confidence interval hypothesis by MA826S

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```									               Estimation and Hypothesis
Lesson Objectives

1. Know what is parameter estimation
2. Understand hypothesis testing & the “types of errors” in
decision making.
3. Learn how to use test statistics to examine hypothesis
Statistic           Parameter

Mean:         X        estimates       ____
Standard
deviation:    s        estimates       ____

Proportion:   p        estimates       ____
from entire
from sample
population
Population     Point estimate Interval estimate
I am 95%
Mean           confident that 
Mean, , is                     is between 40 &
X = 50
unknown                                 60

Sample
CONFIDENCE INTERVAL
Definitions :
1. Estimator :
a formula or process for using sample data to
estimate a population parameter
2. Estimate :
a specific value or range of values used to
approximate some population parameter
3. Point Estimate :
a single value (or point) used to approximate a
population parameter
The sample mean x is the best point estimate of
the population mean µ
4. Confidence Interval (or Interval Estimate) :
a range (or an interval) of values used to
estimate the true value of the population
parameter.

Lower # < population parameter < Upper #

As an example :
Lower # < µ< Upper #
5. Degree of confidence (level of confidence or
confidence coefficient) :
The probability 1 -  (often expressed as the
equivalent percentage value) that is the relative
frequency of times the confidence interval actually
does contain the population parameter, assuming that
the estimation process is repeated a large number of
times.

Usually 95 %, (= 5%) or 99 % (=1 %).
Interpreting a Confidence Interval for  (for example :
Systolic Blood pressure of medical student) :
100 <  < 140

We are 95 % confident that the interval from 100 to 140
actually does contain the true value of . This means
that if we were to select many different samples of
size 100 and construct the confidence interval, 95 % of
them would actually contain the value of the population
mean .

PARAMETER = STATISTIC  ITS ERROR
Standard Error

S
Quantitative Variable   SE (Mean) =    n

p(1-p)
Qualitative Variable    SE (p) =        n
Confidence Interval

α/2                                         α/2
1-α
_
X
SE            SE           Z-axis

95% Samples

X - 1.96 SE       X + 1.96 SE
Confidence Interval

α/2                                       α/2
1-α

SE           SE           Z-axis   p

95% Samples

p - 1.96 SE       p + 1.96 SE
Interpretation of
CI

Probabilistic                         Practical

In repeated sampling 100(1-)%    We are 100(1-)% confident that
of all intervals around sample
the single computed CI contains
means will in the long run
include                           
Example (Sample size≥30)

An epidemiologist studied the blood glucose
level of a random sample of 100 patients. The
mean was 170, with a SD of 10.
 = X + Z SE
SE = s/√n = 10/10 = 1
95
Then CI:                                    %

 = 170 + 1.96  1 168.04   ≤ 171.96
Example (Proportion)
In a survey of 140 asthmatics, 35% had allergy to
house dust. Construct the 95% CI for the
population proportion.
 = p + Z √ p(1-p)/n     SE= √ 0.35(1-0.35)/140 = 0.04

0.35 – 1.96  0.04   0.35 + 1.96  0.04
0.27   ≤ 0.43
27%    43%
HYPOTHESIS

What is Hypothesis ?

Definition :

Hypothesis in statistics, is a claim or statement

Components of formal hypothesis test :
- Null Hypothesis : H0
- Statement about value of population parameter
- Must contain condition of equality : = ; ; or 
- Test the null Hypothesis directly
- Recect H0 or fail to reject H0
Alternative Hypothesis
- Must be true if H0 is false
- ≠; < ; >
- Opposite of null

Note : If you are conducting a study and want to
use a hypothesis test to support your claim, the
claim must be worded so that it becomes the
alternative hypothesis.
LEVEL OF SIGNIFICANCE, 

The probability that the test statistic will fall in
the critical region when the null hypothesis is
actually true. Common choices are 0.05; 0.01
and 0.10.
TYPE ERROR : TYPE I AND TYPE I

Type I error : - the mistake of rejecting the null
hypothesis when it is true.
-  (alpha) is used to represent the
probability of a type I error.

Example : Rejecting a claim that the mean systolic
blood pressure is 110 mmHg when the mean really
does equal 110 mmHg.
Type II error : - the mistake of failing to reject the null
hypothesis when it is false.
-  (beta) is used to represent the
probability of a type II error.

Example : Failling to reject the claim that the mean
systolic blood pressure is 110 mmHg when the mean
really different from 110 mmHg.
Table Type I and Type II Errors

True State of Nature

The null hypothesis is    The null hypothesis is
true                      false
Decision : Type I error              Correct decision
(rejecting a true null
hypothesis)

Correct decision          Type II error
(rejecting a falls null
hypothesis)

p value Test

- Probability of Obtaining a Test Statistic More
Extreme ( or  ) than Actual Sample Value
Given H0 Is True .
- Called Observed Level of Significance
- Used to Make Rejection Decision
* If p value  , Do Not Reject H0
* If p value < , Reject H0
Hypothesis Testing: Steps

Test the Assumption that the true mean SBP of
participants is 120 mmHg.

* State H0                H0 : µ = 120
* State H1               H1 : µ  120
* Choose              = 0.05
* Choose n            n = 100
* Choose Test:        Z, t, X2 Test (or p Value)
* Compute Test Statistic (or compute P value)
* Search for Critical Value
* Make Statistical Decision rule
* Express Decision
Example : One sample-mean Test

- Assumptions
- Population is normally
distributed
- State the null and
alternative hypotheses
H0: µ = µ0
H1: µ  µ0
- t test statistic :
sample mean  null value x   0
t                         
standard error       s
n

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