1036: Probability & Statistics

Prob. & Stat. Lecture10 - one-/two-sample tests of hypotheses (cwliu@twins.ee.nctu.edu.tw)

1036: Probability & 1036: Probability & StatisticsStatistics

Lecture 10 Lecture 10 –– OneOne-- and Twoand Two--Sample Sample Tests of Hypotheses Tests of Hypotheses

Statistical Hypotheses• Decision based on experimental evidence

whether– Coffee drinking increases the risk of cancer in humans.– A person’s blood type or eye color are independent

variables.

• A statistical hypothesis is an assertion or conjecture concerning one or more populations.– True of False is never known with absolute certainty

unless the entire population is examined.– The decision procedure is done with the awareness of

the probability of a wrong conclusion

Role of Probability in Hypothesis Testing

• The acceptance of a hypothesis merely implies that the data do not give sufficient evidence to refute it.

• Rejection means that there is a small probability of obtaining the sample information observed when the hypothesis is true.

• Example: for the conjecture of the fraction defective p = 0.10, a sample of 100 revealing 20 defective items is certainly evidence of rejection– Since the probability of obtaining 20 defectives is

approximately 0.002• The firm conclusion is established by the data analyst when

a hypothesis is rejected

Supporting a Contention• To reject the hypothesis• Contention: coffee drinking increases the

risk of cancer⇒Hypothesis: there is no increase in cancer

risk produced by drinking coffee

• Contention: one kind of gauge is more accurate than another⇒Hypothesis: there is no difference in the

accuracy of the two kinds of gauges

Null and Alternative Hypotheses• Structure of hypothesis

– Null hypothesis, H0• any hypothesis we wish to test

– Alternative hypothesis, H1• the opposite hypothesis to reject H0

• Example– H0 is the null hypothesis p = 0.5 for a binomial

population,– H1 would be one of the following:

p > 0.5, p < 0.5, or p ≠ 0.5

Testing a Statistical Hypothesis• Rejection of the null hypothesis when it is true is called a

type I error (level of significance).• Acceptance of the null hypothesis when it is false is called a

type II error.

• The probability of committing a type I error is denoted by α• The probability of committing a type II error is denoted by β

Example: α and β

0051.0)7.0,20;(

)7.0 when 8()error II type(

2517.0),20;(

) when 8()error II type(

0409.09591.01),20;(1

),20;() when 8()error I type(

=≤==

=−=−=

xbpXPP

H0 : x ≤ 8; H1: x > 8 (p = 1/4, n =20, binomial)

Remarks• Critical value: the last number passing from the

acceptance region into the critical region• For a fixed sample size:

– A reduction in β is always possible by increasing the size of the critical region

– A decrease in the probability of one error usually results in an increase in the probability of the other error

• The probability of committing both types of error can be reduced by increasing the sample size

The Role of α, β, and Sample Size• To determine the probability of committing a type I

error, we shall use the normal-curve approximation with n > 30.

• Example: H0 : p = 1/4; H1: p > 1/4(n = 100, critical value = 36)

0035.0)7.2( ) when 36()error II type(

255.36,5100,50100

0039.09961.01)66.2(1 )66.2() when 36()error I type(

66.233.4

255.36,33.4100,25100

=−<≈

=≤==

−=−

==××===×==

=−=>−=

>≈=>==

==××===×==

ZPpXPP

znpqnp

ZPZPpXPP

znpqnp

Hypothesis Testing with a Continuous Random Variable

• Consider the null hypothesis that the average weight of male students in a certain college is 68 kilograms against the alternative hypothesis that it is unequal to 68.– H0 : µ = 68; H1: µ ≠ 68

0264.0)22.2(2)22.2()22.2(

22.245.0

6869 ,22.245.0

686745.08/6.364, tosize sample increase

0950.0)67.1(2)67.1()67.1(

67.16.06869,67.1

6.06867

)68 when 69()68 when 67()error I type(

6.06/6.3/,36,6.3

=−<=>+−<=

=−=−

=−<=>+−<=

=−=−

=>+=<==

ZPZPZP

nZPZPZP

µµα

σσσ

Hypothesis Testing with a Continuous Random Variable

8661.0)33.3()11.1()11.133.3(

11.145.0

5.6869 ,33.345.0

5.686768.5 hypothesis ealternativ theIf

0132.0)67.6(2)22.2()22.267.6(

22.245.0

7069,67.645.0

7067)70 when 6967()error II type(

6.06/6.3/,36,6.3

=−<−<=<<−=

=−=−

==−<−−<=

−<<−=

−=−

=−=−

=<≤==

ZPZPZP

σσσ

Properties of a Test Hypothesis• The type I error and type II error are related. A decrease

in the probability of one generally results in an increase in the probability of the other

• The size of the critical region, and therefore the probability of committing a type I error, can always be reduced by adjusting the critical value(s).

• An increase in the sample size n will reduce α and βsimultaneously

• If the null hypothesis is false, β is a maximum when the true value of a parameter approaches the hypothesized value. The greater the distance between the true value and the hypothesized value, the smallerβ will be

Power of a Test• The power of a test is the probability of

rejecting H0 given that a specific alternative is true

• The power of a test can be computed as 1 - β.– Previous example: the probability of a type II error is

given by β = 0.8661, thus the power of the test is 1 –0.8661 = 0.1339

– The power is a more succinct measure of how sensitive the test is for detecting differences between a mean of 68 and 68.5

• To produce a desirable power (greater than 0.8), one must either increase α or increase the sample size

One- and Two-Tailed Tests• The null hypothesis H0 is always stated using the equality

sign to specify a single value (easily controlled)• In a hypothesis, the alternative is one-sided, and is called a

one-tailed test.– H0 : θ = θ0; H1: θ > θ0– H0 : θ = θ0; H1: θ < θ0

• In a hypothesis, the alternative is two-sided, and is called a two-tailed test.– H0 : θ = θ0; H1: θ ≠ θ0

Right tail side

left tail side

How are H0 and H1 Chosen?• Example 10.1: A manufacturer of a certain brand of rice cereal

claims that the average saturated fat content does not exceed 1.5 grams. State the null and alternative hypotheses to be used in testing the claim and determine where the critical region is located.– The claim should be rejected only if µ is greater than 1.5– One-tailed test– H0 : µ = 1.5; H1: µ > 1.5

• Example 10.2: A real estate agent claims that 60% of all privateresidences being built today are 3-bedroom homes. State the null and alternative hypotheses to be used in testing the claim and determine the location of the critical region.– The higher or lower test statistic than 0.6 would reject the claim– Two-tailed test– H0 : p = 0.6; H1: p ≠ 0.6

Approach to Hypothesis Testing with Fixed α

1. State the null and alternative hypotheses2. Choose a fixed significance level α.3. Choose an appropriate test statistic and

establish the critical region based on α.4. From the computed test statistic, reject H0 if

the test statistic is in the critical region. Otherwise, do not reject.

5. Draw scientific or engineering conclusions.

P-Values for Decision Making• It had become customary to choose an α of 0.05

or 0.01 and select the critical region accordingly. (to control the type I error)

• However, this approach does not account for values of test statistics that are close to the critical region

• A P-value is the lowest level of significance at which the observed value of the test statistic is significant– no fixed α is determined – The conclusion is made on the basis of p-value in

harmony with the subject judgment of the engineer

P-value ApproachSignificant testing approach– State null and alternative hypotheses.– Choose an appropriate test statistic.– Compute P-value based on computed value of

test statistic. – Use judgment based on P-value and knowledge

of scientific system.

Tests Concerning a Single Mean (Variance Known)

:region Critical

( :region Acceptance

)/,( /

: ofation Standardiz

:,: :Hypothesis

σσµµσ

µµµµ

−<>−

−=<−

<−•

≠=•

Example 10.3• A random sample of 100 recorded deaths in the United

States during the past year showed an average life span of 71.8 years. Assuming a population standard deviation of 8.9 years, does this seem to indicate that the mean life span today is greater than 70 years? Use a 0.05 level of significance.

P = P(Z > 2.02)= 0.0217

Example 10.4• A manufacturer of sports equipment has developed a new synthetic

fishing line that he claims has a mean breaking strength of 8 kilograms with a standard deviation of 0.5 kilogram. Test the hypothesis that μ= 8 kilograms against the alternative that μ≠ 8 kilograms if a random sample of 50 lines is tested and found to have a mean breaking strength of 7.8 kilograms. Use a 0.01 level of significance.

P = P(|Z| > 2.83)= 2 P(Z < -2.83)= 0.0046

Relationship to Confidence Interval Estimation

• For the case of a single population with mean µ and variance σ2 known, both hypothesis testing and confidence interval estimation are based on the R.V.

• We have (1-α)×100% confidence interval on µ• The testing of H0: µ=µ0 against H0: µ≠µ0 at a significance

level α and rejecting H0 if µ0 is not inside the confidence interval

µ−=

σµ α 2/0 −=

nzb σµ α 2/0 +=

Choice of Sample Size• The sample size is usually made to achieve good power for a

fixed α and fixed specific alternative.• Suppose that we wish to test the hypothesis: H0: µ=µ0, H1:

µ>µ0 with a significance level α• For a specific alternative, say µ=µ0+δ, the power of the

test is ) when (1 0 δµµβ +=>=− aXP

⎟⎟⎠

⎞⎜⎜⎝

⎛−<=⎥

⎤⎢⎣

⎡ +−<

σδµ

δµµβ

α)()(

) when (

αβnzz −=−

We can choose the sample size

22)(δ

σβα zzn

1036: Probability & Statistics

Documents

PROBABILITY AND STATISTICS - Programs, Courses AIU ...courses.aiu.edu/STATISTICS/7/SESSION 7.pdf · SESSION 7 Probability and Statistics Probability Line Probability is the chance

Probability Theory Statistics

Statistics and Probability

Probability ans statistics

Summarizing Data. Statistics statistics probability probability vs. statistics sampling inference

Probability & Statistics: Intro, summary statistics

Conditional Probability and Probability Trees Statistics

QBM117 Business Statistics Probability Conditional Probability

∑ STATISTICS AND PROBABILITY

Probability and Statistics in Geologydsw/geomath_lect12_stats.pdf · Probability and Statistics in Geology Probability and statistics are an important aspect of Earth Science.

Dr. Héctor AllendeReview of Probability and Statistics 1 A Review of Probability and Statistics Descriptive statistics Probability Random variables Sampling

Probability & Statistics,

21998904 Probability Statistics

Probability Statistics

Probability and statistics

1 Probability and Statistics What is probability? What is statistics?

Statistics and Probability 13.1 Basic Statistics

Basic Probability Theory for Biomedical Engineers...Probability Theory, Random Processes, Engineering Statistics, Probability and Statistics for Biomedical Engineers, Statistics. P1:

Probability Statistics Advanced

Statistics & probability