Download Chapter 7

Inferences Based on a Single Sample:
Estimation with Confidence Intervals
7.1 Identifying and Estimating the
Target Parameter
Finding a value for Z  or Z
90%
Z  1.645
2
95%
Z  1.96
2
99%
Z  2.576
2
2
The population mean  is an unknown parameter
x is a statistic which estimates 
We call x a point estimate because its value is a point on
the real number line
Unfortunately, if we sample from a continuous
distribution, P( x   )  0
Statisticians prefer interval estimates
7.2 Confidence Interval for a Population
Mean: Normal (z) Statistic
xE
E (the Error Tolerance) depends on the sample size, how
certain we want to be that we are correct (Level of
Confidence), and the amount of variability in the data
When  is known Z 
x

has approximately a
standard normal
n
distribution.
P  z  Z  z   1  
2
2

  1
P x  z 
   x  z 

2
2
n
n

  
  
Therefore, E  Z 
  Z

2 n 
 n
Notice that increasing the level of confidence, decreases
the probability of error, however it also increases E
(creating a wider interval)
Notice as sample size increases, E decreases (creating a
more narrow interval)
Notice the more variability in the population, the larger
E will be (creating a wider interval)
Example
A sample of 100 visa accounts were studied for the
amount of unpaid balance.
x  $645
  $132
Construct and interpret a 95% confidence interval
We are 95% confident that the mean unpaid balance
of visa accounts is between $619.13 and $670.87.
Construct and interpret a 99% confidence interval
We are 99% confident that the mean unpaid balance
of visa accounts is between $611.00 and $679.00.
7.3 Confidence Interval for a Population
Mean: Student’s t-Statistic

is unknown
To avoid the error involved in replacing  by , we will
introduce a new random variable called Student’s t
variable. (t-distribution)
s
If we sample from a normal distribution
x   has a t-distribution with n-1
t
s
degrees of freedom.
n
Properties of the t-distribution
 continuous and symmetric about 0
 more variable and slightly different shape than the
standard normal
 As n becomes large, the t-distribution can be
approximated by the standard normal distribution
(The bottom row of the t-distribution is Z)
With a sample size of 11 and a Confidence level of 95%,
what is the two tailed t value?
t.025,10  2.228
For  unknown,
s
E  t( )
n
Example
Mileage of tires in 1000’s of miles
Sample: 42, 36, 46, 43, 41, 35, 43, 45, 40, 39
Compute and interpret a 95% confidence interval for 
n  10
x  41
s  3.59
t.025,9  2.262
We are 95% confident that the population mean mileage
of tires is between 38,432 and 43,568 miles.
Example
A random sample of 20 apples yields
x  9.2 oz. and s  1.1 oz.
Find and interpret a 99% confidence interval.
We are 99% confident that the population mean weight
of apples is between 8.496 and 9.904 oz.
7.4 Large-Sample Confidence Interval
for a Population Proportion
For large n, Z 
pˆ  p
pˆ qˆ
EZ
n
p̂  E
pq
n
is approximately standard normal.
Example
A survey of 1,200 registered voters yields 540 who plan to
vote for the republican candidate.
p = proportion of all voters who plan to vote for the
republican candidate
Find and interpret a 95% confidence interval for p
We are 95% confident that the population proportion of
voters who plan to vote for the republican candidate is
between 42.2% and 47.8%.
7.5 Determining the Sample Size
In the design stages of statistical research, it is good to
decide the confidence level you wish to use and to
select the error tolerance you want for the project. This
will let us decide how big our sample needs to be.
Sample size for estimating  with known 
 Z 
n

 E 
2
If the value of  is not known, we do preliminary
sampling to approximate it.
Example
We wish to estimate the number of patient-visit hours
per week physicians in solo practice spent. How large
a sample is needed if we want to be 99% confident that
our point estimate is within 1 hour of the population
mean? Assume a standard deviation of 11.97 hours.