Download Section 7

Overview of Chapter 9
Point Estimate – the value of a statistic that estimates the value of a parameter.
x is a point estimate of 
p̂ is a point estimate for p.
s is a point estimate of σ.
Confidence Interval – A range or interval of values used to estimate the value of an
unknown parameter.
The Confidence Interval is a way to estimate a population parameter by creating a range of
values that is likely to contain the true population parameter.
Level of Confidence –It represents the expected proportion of intervals that will contain
the parameter if a large number of different samples is obtained. denoted (1 – α)100%.
The confidence level is sometimes explained as the “success rate of the procedure.”
The most common confidence levels:
90% (.90)
95% (.95)
99% (.99)
A 95% level of confidence implies that if 100 different confidence intervals are constructed,
each based on a different sample from the same population, then we will expect 95 of the
intervals to include the parameter and 5 to not include the parameter.
The level of confidence refers to the confidence in the method, not the specific
interval!
Critical Value – A number that separates sample statistics that are likely to occur from
those that are unlikely to occur.
z / 2 =
90% confidence
 =
/2=
z / 2 =
95% confidence
 =
/2=
99% confidence
 =
/2=
z / 2 =
Margin of Error (E) – A measure of how accurate the point estimate is. It depends on
three factors:
1. Level of confidence – As the level of confidence increases, the margin of error increases.
2. Sample size – As the sample size increases, the margin of error decreases.
3. Standard deviation of the population – The larger the spread in the population, the wider
the interval will be.
Confidence intervals are created by taking the point estimate from your sample and adding
and subtracting the margin of error.
Point Estimate ± Margin of Error
1
Summary of all Estimation Techniques in Chapter 9
Population
Parameter
to Estimate
Section in Point
Formula for Confidence
Chapter 9 Estimate Interval
Type of
Critical
Value
Margin
of
Error
Lowerbound < parameter < Upperbound
Proportion
(p)
9.1
p̂
Mean
( )
9.2
 known
x
Mean
( )
9.2

unknown
9.3
x
s2
9.3
s
Variance
( 2 )
Standard
Deviation
()
p̂  z
x  z
x  t
z
ˆˆ
pq
n

z
n
s
t
n
(n  1)s2
R2
(n  1)s2
R2
 
2

  
(n  1)s2
L2
(n  1)s2
L2
z
z
t
ˆˆ
pq
n

n
s
n
2
2
Section 9.1 – Estimating A Population Proportion
Assumptions: the sample size can be no more that 5% of the population size and npq ≥ 10.
Notation:
p = population proportion
x
p̂ 
sample proportion of x successes in sample size n
n
qˆ  1  pˆ
Point estimate for population proportion p is p̂ .
The sample proportion is the point estimate.
A researcher wants to estimate the true proportion of all likely voters who will vote for
candidate A. If a random sample of 1008 likely voters reveals that 35% of them plan to vote
for candidate A, the sample proportion is ________. The point estimate is ________.
A researcher wants to estimate the true proportion of all college graduates who smoke. If a
random sample of 785 college graduates reveals that 144 of them smoke, then the sample
proportion is ______________. The point estimate is ___________.
Confidence Interval for Proportion:
where E is the margin of error
pˆ  E  p  pˆ  E
ˆˆ
pq
E = z
2
n
2
The margin of error is the maximum error we are willing to accept in the estimation process.
E is the maximum likely difference between the observed sample proportion and the true
population “p”. It is made up of (critical value) x (standard deviation of sample proportions).

ˆˆ
ˆˆ 
pq
pq
, pˆ  z
The formula for the confidence interval for proportion is:  pˆ  z


n
n


Note: p̂ forms the basis of the interval. The margin of error is subtracted from it, and then
added to it.
The sampling distribution of sample proportions is approximately normal with   np and
  npq . (npq ≥ 10) so the critical value used is a Z.
Page 426 Example 1
A sample of 1020 Americans were asked if the amount of federal taxes they paid was too
high. Of the 1020 Americans surveyed, 490 said the amount was too high. If we can say
with 95% confidence the maximum margin of error is plus or minus 4 percentage points,
obtain a point estimate for the number who believe it is too high and find a confidence
interval. Answer: .480 and (.44, .52)
Page 432 Example 4
A survey of 800 16 to 17 year olds found that 272 would text and drive. Obtain a 95%
confidence interval for the proportion of 16 to 17 year olds who text and drive.
Answer: (.307, .373)
Calculator: Stats  Tests A:1Prop Z-Int
Enter x, n and confidence level
Determining the Sample Size needed for estimating the population proportion p:
 z
ˆˆ 2
n = pq
 E





2
(if you know that p̂ is a prior estimate of p)
2
 z 
n = 0.25  2  (if a prior estimate of p is unavailable)
 E 


(if this does not result in a whole number, then round UP to the next highest
whole number)
Page 435 Example 6
An economist wants to know if the proportion of the U.S. population who commutes to work
via car-pooling is on the rise. What sample size should be obtained if the economist wants to
be within 2 percentage points of the true proportion with a 90% confidence if
a. the economist uses the 2006 estimate that 10.7% ? Answer: 647
b. the economist does not use any prior estimates? Answer: 1691.3
3
Section 9.2 – Estimating a Population Mean
**Estimation of the population mean
Assumption: the distribution of the population is approximately normal
The Student’s t Distribution:
t=
x
s
n
Properties of the t-distribution
1. The shape of the distribution is different for different sample sizes.
2. The distribution has mean 0 and is symmetric about 0.
3. The distribution has standard deviation greater than 1. (more variation)
4. As n gets larger, the t-distribution gets closer to the standard normal distribution.
Use the t-distribution when σ is unknown and EITHER n > 30 or the population is bellshaped.
Point estimate for population mean is x .
Confidence Interval for a Population Mean μ where σ is unknown:
x E
  
x E
 s 
where E  t  

2  n 
E is the margin of error
The margin of error is the maximum error we are willing to accept in the estimation process.
E is the maximum likely difference between the observed sample mean and the true
population mean. It is made up of (critical value) x (standard deviation of sample means).
The formula for the confidence interval for the mean when the population standard deviation

s
s 
is unknown:  x  E, x  E =  x  t
, x t

n
n

Note: x forms the basis of the interval. The margin of error is subtracted from it, and then
added to it.
For sample means when the population standard deviation is not known, the distribution
follows a t-distribution. It is bell-shaped and behaves like the normal distribution, but
changes based on the sample size.
Calculator: Stat  Tests 8:T-Interval Choose either Data or Stats
4
If Data, enter confidence level and calculate.
If Stats, enter x , s, n and confidence level then calculate.
Ex ample. Page 447 #7
a. Find the t-value such that the area in the right tail is .10 with 25 degrees of freedom.
Answer: t = 1.316
b. Find the t-value such that the area in the right tail is .05 with 30 degrees of freedom.
Answer: t = 1.697
c. Find t-value such that the area left of the t-value is .01 with 18 degrees of freedom.
Answer: t = -2.552
d. Find the critical t-value that corresponds to 90% confidence. Assume 20 degrees of
freedom. Answer: t = .1725
Page 448 #18
Determine the point estimate of the population mean and the error for the
confidence level with lower bound equal to 15 and upper bound equal to 35.
Page 449 #24
A 90% confidence interval for the number of hours full-time college sleep during the
weekday is lower bound 7.8 hours and upper bound 8.8 hours.
Interpret the meaning of the sentence.
Page 449 # 32
A Gallup poll conducted July 21 – August 14, 1978, asked 1006 Americans “During the past
year about how many books did you read either all or part of the way through? Results of
the survey indicated the mean was 18.8 books and the standard deviation was 19.8 books.
a. Construct a 99% confidence interval for the mean number of books read and interpret the
interval. Answer: (17.19, 20.41)
Page 450 #38
The following data represent the age in weeks at which babies first crawl. The data is based
on a survey. Construct a 95% confidence interval for the mean age at which a baby first
crawls. Answer: (31.9, 44.6)
Survey results: 52, 30, 44, 35, 39, 26, 47, 37, 56, 26, 39, 28
Section 9.3 – Estimating a Population Standard Deviation
5
**Estimation of population proportion
Section 9.4 – Confidence Intervals for a Population Standard Deviation
**Estimation of population variance or standard deviation.
Assumptions: Must have simple random sample and the sample must come from a fairly
normal distribution. Under these conditions sample variances have a distribution called a
 2 (Chi-Squared) distribution.
Chi-Square Distribution:
Characteristics:
1.
2.
3.
4.
 
2
(n  1) s 2
(degrees of freedom: n – 1)
2
Not symmetric
Shape of the distribution depends on degrees of freedom
More nearly symmetric as degrees of freedom increases
No negative values
Point estimate for population variance  2 is s2.
Point estimate for population standard deviation  is s.
(n  1)s2
R2
Confidence Interval for Variance:
Confidence Interval for Standard Deviation:
(n  1)s2
R2
 2 
(n  1)s2
L2
  
The critical value for this confidence interval is a  2 value from Table VII.
To Find the Chi-Square Critical Values from Table VII:
1. Find 
2. Calculate 
2
3. Locate the two columns corresponding to 
4. Locate the degrees of freedom row (n – 1)
5. This will give you  L2 and  R2
Find the critical values:
6
2
(n  1)s2
L2
1. 95% confidence interval with n = 25
Answer: 12.401 and 39.364
2. 90% confidence interval with n = 14
Answer: 3.565 and 29.819
Page 459
10. A simple random sample of size n is drawn from a population that is known to be
normally distributed. The sample variation is determined to be 19.8.
a. Construct a 95% confidence interval for the sample variation if n = 10. Answer: (9.37,
66.00)
b. Construct a 95% confidence interval for the sample variation if n = 25. What happens to
the confidence interval as n increases? Answer: (12.07, 38.32)
14. The following data represent the age in weeks at which babies first crawl. The data is
based on a survey. It was verified that the data are normally distributed and s = 10 weeks.
Construct and interpret a 95% confidence interval for the population standard deviation.
Survey results: 52, 30, 44, 35, 39, 26, 47, 37, 56, 26, 39, 28
Answer: (7.08, 16.98)
7