Rule of thumb(s)
Explain why you cannot use the methods in ch9.1 on this problem
A factory employs 3000 unionized workers, of whom
30% are hispanic. The 15-member union executive
committee contains 3 hispanics. What would be the
probability of 3 or fewer Hispanics if the executive
committee were chosen at random from all the worker.
np≧10 and n(1-p)≧10
15(.30)≧10 and 15(1-30)≧10
4.5≧10 and 10.5≧10
Mean and Standard Deviation
of a sample mean
Mean of sampling distribution: ℳx = ℳ
Standard Deviation of sampling distribution:
σx = σ / √n
Mean and Standard Deviation
of a sample mean
Investors remember 1987 as the year the stocks lost 20%
of their value in a single day. For 1987 as a whole, the
mean return of all common stocks on the NYSE was ℳ=3.75% and the standard deviation of the returns was
about σ=26%. What are the mean and standard
deviation of the distribution for all possible samples of 5
stocks?
ℳ= -3.75%
or
-.00375
σx = σ / √n
26 / √5
σ= 11.6376%
σ=
Example: Servicing Air conditioners
The average of servicing an air conditioning unit in a
certain company is 1 hour with a standard deviation
of 1 hour as well. The company has been contracted
to maintain 70 of these units in an apartment building.
You must schedule technicians’ time for a visit to this
building. Is it safe to budget an average of 1.1 hours
for each unit? Or should you budget 1.25 hours?
ℳ=1, σ=1
Sd=σ/√n
1/√70=0.120hrs
ℳ=1 hr.
σ=.120 hr.
P(x-bar>1.1 hrs.)
normalcdf(1.1 , +∞ , 1, .120)
= 0.2033
If you budget 1.1 hrs, there is a
20% chance that the
technicians will not be able to
complete the work
N(1, .120)
P(x-bar>1.25 hrs.)
normalcdf(1.25 , +∞ , 1, .120)
= 0.0182
If you budget 1.25 hrs, there is
a 2% chance that the
technicians will not be able to
complete the work
What you should have learned
A. Sampling Distribution
1. Identify parameters and statistics in sample
experiment.
2. Recognize the fact of sampling variability.
3. Interpreting sampling distribution.
4. Describe the bias and variability of statistic in terms of
the mean and its spread.
5. Understand the variability of a statistic. Statistic from
larger samples are less variable.
B. Sample Proportions
1. Recognize when a problem involves a sample
proportion ƥ
2. Find the mean and standard deviation of sampling
distribution
3. Know that as the spread gets smaller the sample size
gets bigger.
4. Recognize a reliable conclusion by verifying the rule of
thumbs: N≥10n
and
np≥10, nq≥10
C. Sample Means
1. Recognize when a problem involves the mean of the
sample. (x-bar)
2. Find the mean and sd of the sampling distribution when
the ℳ and σ of the population are known.
3. Know that as the spread gets smaller the sample size
gets bigger
4. X-bar is approximately Normal when the sample size is
large (CLT)