Post on 27-May-2018
Overview
8.1 Z Interval for the Mean
8.2 t Interval for the Mean
8.3 Z Interval for a Population Proportion
8.4 Confidence Intervals for the Population
Variance and Standard Deviation
8.5 Sample Size Considerations
8.1 Z Interval for the Mean
Objectives:
By the end of this section, I will be
able to…
1)Explain how the formula is developed for the Z interval for the population mean μ.
2)Interpret the meaning of a confidence interval.
3)Calculate and interpret a Z interval for the population mean μ, when σ is known, for two different cases.
4)Explain the meaning of the margin of error.
Introduction to confidence intervals
Confidence interval estimate - consists of an
interval of numbers generated by a point
estimate together with an associated
confidence level specifying the probability
that the interval contains the parameter
FIGURE 8.2 Z is the value of Z that has area /2 to the right of it for . /2
2/zThe symbol represents the value of z
in the standard normal distribution so that the
area to the right of z is 2/
Recall from chapter 7, Central Limit Theorem
The sampling distribution of the sample mean for a normal population or any population if sample size is at least 30 is distributed as normal with mean:
And standard deviation:
x
nx /
Recall from chapter 7.1
When the sampling distribution of
is normal, we may standardize to produce
the standard normal random variable Z as follows:
where is the population mean, is the
population standard deviation, and n
is the sample size.
/
x
x
x xZ
n
x
Derive Z Interval for the Population Mean
Substituting into the left side:
and solving for we get the lower bound for the Z interval for the population mean
2/z
nZx 2/
n
xZ
/2/
Derive Z Interval for the Population Mean
Substituting into the left side:
and solving for we get the upper bound for the Z interval for the population mean
2/z
nZx 2/
n
xZ
/2/
Z Interval for the Population Mean μ
When a random sample of size n is taken from a population, a 100(1- )% confidence interval for μ is given by
lower bound = x –
upper bound = x +
where 1- is the confidence level.
/ n
/ n
Z Interval for the Population Mean μ
The Z interval can also be written as confidence interval (CI)
nZx
nZxCI 2/2/ ,
Z Interval for the Population Mean μ
Used only under certain conditions
Case 1:
Population is normally distributed
The value of σ is known
Case 2:
n ≥ 30
The value of σ is known
Margin of Error
Denoted as E
Measure of the precision of the confidence interval estimate
For the Z interval
nZE 2/
Interpreting the Margin of Error
For a (1 - )100% confidence interval for μ
“We can estimate μ to within E units with
(1 - )100% confidence.”
Solutions
(e) We interpret CI as follows:
We are 95% confident that the population mean lies between 18.04 and 21.96.
Example
Calculate margin of error for each part:
(a)
(b)
(c)
Example
8225.0100
5645.12/
nZE
98.0100
596.12/
nZE
288.1100
5576.22/
nZE
Solutions
(a) We interpret CI as follows:
We are 90% confident that the true mean lies between 49.1775 and 50.8225.
Example
Solutions
(b) We interpret CI as follows:
We are 95% confident that the true mean lies between 49.02 and 50.98.
Example
Solutions
(b) We interpret CI as follows:
We are 99% confident that the true mean lies between 48.712 and 51.288.
Example
Solutions
(d) The confidence interval for a given sample size gets wider as the confidence level increases
(e) Case 2: the sample size is large, and the value of σ is known
Example
(d) Margin of error
We can estimate the population mean to within 7.16 gallons with 95% confidence.
Example
gallons 16.7)651.3(96.12/n
ZE
Solutions
(e) We interpret CI as follows:
We are 95% confident that the true mean lies between 61.84 gallons and 76.16 gallons.
Example
Summary
A confidence interval estimate of a parameter consists of an interval of numbers generated by a point estimate, together with an associated confidence level specifying the probability that the interval contains the parameter.
The meaning of a 100(1 confidence interval is as follows: If we take sample after sample for a very long time, then in the long run, the proportion of intervals that will contain the parameter μ will equal 100 100(1 - )%.
Summary
The 100(1 - )% confidence interval for μ is given by the interval
lower bound =
upper bound =
where 1 – is the confidence level.
/2 / nx Z
/2 / nx Z
Summary
The conditions for applying this confidence interval are as follows:
Case 1:
The original population is normal and σ is
known.
Case 2:
The sample size is large (n ≥ 30) and σ is known.
Summary
If σ is not known, then the Z interval cannot be used.
The margin of error E is a measure of the precision of the confidence interval estimate.
For the Z interval, the margin of error takes the form
2
E Zn
Summary
We interpret the margin of error E for a (1 - )100% confidence interval for μ as follows:
“We can estimate to within E with (1 - )100% confidence.”
Confidence intervals often take the form
point estimate ± margin of error
8.2 t Interval for the Mean
Objectives:
By the end of this section, I will be
able to…
1) Describe the characteristics of the t distribution.
2) Calculate and interpret a t interval for the mean, for either of two cases.
3) Compute and interpret the standard error.
t Distribution
In real-world problems, σ is often unknown
Use s to estimate the value of σ
For a normal population
follows a t distribution
/
xt
s n
t Distribution continued
n - 1 degrees of freedom (df)
Where x is the sample mean
μ is the unknown population mean
s is the sample standard deviation
n is the sample size
Characteristics of the t Distribution
Centered at zero
The mean of t is zero
Symmetric about its mean zero
As df decreases, the t curve gets flatter, and the area under the t curve decreases in the center and increases in the tails.
As df increases toward infinity, the t curve approaches the Z curve, and the area under the t curve increases in the center and decreases in the tails.
Procedure for Finding t
Step 1
Go across the row marked “Confidence level” in the t table (Table D in the Appendix, page T-11) until you find the column with the desired confidence level at the top.
The t value is in this column somewhere.
Step 2
Go down the column to find df=n-1 on the left.
The number in that row and column is the desired value of t .
Example 8.11 - Finding t
Find the value of t that will produce a 95%
confidence interval for μ if the sample
size is n = 20.
Example 8.11 continued
Solution
Step 1
Go across the row labeled “Confidence level” in the t table (Figure 8.15) until we see the 95% confidence level.
t is somewhere in this column.
Example 8.11 continued
Solution
Step 2
df = n - 1 = 20 - 1 = 19.
Go down the column until you see 19 on the left.
The number in that row is t , 2.093.
t Interval for μ
Random sample of size n
Unknown mean μ
Confidence interval for μ
lower bound ,
upper bound
x is the sample mean
t is associated with the confidence level
n - 1 degrees of freedom
s is the sample standard deviation.
/2 /x t s n
/2 /x t s n
The t interval may also be written as confidence interval (CI)
n
stx
n
stxCI 2/2/ ,
t Interval for μ
t Interval for μ continued
The t interval applies whenever either of the following conditions is met:
Case 1:
The population is normal.
Case 2:
The sample size is large (n ≥ 30).
Margin of Error for the t Interval
The margin of error E for a (1- )100%
t interval for μ can be interpreted as follows:
“We can estimate to within E units with (1- )100% confidence.”
n
stE 2/
(a) Use t-Distribution table with confidence level 90% and degrees of freedom df=8
(b)
Example
860.116
4860.12/
n
stE
860.12/t
Solution:
Case 1 does not apply: we are not told the distribution is normal. Case 2 does not apply since the sample size must be 30 or larger. The sample size is too small (only 16) and we cannot construct the confidence interval.
Example
Solution:
Case 1 does apply: we are told the distribution is normal. We can construct the confidence interval.
Example
(a) Use t-Distribution table with confidence level 95% and degrees of freedom df=24
Margin of error:
Example
128.425
10064.22/
n
stE
064.22/t
Solution:
a) The graph is symmetric with the highest frequency occurring for the value in the middle. It appears to be a normal distribution and we can use the t-interval (by case 1)
Example
Solution:
parts (b) and (c) can be determined by entering the data in a list in the calculator and using:
STAT CALC 1:1-Var Stats
b) the sample mean is
c) sample standard deviation:
Example
25.2392x
59.274s
(d) Use t-Distribution table with confidence level 90% and degrees of freedom df=7
Margin of error:
Example
97.1838
59.274895.12/
n
stE
895.12/t
e) We are 90% confident that the population mean lies between 2208.28 and 2576.22 cigarettes per person.
Example
Summary
For a normal population, the distribution of
follows a t distribution, with n - 1 degrees of freedom, where x is the sample mean, μ is the unknown population mean, s is the sample standard deviation, and n is the sample size.
/
xt
s n
Summary
The t distribution is symmetric about its mean 0, just like the Z distribution.
t distribution is flatter.
A 100(1- )% confidence interval for μ is
given by the interval
where x is the sample mean, t is
associated with the confidence level and
n - 1 degrees of freedom, s is the sample standard deviation, and n is the sample size.
/2 /x t s n
Summary
We can construct a t interval whenever either of the following conditions is met:
Case 1:
The population is normal, or
Case 2:
The sample size is large (n ≥ 30).
The margin of error for the t interval is given by
2
sE t
n
8.3 Z Interval for a Population Proportion Objectives:
By the end of this section, I will be
able to…
1) Calculate the point estimate of the population proportion p.
2) Construct and interpret a Z interval for the population proportion p.
3) Compute and interpret the margin of error for the Z interval for p.
p̂
Point Estimate
Recall from Section 7.1 that the sample proportion of successes
is a point estimate of the population proportion p.
p̂
number of successesˆ
sample size
xp
n
Central Limit Theorem for Proportions
The sampling distribution of the sample proportion p follows an approximately normal distribution with mean μp = p
standard deviation
When both the following conditions are satisfied: (1) np ≥ 5 and (2) n(1 - p) ≥ 5.
1p
p p
n
Z Interval for p
May be performed only if both the following conditions apply: np ≥ 5 and n(1 - p) ≥ 5
Random sample of size n is taken from a binomial population with unknown population proportion p
The 100(1 - )% confidence interval for p is given by
/2
ˆ ˆ1ˆlower bound=p
p pZ
n/2
ˆ ˆ1ˆupper bound=p
p pZ
n
Z Interval for p continued
Alternatively
Where p is the sample proportion of successes, n is the sample size, and
depends on the confidence level
/2
ˆ ˆ1ˆ
p pp Z
n
Margin of Error for the Z Interval for p
The margin of error E for a (1- )100% Z interval for p can be interpreted as follows:
“We can estimate p to within E with (1- )100% confidence.”
/2
ˆ ˆ1p pE Z
n
Solutions: continuing we get the following answers
(a) E=0.0588
(b) E=0.0784
(c) E=0.0898
(d) E=0.0960
(e) E=0.0980
Example
10.0p̂
20.0p̂
30.0p̂
40.0p̂
50.0p̂
Solutions
(a)The margin of error increases as the sample proportion approaches 0.5 and the sample size remains constant.
(b) The width of the confidence interval is twice the margin of error so this width also increases.
Example
(b)
Example
571)0190.0(3733p̂n
0190.03733
71p̂
53662)981.0(3733)0190.01(3733q̂n
We can use the Z interval for the sample proportion.
(c)
We can estimate the population proportion
within 0.0058 with 99% confidence.
Example
0058.03733
)981.0(0190.0576.2
ˆˆ2/
n
qpZE
0190.03733
71p̂
(d) We interpret CI as follows:
We are 99% confident that the
population proportion of all Americans who have gotten most of their information about the September 11 events from the internet lies between 0.0132 (1.32%) and 0.0248 (2.48%).
Example
Summary
The sample proportion of successes
is a point estimate of the population
proportion p.
The 100(1- )% confidence interval for the population proportion p, is given by
number of successesˆ
sample size
xp
n
/2
ˆ ˆ1ˆ
p pp Z
n
Summary
Where p is the sample proportion of
successes, n is the sample size, and
depends on the confidence level.
The Z interval for p may be constructed only
if both the following conditions apply:
n p ≥ 5 and n(1 - p) ≥ 5.