Testing Hypothesis Nutan S. Mishra Department of Mathematics and Statistics University of South Alabama

Description of the problem The population parameter(s) is unknown. Some one (say person A) has some claim about the value of this unknown parameter. Another person (say person B) wants to test how valid is this claim. The person B collects a sample and gathers sample data And proceeds to test the claim.

Example Manufacturer of certain cereal claims that his boxes contain 16oz on the average. He does not know the true average. He could claim this because he has set up the filling machines to pour 16oz of material. A consumer protection agency received multiple complaints over a period of time that this brand of boxes contain less amount then claimed. So the consumer protection agency wants to test the claim of the manufacturer. To start with consumer protection agency thinks that on the average boxes contain less than 16oz.

Example 1 So there are two parties who claim differently about the population parameter. The Manufacturer says average = 16oz This a prevailing claim in the market. The Consumer Protection Agency says the average is less than 16 oz. This is the doubt raised by the consumer so its consumer protection agency’s responsibility to prove the claim. The consumer protection agency proceeds to test the validity of the claims.

Notations and definitions A claim about the population parameter is called Statistical hypothesis Example1: µ = 16oz Example2: µ< 16oz The prevailing belief in the market is that the box contains 16 oz cereal. The claim about the population parameter which is a prevailing belief is called Null Hypothesis Example: Ho : µ = 16oz On the other hand a claim made by another agency is called Alternative hypothesis Example: H1: µ< 16oz Some times the other claims are also made by a researcher in a given problem. Thus alternative hypothesis is also known as researcher’s hypothesis.

Example 2 Consider the claim of an ambulance company that on the average their vehicle reaches on site within 10 minutes. Whereas the consumer protection agency received the complaint that they take longer time. X = time taken by an ambulance to get there H0: µ= 10 vs H1: µ > 10 minutes

Example 3 A company manufactures and supplies the hexanuts with average inner diameter as 7.00mm. The customer wants to test if the hexanuts supplied are according to specification or not. H0: µ= 7.00mm H1: µ ≠ 7.00 mm

Basic philosophy Burden of the proof lies with the agency/person who raises the doubts against a prevailing belief. Example: The consumer protection agency needs to provide enough evidences against the prevailing belief or Null hypothesis H0: µ = 16oz In order to collect the evidences, the consumer protection agency/ researcher collects a sample. And uses the information contained in the sample to disprove the Null hypothesis. Note that Null hypothesis is the target. So he starts with assumption that Null hypothesis is true. The procedure of testing of hypothesis is developed under the assumption that Null hypothesis is true.

Definitions A null hypothesis is a claim about the population parameter that is assumed to be true until it is declared to be false. An alternative hypothesis is a claim about population parameter that will be true if null hypothesis is false.

Method for population mean Step 1 : define null and alternative hypotheses. Step2 :Collect a sample of size n Step3 :Compute the sample mean. Step4: compare the sample mean with the critical value Recall that if we collect different samples, value of the sample mean will be different. And there is always a difference between sample mean and population mean This difference occurs due to: –Sampling errors (chance errors which are inherent) –Non sampling errors ( due to some assignable cause) If the difference is too big, its easy to make a decision on H0. When the difference is small we need to analyze it carefully. We want to find out whether the difference in sample mean and claimed value of population mean has occurred just due to chance or is there any systematic cause behind this difference?

Method for population mean Step 1 : define null and alternative hypotheses. Step2 :Collect a sample of size n Step3 :Compute the sample mean. Step4: compare the sample mean with the critical value Question: How to find the critical value? To find the critical value we need to take into account two types of errors which may incur in our decision.

Two types of errors In testing of hypothesis we make a decision on the basis of sample evidence (that is sample data). Thus we may commit two types of errors in our decision Actual situation H0 is trueH0 is false Our Decision Reject H0Type I error Correct decision Do not reject H0 Correct decision Type II error

Two types of errors Type I error = (rejecting H0| H0 is true) Type II error = (not rejecting H0| H0 is false) These two types of errors are measured in terms of probability α = P(committing type I error) α = P(rejecting H0| H0 is true) α is called level of significance of a test β = P(committing type II error) β = P(not rejecting H0| H0 is false) 1- β is called power of a test In our decision process we want to minimize these two types of errors These two errors can not be minimized simultenuously.

Examples H0:µ = 16 vs H1: µ<16Actual situation H0 is trueH0 is false Our Decision Reject H0Type I errorCorrect decision Do not reject H0Correct decisionType II error Type I error = declaring the product faulty when in fact the manufacturer is producing the boxes with average content 16oz Type II error = manufacture is producing a faulty product but because of lack of enough evidence the product is declared OK.

Examples H0 : person is innocent H1: Person is guilty Actual situation Person is innocentPerson is guilty Our Decision Person is guiltyType I errorCorrect decision Person is not guiltyCorrect decisionType II error Type I error: Declaring an innocent person guilty (based on evidences available) Type II error : the person has committed the crime but due to lack of evidences is declared not guilty.

Critical value How much evidence is enough to declare a person guilty? How small should be the sample mean in order to reject the manufacturer’s claim µ=16oz? A fixed value c such that all the values of sample means below c means reject null. C defines the rejection region and non rejection region Then such a c is called critical value In our first example what should be the value of c? or or 15.97?

Three types of rejection regions In example 1 H0: µ = 16 oz vs H1: µ< 16oz If the value of sample mean lies on the left of critical value we will reject H0. This is called left sided rejection region. And the alternative hypothesis is called left sided and the test is called left tailed test In example 2 H0: µ= 10 vs H1: µ > 10 minutes If the value of sample mean lies on the right of the critical value we will reject H0. This is called right sided critical region. And the alternative hypothesis is called right sided. The test is called right-tailed test. In example 3 H0: µ= 7.00mm vs H1: µ ≠ 7.00 mm In this case there are two critical values c1 and c2 16 Critical value 16 10Critical value7Critical value c1 Critical value c2

Tails of a test When the alternative hypothesis is of the type µ ≠ µ0 then the test is called two tailed test because the rejection region lies at the left tail as well as right tail of the distribution of the mean. When alternative hypothesis is of the type µ > µ0, then the test is right tailed test because the rejection region lies on the right side of the critical value, that is on the right tail of the curve of sample mean. When alternative is of the type µ < µ0, then the test is left tailed test because the rejection region lies on the left side of the critical value, that is on the left tail of the curve of sample mean.

Large sample cases Recall that for large samples, distribution of is approximately normal with parameters ~ N( µ, σ/√n) and hence Z = ~ N(0,1) (this z-value we compute using the sample information) The idea is on the normal curve we can compare the value with the critical value(s) or the corresponding z-values on the z-curve

Rejection region of a left tailed test A test is left tailed if the alternative is of the form H1 : µ< µ 0 For example H 0 : µ= 16oz vs H 1 : µ< 160z Let α be the level of significance then the rejection region is shown as follows If the value of sample mean lies towards the left of c then reject the null If p-value < α reject null hypothesis

Rejection region of a right tailed test A test is right tailed if the alternative is of the form H1 : µ > µ 0 For example H 0 : µ= 10 minutes vs H 1 : µ >10 minutes Let α be the level of significance then the rejection region is shown as follows If the value of sample mean lies towards the right of c then reject the null. If p-value < α reject null hypothesis c

Rejection region of a two tailed test A test is two tailed if the alternative is of the form H1 : µ ≠ µ 0 For example H 0 : µ= 7 mm vs H 1 : µ ≠ 7 mm Let α be the level of significance then the rejection region is shown as follows If the value of sample mean lies towards the right of c 1 or left of c 2 then reject the null. If p-value < α reject null hypothesis c2c2 c1c1

About p-value We have seen α and p values are areas on the curve corresponding to critical value c and the computed value of sample mean To make a decision we can compare either the areas or the values. α < p-value is equivalent to c < Note that c and can be transformed in to corresponding z-values Thus instead of comparing areas on -curve we may compare the corresponding areas on the z- curve

Exercise 9.9 a.X = hours spent working per week by students H0: µ = 20 hrs vs H1: µ ≠ 20 hrs. b.X = #hours banks ATM was out of service/month H0: µ = 10 hrs vs H1: µ > 10 hrs. c.X= length of experience of security guard H0: µ = 3 years vs H1: µ ≠ 3 hrs. d.X= credit card debt of a college senior H0: µ = $1000 vs H1: µ < $1000