Download Introduction to inference Confidence Intervals Statistical inference

Introduction to inference Confidence Intervals
Statistical inference – provides methods for drawing conclusions about a
population from sample data.
10.1 Estimating with confidence
SAT σ = 100
n = 500
For sample (σx = σ/
) σx = 100/
µ = 461
= 4.5
___________________________
µ-9
µ
µ+9
95 %
95% of the samples of size 500 will capture µ between x
461 – 9 = 452
9
461 + 9 = 470
95% between 452 and 470
*******We are 95% confident that the true mean of the SAT for
California falls between 452 and 470. *****
Margin of error – how accurate we believe our guess is.
Confidence interval
A level C confidence interval for a parameter has two parts
1. An interval calculated from the data, usually of the form
Estimate
x
margin of error
2 standard deviations
2. A confidence level, C, which gives the probability that the interval
will capture the true parameter value in repeated samples.
C
.9 or 90 %
Page 541 – picture (need to look at)
Homework read pages 542 -543 do problems 1- 4
Confidence interval for a population mean with known σ
Conditions for constructing a confidence interval for µ
1. Data comes from SRS of the population of interest
2. Sampling distribution of x is approximately normal.
__.10__________0.8
-1.28
.10___
1.28
_.05______0.9_ _____.05_
-1.645
1.645
Need to know
Confidence
Tail area
Z*
80%
.10
1.28
90%
.05
1.645
95%
.025
1.960
99%
.005
2.576
Can find the z* at the bottom of the table in the back of book.
Z* is called critical value (on the handout z table Z* is noted as
)
Critical values
The number z* with probability p lying to its right under the standard
normal curve is called the upper p critical value of the standard normal
distribution
Probability p
__________________________
Z*
Confidence interval for a population mean
Choose an SRS of size n from population having unknown µ and known
σ. A level C confidence interval for µ is
X
z* ( )
Where z* is the value with an area C between –z* and z* under the
standard normal curve.
Confidence intervals
1. Identify population of interest and the parameter
2. Choose the appropriate inference procedure. Verify the conditions
for using the procedure.
3. If conditions are met, do procedure
CI = estimate margin of error
4. Interpret results ---Context!!!!!
Problems chapter 10 page 550-551 8 – 10
Margin of error gets smaller when
Z* gets smaller
σ gets smaller
n gets larger
choosing sample size
m = z*(
)
95% CI m
5
5 σ = 43
(1.96) (43/
)
5
(1.96) (43)
5
84.28
16.856
N
284.125
n
285
Cautions page 553
Homework problems
pages 552-557 12, 13 σ = 3.2, 14 σ=0.60, 20 c, 22 a,b
Null Hypothesis – states there is no change or effect on the population
Alternate Hypothesis – there is a change
Null Hypothesis Ho: µ = # (This is what you are really trying to disprove)
Alternate Hypothesis Ha: µ # (this is really what you want)
α=level of significance
95% confidence interval α = .05
99% confidence interval α = .01
If mean is in the range than we say “ at the 5% significance level we fail
to reject the claim that (whatever the null is)”
If mean is outside of the range then we say “at the 5% significance level
we reject the claim that (whatever the null is)”
Problems 79a, b, 80, 87
Worksheet 1,2 and 6
Inference for the mean of a population with unknown σ
Last section- We did not know the true mean but claimed to know the
standard deviation for the population.
This section- We do not know the population mean or standard
deviation of the mean
Conditions
1. SRS
2. Normal distribution
We will now use S for standard deviation instead of σ
Standard error of the statistic is
When σ is known we use Z-table
When we switch to
normal distribution)
How to use t-table
80% CI with n = 25
99% CI with n = 12
95% CI with n = 62
90% CI with n = 148
we switch to t-distribution (this does not have a
t confidence intervals and tests
Confidence intervals
x
t*
Construct a 95% CI
x= 1.329
n= 46
s= 0.484
x
t* (
)
1.329
(2.021)(
1.329
.1442
)
1.1848 – 1.4732
We are 95% confident that the true mean level of nitrogen oxides
emitted by this type of light duty engine is between 1.1848 and 1.4732
grams/mi
Formula
Estimate
x
t* SEstimate
11.4 and 11.9a-c
Confidence interval (matched pair)
x
t*(s/
)
Mean difference
(X1-X2)
t*(s/
Standard deviation of the differences
)
This is the list you want x and s
Test Test Test 1 1
2
Test 2
Need to know these
Using t-procedures
SRS-more important than normal
n < 15 use t-procedure if close to normal
n
15 use t-procedure except is strong outlier or strong skewness
n
40 can always use t-procedure
Do problems chapter 11 13a,b,d, 15a,b,c
Review problems 11.17, 11.18, 11.19
Review 11.27, 11.28, 11.29, 11.31 (95% CI)
confidence interval (x1 – x2)
t*(
degrees of freedom = n – 1 for the smallest n
problems 11.40 a,c, 11.40 b, 11.42 a,b,c,d, 11.43 (90% CI)
Review 11.47a, 11.53 a, CI 99%, c, 11.55 b, 11.56 c, 11.63 b,
****11.64 a – e****, 11.72 b,c