Statistical Inference

Hypothesis Testing

• What is a statistical hypothesis? • What is so important about it? • What is a rejection region? • What does it mean to say that a finding is

statistically significant? • Describe Type I and Type II errors.

Hypothesis Testing• Task: Statement about unknown vales of population

parameters in terms of sample information• Elements of a hypothesis test:

– Null hypothesis (H0 ) - Statement on the value(s) of unknown parameter(s);

– Alternative hypothesis - Statement contradictory to the null hypothesis

– Test statistic – Estimate based on sample information and null hypothesis used to test between null and alternative hypotheses

– Rejection (Critical) region – Range of value on the test statistic for which we reject the null in favor of the alternative hypothesis

Hypothesis Testing True State

Decision

H0 True HA True

Do not reject Null Correct Decision

Type II ErrorP(Type II)=

Reject Null Type I Errorp(Type I)=

Correct DecisionPower=1-

Critical Value

• Critical value: Value that separates the critical (rejection) region from the values of the test statistic that do not lead to rejection of the null hypothesis, given the sampling distribution and the significance level .

Significance Level

• The significance level (): Probability of the test statistic falling in the critical region under a valid null hypothesis.

• : Conventional Choices for are 0.05, 0.01, and 0.10.

• p-value: Probability of observing the test statistic under the null hypothesis

Significance Level and Power

• Level of significance: Probability that the test rejects the null hypothesis on the assumption that the null hypothesis is true.

• Power of a test: Probability that that the test rejects the null hypothesis on the assumption that that alternative hypothesis is true.

Test of Hypothesis: Interpretations

• Rejecting the null hypothesis • Do not reject H0 : Does not mean that H0 is true;

nor that the data supports H0.If the observations are few, the test would not have the power, that is, difficult for a test to reject a false null H0.

• Reject H0 does not mean that HA is true. It means that either H0 is false or the event has probability no larger than the significance level.

Statistical Significance vs. Practical Importance

• An effect may be of importance but not statistically significant because of sample limitations (poor quality or few observations).

• The effect may not be of much policy significance in terms of impact but still statistically significant due to high quality of data.

Hypothesis Testing: Steps• State the maintained hypothesis.• State the null & alternative hypotheses.• Choose the test statistic and estimate its value.• Specify the sampling distribution of the test statistics

under the null hypothesis.• Determine the critical value(s) corresponding to a

significance level.• Determine the p-value for the test statistic. • State the conclusion of a hypothesis test in simple,

nontechnical terms.

Hypothesis Testing: Rationale

• We infer that the assumption is probably incorrect given the maintained and null hypotheses, if the probability of getting the sample is exceptionally small.

• Please note that the null hypothesis contains equality. Comparing the assumption and the sample results, we infer one of the following:

Hypothesis Testing: Rationale

• Under the null hypothesis, if the probability of observing the sample estimate is high, discrepancy between the assumption and the sample estimate, if any, is due to chance.

• If this probability is very low, even relatively large discrepancy between the assumption and the sample is due to invalid null hypothesis.

Test of Hypothesis: Population Means

• Assumptions:1) Simple random sample2) Population variance is known3) Population distribution is normal or sample

size is more than 30

Test of Hypothesis: Population MeansKnown population variance

• Test statistic

n

x – µxz =

Two Tail Test

One- and Two Tail TestsOne-Tail Test

(left tail)Two-Tail Test One-Tail Test

(right tail)

Test of Hypothesis: Population Means

• Assumptions:1) Simple random sample2) Population variance is unknown3) Population distribution is normal or sample

size is more than 30

Test of Hypothesis: Population MeansUnknown population variance

• Test statistic

t = x –µxsn

Student’s t-test: An Illustration

• Question: The diameter of some ball for study is specified to be one meter. A random sample of 10 such balls is selected to check the specification. The sample selected gave the following measurements:1.01, 1.01, 1.02, 1.00, 0.99, 0.99, 1.02, 1.02, 1.00, 1.02

• Is there any reason to believe that there has been a change in the average diameter?

Student’s t-test

• Level of significance = 0.05• Maintained hypothesis: Distribution of

diameters is normal• n = 10• H0 : m = 1.0• HA : m <> 1.0• Sample mean = 1.008

Student’s t-test

• Estimate of population variance = 0.000151• Std. deviation = 0.012288• t-statistic = 1.953125 (9 d.f.)• t(9,0.05) = 2.262• Computed t < tabulated t• Do not reject H0 • Conclusion: Sample information supports the

hypothesis that the average diameter of the ball is one meter.

What would be the sampling distribution of a sample mean from a normally distributed

population?

Sample mean from a normal population will also be normally distributed for any sample size n

Central Limit Theorem

• The sampling distribution of mean of n sample observations from any population would be approximately normal when n is sufficiently large.