Statistical Inference Making decisions regarding the population base on a sample.

Statistical Inference

Making decisions regarding the population base on a sample

Decision Types• Estimation

– Deciding on the value of an unknown parameter

• Hypothesis Testing– Deciding a statement regarding an unknown parameter

is true of false

• All decisions will be based on the values of statistics

• Prediction– Deciding the future value of a random variable

Estimation

• Definitions

– An estimator of an unknown parameter is a sample statistic used for this purpose

– An estimate is the value of the estimator after the data is collected

• The performance of an estimator is assessed by determining its sampling distribution and measuring its closeness to the parameter being estimated

Examples of Estimators

The Sample Proportion

Let p = population proportion of interest or binomial probability of success.

Let

trialsbimomial of no.

succeses of no.ˆ

n

Xp

p̂ ofon distributi sampling Then the

pp ˆmean

n

ppp

)1(ˆ

is a normal distribution with

= sample proportion or proportion of successes.

p̂ ofon distributi Sampling

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c

pp ˆ

The Sample Mean

Let x1, x2, x3, …, xn denote a sample of size n from a normal distribution with mean and standard deviation .Let

mean sample1

n

xx

n

ii

x ofon distributi sampling Then the

xmean

nx

is a normal distribution with

0

0.05

0.1

0.15

0.2

0.25

0.3

80 90 100 110 120

population

n = 5

n = 10

n = 15

n = 20c

x

x ofon distributi Sampling

Confidence Intervals

Estimation by Confidence Intervals• Definition

– An (100) P% confidence interval of an unknown parameter is a pair of sample statistics (t1 and t2) having the following properties:

1. P[t1 < t2] = 1. That is t1 is always smaller than t2.

2. P[the unknown parameter lies between t1 and t2] = P.

• the statistics t1 and t2 are random variables

• Property 2. states that the probability that the unknown parameter is bounded by the two statistics t1 and t2 is P.

Critical values for a distribution• The upper critical value for a any distribution

is the point x underneath the distribution such that P[X > x] =

x

Critical values for the standard Normal distribution

P[Z > z] =

z

Critical values for the standard Normal distribution

P[Z > z] =

Confidence Intervals for a proportion p

Then t1 to t2 is a (1 – )100% = P100% confidence interval for p

n

ppzpzpt p

1ˆˆ 2/ˆ2/1

n

ppzp

ˆ1ˆˆ 2/

and

n

ppzpzpt p

1ˆˆ 2/ˆ2/2

n

ppzp

ˆ1ˆˆ 2/

Let

Logic:

Thus t1 to t2 is a (1 – )100% = P100% confidence interval for p

p

ppz

ˆ

ˆ

has a Standard Normal distribution

and

1

ˆ2/

ˆ2/ z

ppzP

p

PzzzP 1Then

Hence 1ˆˆ ˆ2/ˆ2/ pp zppzpP

121 tptP

ˆ ˆ/ 2 / 2ˆ 1p pP z p p z

ˆ ˆ/ 2 / 2ˆ 1p pP z p p z

Example• Suppose we are interested in determining the success rate

of a new drug for reducing Blood Pressure

• The new drug is given to n = 70 patients with abnormally high Blood Pressure

• Of these patients to X = 63 were able to reduce the abnormally high level of Blood Pressure

900.070

63ˆ

n

Xp

• The proportion of patients able to reduce the abnormally high level of Blood Pressure was

This is an estimate of p.

Then

Thus a 95% confidence interval for p is 0.8297 to 0.9703

n

ppzpt

ˆ1ˆˆ 2/1

and

If P = 1 – = 0.95 then /2 = .025 and z = 1.960

70

10.090.0)960.1()90.0(

8297.00703.)90.0(

n

ppzpt

ˆ1ˆˆ 2/2

70

10.090.0)960.1()90.0(

9703.00703.)90.0(

This comes from the Table

What is the probability that p is beween 0.8297 and 0.9703?

Is it 95% ?

Answer: p (unknown) , 0.8297 and 0.9703 are numbers.

Either p is between 0.8297 and 0.9703 or it is not.

The 95% refers to success of confidence interval procedure prior to the collection of the data.

After the data is collected it was either successful in capturing p or it was not.

Error Bound

For a (1 – )% confidence level, the approximate margin of error in a sample proportion is

ˆ ˆ1Error Bound

p pz

n

Factors that Determine the Error Bound

1. The sample size, n. When sample size increases, margin of error decreases.

p̂2. The sample proportion, . If the proportion is close to either 1 or 0 most individuals have the same trait or opinion, so there is little natural variability and the margin of error is smaller than if the proportion is near 0.5.

3. The “multiplier” z/2. Connected to the “(1 – )%” level of confindence of the Error Bound. The value of z/2 for a 95% level of

confidence is 1.96 This value is changed to change the level of confidence.

Determination of Sample Size

In almost all research situations the researcher is interested in the question:

How large should the sample be?

Answer:

Depends on:

• How accurate you want the answer.

Accuracy is specified by:

• Specifying the magnitude of the error bound

• Level of confidence

Error Bound:

• If we have specified the level of confidence then the value of za/2 will be known.

• If we have specified the magnitude of B, it will also be known

n

ppz

n

ppzB aa

ˆ1ˆ12/2/

Solving for n we get:

2

22/

2

22/ *1*1

B

ppz

B

ppzn aa

Summarizing:

The sample size that will estimate p with an Error Bound B and level of confidence P = 1 – is:

where:• B is the desired Error Bound• z is the /2 critical value for the standard normal

distribution• p* is some preliminary estimate of p.If you do not have a preliminary estimate of p, use p* = 0.50

2

22/ *1*

B

ppzn a

Reason

For p* = 0.50 2

22/ *1*

B

ppzn a

0

500

1000

1500

2000

2500

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

n

*p

n will take on the largest value.

Thus using p* = 0.50, n may be larger than required if p is not 0.50. but will give the desired accuracy or better for all values of p.

Example• Suppose that I want to conduct a survey and want to estimate

p = proportion of voters who favour a downtown location for a casino:

I know that the approximate value of p is• p* = 0.50. This is also a good choice for p if one has no

preliminary estimate of its value.• I want the survey to estimate p with an error bound B = 0.01

(1 percentage point)• I want the level of confidence to be 95% (i.e. = 0.05 and

z = z = 1.960Then

9604

01.0

50.050.0960.12

2

n


for the mean , ,

of a Normal Population

Confidence Intervals for the mean of a Normal Population,

Then t1 to t2 is a (1 – )100% = P100% confidence interval for

nzxzxt x

2/2/1 Let

andn

zxzxt x

2/2/2

Logic:

Thus t1 to t2 is a (1 – )100% = P100% confidence interval for

x

xz

has a Standard Normal distribution

and

12/2/ z

xzP

x

PzzzP 1Then

Hence 12/2/ xx zxzxP

121 ttP

Example• Suppose we are interested average Bone Mass Density

(BMD) for women aged 70-75

• A sample n = 100 women aged 70-75 are selected and BMD is measured for eahc individual in the sample.

• The average BMD for these individuals is:

63.25x• The standard deviation (s) of BMD for these individuals

is: 82.7s

Then

Thus a 95% confidence interval for is 24.10 to 27.16

n

szx

nzxt 2/2/1

and

If P = 1 – = 0.95 then /2 = .025 and z = 1.960

10.2453.163.25100

82.7960.163.25

n

szx

nzxt 2/2/2

16.2753.163.25100

82.7960.163.25


Again a question to be asked:

How large should the sample be?

Answer:

Depends on:

• How accurate you want the answer.

Accuracy is specified by:

• Specifying the magnitude of the error bound

• Level of confidence

Error Bound:

• If we have specified the level of confidence then the value of z/2 will be known.

• If we have specified the magnitude of B, it will also be known

nzB a

2/

Solving for n we get:

2

222/

2

222/ *

B

sz

B

zn aa

Summarizing:

The sample size that will estimate with an Error Bound B and level of confidence P = 1 – is:


distribution• s* is some preliminary estimate of .

2

222/

2

222/ *

B

sz

B

zn aa

Notes:

• n increases as B, the desired Error Bound, decreases– Larger sample size required for higher level of

accuracy• n increases as the level of confidence, (1 – ), increases

– z increases as /2 becomes closer to zero.– Larger sample size required for higher level of

confidence• n increases as the standard deviation, , of the population

increases.– If the population is more variable then a larger sample

size required

2

222/

2

222/ *

B

sz

B

zn aa

Summary:

The sample size n depends on: • Desired level of accuracy• Desired level of confidence• Variability of the population

Example• Suppose that one is interested in estimating the average

number of grams of fat (m) in one kilogram of lean beef hamburger :

This will be estimated by:• randomly selecting one kilogram samples, then • Measuring the fat content for each sample.• Preliminary estimates of and indicate:

– that and are approximately 220 and 40 respectively.

• I want the study to estimate with an error bound 5 and • a level of confidence to be 95% (i.e. = 0.05 and z =

z = 1.960)

Solution

2469.2455

40960.12

22

n

Hence n = 246 one kilogram samples are required to estimate within B = 5 gms with a 95% level of confidence.

Statistical Inference

Making decisions regarding the population base on a sample

Decision Types• Estimation

– Deciding on the value of an unknown parameter

• Hypothesis Testing– Deciding a statement regarding an unknown parameter

is true of false

• All decisions will be based on the values of statistics

• Prediction– Deciding the future value of a random variable

Estimation

• Definitions

– An estimator of an unknown parameter is a sample statistic used for this purpose

– An estimate is the value of the estimator after the data is collected

• The performance of an estimator is assessed by determining its sampling distribution and measuring its closeness to the parameter being estimated

Comments

• When you use a single statistic to estimate a parameter it is called a point estimator

• The estimate is a single value• The accuracy of this estimate cannot be

determined from this value• A better way to estimate is with a confidence

interval.• The width of this interval gives information on

its accuracy

Estimation by Confidence Intervals• Definition

– An (100) P% confidence interval of an unknown parameter is a pair of sample statistics (t1 and t2) having the following properties:

1. P[t1 < t2] = 1. That is t1 is always smaller than t2.

2. P[the unknown parameter lies between t1 and t2] = P.

• the statistics t1 and t2 are random variables

• Property 2. states that the probability that the unknown parameter is bounded by the two statistics t1 and t2 is P.


Summary

Confidence Interval for a Proportion

pzp ˆ2/ˆ

n

pp

n

ppp

ˆ1ˆ1ˆ

point critical 2/upper 2/ z

ndistribtio normal standard theof

ˆ/ 2 / 2 / 2

ˆ ˆ1 1p

p p p pB z z z

n n

Error Bound

The sample size that will estimate p with an Error Bound B and level of confidence P = 1 – is:


distribution• p* is some preliminary estimate of p.

2

22/ *1*

B

ppzn a


Confidence Intervals for the mean of a Normal Population,

/ 2 xx z

/ 2or x zn

/ 2or s

x zn

sample meanx point critical 2/upper 2/ z

ndistribtio normal standard theof sample standard deviation s

The sample size that will estimate with an Error Bound B and level of confidence P = 1 – is:


distribution• s* is some preliminary estimate of s.

2

222/

2

222/ *

B

sz

B

zn aa


Hypothesis Testing

An important area of statistical inference

Definition

Hypothesis (H)– Statement about the parameters of the population

• In hypothesis testing there are two hypotheses of interest.– The null hypothesis (H0)

– The alternative hypothesis (HA)

• There two possible reasons for the term Null hypothesis (H0)

1. It is usually the hypothesis of • No difference , no effect

2. It is also the hypothesis to be disproved (nullified)

Either

– null hypothesis (H0) is true or

– the alternative hypothesis (HA) is true.

But not both

We say that are mutually exclusive and exhaustive.

One has to make a decision – to either to accept null hypothesis

(equivalent to rejecting HA)

or– to reject null hypothesis (equivalent to

accepting HA)

There are two possible errors that can be made.

1. Rejecting the null hypothesis when it is true. (type I error)

2. accepting the null hypothesis when it is false (type II error)

An analogy – a jury trial

The two possible decisions are

– Declare the accused innocent.

– Declare the accused guilty.

The null hypothesis (H0) – the accused is innocent

The alternative hypothesis (HA) – the accused is guilty

Hypothesis Testing

An important area of statistical inference

Definition

Hypothesis (H)– Statement about the parameters of the population

• In hypothesis testing there are two hypotheses of interest.– The null hypothesis (H0)


• There two possible reasons for the term Null hypothesis (H0)

1. It is usually the hypothesis of • No difference , no effect

2. It is also the hypothesis to be disproved (nullified)

Either

– null hypothesis (H0) is true or

– the alternative hypothesis (HA) is true.

But not both

We say that are mutually exclusive and exhaustive.

One has to make a decision – to either to accept null hypothesis

(equivalent to rejecting HA)

or– to reject null hypothesis (equivalent to

accepting HA)

There are two possible errors that can be made.

1. Rejecting the null hypothesis when it is true. (type I error)

2. accepting the null hypothesis when it is false (type II error)

Decision Table showing types of Error

H0 is True H0 is False

Correct Decision

Correct Decision

Type I Error

Type II Error

Accept H0

Reject H0

To define a statistical Test we

1. Choose a statistic (called the test statistic)

2. Divide the range of possible values for the test statistic into two parts

• The Acceptance Region

• The Critical Region

To perform a statistical Test we

1. Collect the data.

2. Compute the value of the test statistic.

3. Make the Decision:

• If the value of the test statistic is in the Acceptance Region we decide to accept H0 .

• If the value of the test statistic is in the Critical Region we decide to reject H0 .

Example

We are interested in determining if a coin is fair.

i.e. H0 : p = probability of tossing a head = ½.

To test this we will toss the coin n = 10 times.

The test statistic is x = the number of heads.

This statistic will have a binomial distribution with p = ½ and n = 10 if the null hypothesis is true.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

Sampling distribution of x when H0 is true

Note

We would expect the test statistic x to be around 5 if H0 : p = ½ is true.

Acceptance Region = {3, 4, 5, 6, 7}.

Critical Region = {0, 1, 2, 8, 9, 10}.

The reason for the choice of the Acceptance region:

Contains the values that we would expect for x if the null hypothesis is true.

Definitions: For any statistical testing procedure define

1 = P[Rejecting the null hypothesis when it is true] = P[ type I error]

= P[accepting the null hypothesis when it is false] = P[ type II error]

In the last example

1 = P[ type I error] = p(0) + p(1) + p(2) + p(8) + p(9) + p(10) = 0.109, where p(x) are binomial probabilities with p = ½ and n = 10 .

= P[ type II error] = p(3) + p(4) + p(5) + p(6) + p(7), where p(x) are binomial probabilities with p (not equal to ½) and n = 10. Note: these will depend on the value of p.

Table: Probability of a Type II error, vs. p

p 0.1 0.0700.2 0.3220.3 0.6160.4 0.8200.6 0.8200.7 0.6160.8 0.3220.9 0.070

Note: the magnitude of increases as p gets closer to ½.

Comments: 1. You can control = P[ type I error] and = P[

type II error] by widening or narrowing the acceptance region. .

2. Widening the acceptance region decreases = P[ type I error] but increases = P[ type II error].

3. Narrowing the acceptance region increases = P[ type I error] but decreases = P[ type II error].

Example – Widening the Acceptance Region

1. Suppose the Acceptance Region includes in addition to its previous values 2 and 8 then = P[ type I error] = p(0) + p(1) + p(9) + p(10) = 0.021, where again p(x) are binomial probabilities with p = ½ and n = 10 .

= P[ type II error] = p(2) + p(3) + p(4) + p(5) + p(6) + p(7) + p(8). Tabled values of are given on the next page.


p 0.1 0.2640.2 0.6240.3 0.8510.4 0.9520.6 0.9520.7 0.8510.8 0.6240.9 0.264

Note: Compare these values with the previous definition of the Acceptance Region. They have increased,

Example – Narrowing the Acceptance Region

1. Suppose the original Acceptance Region excludes the values 3 and 7. That is the Acceptance Region is {4,5,6}. Then = P[ type I error] = p(0) + p(1) + p(2) + p(3) + p(7) + p(8) +p(9) + p(10) = 0.344.

= P[ type II error] = p(4) + p(5) + p(6) . Tabled values of are given on the next page.


p 0.1 0.0130.2 0.1200.3 0.3400.4 0.5630.6 0.5630.7 0.3400.8 0.1200.9 0.013

Note: Compare these values with the otiginal definition of the Acceptance Region. They have decreased,

p 0.1 0.0130.2 0.1200.3 0.3400.4 0.5630.6 0.5630.7 0.3400.8 0.1200.9 0.013

p 0.1 0.2640.2 0.6240.3 0.8510.4 0.9520.6 0.9520.7 0.8510.8 0.6240.9 0.264

p 0.1 0.0700.2 0.3220.3 0.6160.4 0.8200.6 0.8200.7 0.6160.8 0.3220.9 0.070

Acceptance Region

{4,5,6}.

Acceptance Region

{3,4,5,6,7}.

Acceptance Region

{2,3,4,5,6,7,8}.

= 0.344 = 0.109 = 0.021

Hypothesis Testing

Definition• The two hypotheses of interest.

– The null hypothesis (H0)


• The decision to be made– Accept the null hypothesis (H0)

– Reject the null hypothesis (H0)

To define a statistical Test we

1. Choose a statistic (called the test statistic)

2. Divide the range of possible values for the test statistic into two parts



The Approach in Statistical Testing is:

• Set up the Acceptance Region so that is close to some predetermine value (the usual values are 0.05 or 0.01)

• The predetermine value of (0.05 or 0.01) is called the significance level of the test.

• The significance level of the test is = P[test makes a type I error]

The two types of error

Definitions:

For any statistical testing procedure define

1 = P[Rejecting the null hypothesis when it is true] = P[ type I error]

= P[accepting the null hypothesis when it is false] = P[ type II error]

Determining the Critical Region

1. The Critical Region should consist of values of the test statistic that indicate that HA is true. (hence H0 should be rejected).

2. The size of the Critical Region is determined so that the probability of making a type I error, , is at some pre-determined level. (usually 0.05 or 0.01). This value is called the significance level of the test.

Significance level = P[test makes type I error]

To find the Critical Region

1. Find the sampling distribution of the test statistic when is H0 true.

2. Locate the Critical Region in the tails (either left or right or both) of the sampling distribution of the test statistic when is H0 true.

Whether you locate the critical region in the left tail or right tail or both tails depends on which values indicate HA is true.

The tails chosen = values indicating HA.

3. the size of the Critical Region is chosen so that the area over the critical region and under the sampling distribution of the test statistic when is H0 true is the desired level of =P[type I error]

Sampling distribution of test statistic when H0

is true

Critical Region - Area =

The z-test for Proportions

Testing the probability of success in a binomial experiment

Situation

• A success-failure experiment has been repeated n times

• The probability of success p is unknown. We want to test – H0: p = p0 (some specified value of p)

Against

– HA:0pp

The Data

• The success-failure experiment has been repeated n times

• The number of successes x is observed.

• Obviously if this proportion is close to p0 the Null Hypothesis should be accepted otherwise the null Hypothesis should be rejected.

successes ofpoportion theˆ n

xp

The Test Statistic• To decide to accept or reject the Null Hypothesis

(H0) we will use the test statistic

n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ

• If H0 is true we should expect the test statistic z to be close to zero.

• If H0 is true we should expect the test statistic z to have a standard normal distribution.

• If HA is true we should expect the test statistic z to be different from zero.

The sampling distribution of z when H0 is true:

The Standard Normal distribution

0 z

Accept H0Reject H0 Reject H0

The Acceptance region:

1 when trueAccept 2/2/0 zzzPHP

/2

0 z

/2

2/z 2/z

Accept H0

Reject H0 Reject H0

2/2/0 or when trueReject zzzzPHP

• Acceptance Region– Accept H0 if:

• Critical Region– Reject H0 if:

2/2/ zzz

2/2/ or zzzz

when trueReject Error I Type 0HPP

• With this Choice

2/2/ or zzzzP

Summary

To Test for a binomial probability p

H0: p = p0 (some specified value of p)Against

HA:

we

0pp

1. Decide on = P[Type I Error] = the significance level of the test (usual choices 0.05 or 0.01)

2. Collect the data

3. Compute the test statistic

n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ

4. Make the Decision• Accept H0 if:

• Reject H0 if:

2/2/ zzz

2/2/ or zzzz

Example

• In the last provincial election the proportion of the voters who voted for the Liberal party was 0.08 (8 %)

• The party is interested in determining if that percentage has changed

• A sample of n = 800 voters are surveyed

We want to test

– H0: p = 0.08 (8%)

Against

– HA: %)8( 08.0p

Summary

1. Decide on = P[Type I Error] = the significance level of the test

Choose ( = 0.05)

2. Collect the data

• The number in the sample that support the liberal party is x = 92

(11.5%) 115.0800

92ˆ

n

xp


n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ


• Reject H0 if:

960.1960.1 z

960.1or 960.1 zz

649.3

80080.0180.0

80.0115.0

960.1025.02/ zz

Since the test statistic is in the Critical region we decide to Reject H0

Conclude that H0: p = 0.08 (8%) is false

There is a significant difference ( = 5%) in the proportion of the voters supporting the liberal party in this election than in the last election

The two-tailed z-test for Proportions


Situation



Against

– HA:0pp



n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ



2/2/ zzz

2/2/ or zzzz



2/2/ or zzzzP



/2

0 z

/2

2/z 2/z

Accept H0

Reject H0 Reject H0


The one tailed z-test


• The probability of success p is unknown. We want to test – H0: (some specified value of p)Against– HA:

• The alternative hypothesis is in this case called a one-sided alternative

0pp

0pp



n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ

• If H0 is true we should expect the test statistic z to be close to zero or negative

• If p = p0 we should expect the test statistic z to have a standard normal distribution.

• If HA is true we should expect the test statistic z to be a positive number.

The sampling distribution of z when p = p0 :


0 z

Accept H0Reject H0

The Acceptance and Critical region:

1 when trueAccept 0 zzPHP

0 zz

Accept H0

Reject H0

zzPHP when trueReject 0



zz

zz


• The Critical Region is called one-tailed


zzP

Example• A new surgical procedure is developed for

correcting heart defects infants before the age of one month.

• Previously the procedure was used on infants that were older than one month and the success rate was 91%

• A study is conducted to determine if the success rate of the new procedure is greater than 91% (n = 200)

We want to test

– H0:

Against

– HA: %)91( 91.0p

%)91( 91.0p

procedure new theof rate success thep

Summary

1. Decide on = P[Type I Error] = the significance level of the test

Choose ( = 0.05)

2. Collect the data

• The number of successful operations in the sample of 200 cases is x = 187

(93.5%) 935.0200

187ˆ

n

xp


n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ


• Reject H0 if:

645.1z

645.1z

235.1

20091.0191.0

91.0935.0

645.105.0 zz

Since the test statistic is in the Acceptance region we decide to Accept H0

There is a no significant ( = 5%) increase in the success rate of the new procedure over the older procedure

Conclude that H0: is true

More precisely H0 can’t be rejected

%)91( 91.0p

Comments

• When the decision is made to accept H0 is made one should not conclude that we have proven H0.

• This is because when setting up the test we have not controlled = P[type II error] = P[accepting H0 when H0 is FALSE]

• Whenever H0 is accepted there is a possibility that a type II error has been made.

In the last example

The conclusion that there is a no significant ( = 5%) increase in the success rate of the new procedure over the older procedure should be interpreted:

We have been unable to proof that the new procedure is better than the old procedure

The two-tailed z-test for Proportions


Situation



Against

– HA: 0pp



n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ



2/2/ zzz

2/2/ or zzzz



2/2/ or zzzzP

The one tailed z-test



• The alternative hypothesis is in this case called a one-sided alternative

0pp

0pp



n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ

• If H0 is true we should expect the test statistic z to be close to zero or negative


• If HA is true we should expect the test statistic z to be a positive number.



zz

zz




zzP


1 when trueAccept 0 zzPHP

0 zz

Accept H0

Reject H0

zzPHP when trueReject 0

The one tailed z-test (lower tail)



• The alternative hypothesis is in this case also one-sided alternative

0p p

0p p



n

pp

ppppz

p 00

0

ˆ

0

1

ˆ

ˆ

• If H0 is true we should expect the test statistic z to be close to zero or positive


• If HA is true we should expect the test statistic z to be a negative number.



z z

z z




P z z


0 zzAccept H0

Reject H0

Some comments:

When does one use a two-tailed test?

When does one use a one tailed test?

Answer: This depends on the alternative hypothesis HA.

Critical Region = values that indicate HA

Thus if only the upper tail indicates HA, the test is one tailed.

If both tails indicate HA, the test is two tailed.

Also:

The alternative hypothesis HA usually corresponds to the research hypothesis (the hypothesis that the researcher is trying to prove)

1. The new procedure is better

2. The drug is effective in reducing levels of cholesterol.

3. There has a change in political opinion from the time the survey was taken till the present time (time of current survey).

The z-test for the Mean of a Normal Population

We want to test, , denote the mean of a normal population

Situation

• A sample of n observations are collected from a Normal distribution

• The mean of the Normal distribution, , is unknown. We want to test

– H0: = 0 (some specified value of )

Against

– HA: 0

The Data

• Let x1, x2, x3 , … , xn denote a sample from a normal population with mean and standard deviation .

• Let

• we want to test if the mean, , is equal to some given value 0.

• Obviously if the sample mean is close to 0 the Null Hypothesis should be accepted otherwise the null Hypothesis should be rejected.

mean sample the1

n

xx

n

ii



s

xn

xn

n

xxz

x

0000

• If H0 is true we should expect the test statistic z to be close to zero.

• If H0 is true we should expect the test statistic z to have a standard normal distribution.

• If HA is true we should expect the test statistic z to be different from zero.

The sampling distribution of z when H0 is true:


0 z

Accept H0Reject H0 Reject H0



/2

0 z

/2

2/z 2/z

Accept H0

Reject H0 Reject H0




2/2/ zzz

2/2/ or zzzz



2/2/ or zzzzP

Summary

To Test for mean , of a normal population

H0: = 0 (some specified value of )

Against

HA: 0

1. Decide on = P[Type I Error] = the significance level of the test (usual choices 0.05 or 0.01)

2. Collect the data



• Reject H0 if:

2/2/ zzz

2/2/ or zzzz

s

xn

xnz 00

Example

A manufacturer Glucosamine capsules claims that each capsule contains on the average:

• 500 mg of glucosamine

To test this claim n = 40 capsules were selected and amount of glucosamine (X) measured in each capsule.

Summary statistics:

496.3 and 8.5x s

We want to test:

Manufacturers claim is correct

against

0 :H

:AH Manufacturers claim is not correct

The Test Statistic

s

xn

xn

n

xxz

x

0000

496.3 500 40

8.52.75

The Critical Region and Acceptance Region

Using = 0.05

We accept H0 if-1.960 ≤ z ≤ 1.960

z/2 = z0.025 = 1.960

reject H0 ifz < -1.960 or z > 1.960

The Decision

Sincez= -2.75 < -1.960

We reject H0

Conclude: the manufacturers’s claim is incorrect:

Hypothesis Testing

A review of the concepts

In hypotheses testing there are two hypotheses

1.The Null Hypothesis (H0)

2.The Alternative Hypothesis (HA)

• The alternative hypothesis is usually the research hypothesis - the hypothesis that the researcher is trying to prove.

• The null hypothesis is the hypothesis that the research hypothesis is not true.

A statistical Test is defined by

1. Choosing a statistic (called the test statistic)

2. Dividing the range of possible values for the test statistic into two parts



To perform a statistical Test we

1. Collect the data.

2. Compute the value of the test statistic.

3. Make the Decision:

• If the value of the test statistic is in the Acceptance Region we decide to accept H0 .

• If the value of the test statistic is in the Critical Region we decide to reject H0 .

The z-testsTesting the probability of success

0 0

ˆ 0 0

ˆ ˆ

1p

p p p pz

p p

n

Testing the mean of a Normal Population

s

xn

xn

n

xxz

x

0000

The Alternative Hypothesis HA

The Critical Region

0:AH p p

0:AH p p

0:AH p p

/ 2 / 2 or z z z z

z z

z z

Critical Regions for testing the probability of success, p

The Alternative Hypothesis HA

The Critical Region

0: AH

0: AH

0: AH

/ 2 / 2 or z z z z

z z

z z

Critical Regions for testing mean, of a normal population

ExampleNormal body temperature has a mean of

0 = 37.0 degreesA researcher is interested in the question – “Does average normal body temperature increase after heavy excercise?”

A sample of n = 50 subjects were asked to perform heavy excercise after which body temperatures were measured.

Summary statistics:

38.1 and 0.95x s

We want to test:

Average body temperature is normal or even lower than normal after heavy exercise.against

0 : 37.0H

: 37.0AH Average body temperature is higher than normal after heavy exercise.

The Test Statistic

s

xn

xn

n

xxz

x

0000

38.1 37.0 50

0.958.19

The Critical Region and Acceptance Region

Using = 0.05

We accept H0 ifz ≤ 1.645

z = z0.05 = 1.645

reject H0 ifz > 1.645

Comment: A one-tailed test is used. Reason: only the positive tail indicates HA.

The Decision

Sincez= 8.19 > 1.645

We reject H0

Conclude: There is a significant increase in body temperature after heavy excercise.

• You can compare a statistical test to a meter

Value of test statistic

Acceptance Region

Critical

Region

Critical

Region

Critical Region is the red zone of the meter


Acceptance Region

Critical

Region

Critical

Region

Accept H0


Acceptance Region

Critical

Region

Critical

Region

Reject H0

Acceptance Region

Critical

Region

Sometimes the critical region is located on one side. These tests are called one tailed tests.

Whether you use a one tailed test or a two tailed test depends on:

1. The hypotheses being tested (H0 and HA).

2. The test statistic.

If only large positive values of the test statistic indicate HA then the critical region should be located in the positive tail. (1 tailed test)

If only large negative values of the test statistic indicate HA then the critical region should be located in the negative tail. (1 tailed test)

If both large positive and large negative values of the test statistic indicate HA then the critical region should be located both the positive and negative tail. (2 tailed test)

Usually 1 tailed tests are appropriate if HA is one-sided.

Two tailed tests are appropriate if HA is two -sided.

But not always

Once the test statistic is determined, to set up the critical region we have to find the sampling distribution of the test statistic when H0 is true

This describes the behaviour of the test statistic when H0 is true

We then locate the critical region in the tails of the sampling distribution of the test statistic when H0 is true

The size of the critical region is chosen so that the area over the critical region is .

/2 /2

This ensures that the P[type I error] = P[rejecting H0 when true] =

/2 /2

To find P[type II error] = P[ accepting H0 when false] = we need to find the sampling distribution of the test statistic when H0 is false

/2 /2

sampling distribution of the test statistic when H0 is false

sampling distribution of the test statistic when H0 is true

The p-value approach to Hypothesis Testing

1. A test statistic

2. A Critical and Acceptance region for the test statistic

In hypothesis testing we need

The Critical Region is set up under the sampling distribution of the test statistic.

Area = (0.05 or 0.01) above the critical region. The critical region may be one tailed or two tailed

The Critical region:


/2

0 z

/2

2/z 2/z

Accept H0

Reject H0 Reject H0


1. Computing the value of the test statistic

2. Making the decisiona. Reject if the value is in the Critical

region and b. Accept if the value is in the

Acceptance region.

The test is carried out by

The value of the test statistic may be in the Acceptance region but close to being in the Critical region, or

It may be in the Critical region but close to being in the Acceptance region.

To measure this we compute the p-value.

Definition – Once the test statistic has been computed form the data the p-value is defined to be:

p-value = P[the test statistic is as or more extreme than the observed value of the test statistic]

more extreme means giving stronger evidence to rejecting H0

Example – Suppose we are using the z –test for the mean of a normal population and = 0.05.

Z0.025 = 1.960


= P [ z > 2.3] + P[z < -2.3]

= 0.0107 + 0.0107 = 0.0214

Thus the critical region is to reject H0 if

Z < -1.960 or Z > 1.960 .

Suppose the z = 2.3, then we reject H0

p - value

2.3-2.3

Graph


= P [ z > 1.2] + P[z < -1.2]

= 0.1151 + 0.1151 = 0.2302

If the value of z = 1.2, then we accept H0

23.02% chance that the test statistic is as or more extreme than 1.2. Fairly high, hence 1.2 is not very extreme

p - value

1.2-1.2

Graph

Properties of the p -value

1. If the p-value is small (<0.05 or 0.01) H0 should be rejected.

2. The p-value measures the plausibility of H0.

3. If the test is two tailed the p-value should be two tailed.

4. If the test is one tailed the p-value should be one tailed.

5. It is customary to report p-values when reporting the results. This gives the reader some idea of the strength of the evidence for rejecting H0

Summary

• A common way to report statistical tests is to compute the p-value.

• If the p-value is small ( < 0.05 or < 0.01) then H0 is rejected.

• If the p-value is extremely small this gives a strong indication that HA is true.

• If the p-value is marginally above the threshold 0.05 then we cannot reject H0 but there would be a suspicion that H0 is false.

Next topic: Student’s t - test

Statistical Inference Making decisions regarding the population base on a sample.

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Transcript of Statistical Inference Making decisions regarding the population base on a sample.