Hypothesis Testing

 Hypothesis is a claim or statement about a property of a population.  Hypothesis Testing is to test the claim or statement  Example : A conjecture is made that “the average starting salary for computer science gradate is Rs 45,000 per month”. Hypothesis Testing

 Null Hypothesis (H 0 ): is the statement being tested in a test of hypothesis.  Alternative Hypothesis (H 1 ): is what is believe to be true if the null hypothesis is false.

Null Hypothesis  Must contain condition of equality =, , or   Test the Null Hypothesis directly  Reject H 0 or fail to reject H 0

Alternative Hypothesis  Must be true if H 0 is false  ,  ‘opposite’ of Null Hypothesis Example H 0 : µ = 30 versus H 1 : µ > 30 or H 1 : µ < 30

 State the Null Hypothesis (H 0 :   3)  State its opposite, the Alternative Hypothesis (H 1 :  < 3)  Hypotheses are mutually exclusive Identify the Problem

Population Assume the population mean age is 50. (Null Hypothesis) REJECT The Sample Mean Is 20 Sample Null Hypothesis Hypothesis Testing Process No, not likely!

Sample Mean  = 50 Sampling Distribution It is unlikely that we would get a sample mean of this value if in fact this were the population mean.... Therefore, we reject the null hypothesis that  = H0H0 Reason for Rejecting H 0

Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True  Called Rejection Region of Sampling Distribution Designated  (alpha)  Typical values are 0.01, 0.05, 0.10 Selected by the Researcher at the Start Provides the Critical Value(s) of the Test Level of Significance, 

Level of Significance  and the Rejection Region H 0 :   3 H 1 :  < H 0 :   3 H 1 :  > 3 H 0 :   3 H 1 :   3    /2 Critical Value(s) Rejection Regions

 Type I Error  Reject True Null Hypothesis  Has Serious Consequences  Probability of Type I Error is  Called Level of Significance  Type II Error  Do Not Reject False Null Hypothesis  Probability of Type II Error Is  (Beta) Errors in Making Decisions

H 0 : Innocent Jury Trial Hypothesis Test Actual Situation Verdict InnocentGuilty Decision H 0 TrueH 0 False Innocent CorrectError Do Not Reject H  Type II Error (  ) Guilty Error Correct Reject H 0 Type I Error (  ) (1 -  ) Result Possibilities

Type I Error  The mistake of rejecting the null hypothesis when it is true.  The probability of doing this is called the significance level, denoted by   (alpha).  Common choices for  : 0.05 and 0.01  Example: rejecting a perfectly good parachute and refusing to jump

Type II Error  The mistake of failing to reject the null hypothesis when it is false.  Denoted by ß (beta)  Example:Failing to reject a defective parachute and jumping out of a plane with it.

Critical Region  Set of all values of the test statistic that would cause a rejection of the null hypothesis. Critical Region

Critical Region  Set of all values of the test statistic that would cause a rejection of the null hypothesis. Critical Region

Critical Region  Set of all values of the test statistic that would cause a rejection of the null hypothesis. Critical Regions

Critical Value Value (s) that separates the critical region from the values that would not lead to a rejection of H 0. Critical Value ( z score ) Fail to reject H 0 Reject H 0

Original claim is H 0 Conclusions in Hypothesis Tests Do you reject H 0 ?. Yes (Reject H 0 ) “There is sufficient evidence to warrant rejection of the claim that... (original claim).” “There is not sufficient evidence to warrant rejection of the claim that... (original claim).” “The sample data supports the claim that... (original claim).” “There is not sufficient evidence to support the claim that... (original claim).” Do you reject H 0 ? Yes (Reject H 0 ) No (Fail to reject H 0 ) No (Fail to reject H 0 ) (This is the only case in which the original claim is rejected). (This is the only case in which the original claim is supported). Original claim is H 1

Left-tailed Test H 0 : µ  200 H 1 : µ < Values that differ significantly from 200 Fail to reject H 0 Reject H 0 Points Left

Right-tailed Test H 0 : µ  200 H 1 : µ > 200 Values that differ significantly from Fail to reject H 0 Reject H 0 Points Right

Two-tailed Test H 0 : µ = 200 H 1 : µ  200 Means less than or greater than Fail to reject H 0 Reject H Values that differ significantly from 200  is divided equally between the two tails of the critical region

Definition Test Statistic: is a sample statistic or value based on sample data Example: z =z = x – µxx – µx   n

Question: How can we justify/test this conjecture? A. What do we need to know to justify this conjecture? B. Based on what we know, how should we justify this conjecture?

Answer to A: Randomly select, say 100, computer science graduates and find out their annual salaries ---- We need to have some sample observations, i.e., a sample set!

Answer to B: That is what we will learn in this chapter ---- Make conclusions based on the sample observations

Statistical Reasoning Analyze the sample set in an attempt to distinguish between results that can easily occur and results that are highly unlikely.

Figure 7-1 Central Limit Theorem:

Distribution of Sample Means µ x = 30k Likely sample means Assume the conjecture is true!

Figure 7-1 Central Limit Theorem: Distribution of Sample Means z = –1.96 x = 20.2k or z = 1.96 x = 39.8k or µ x = 30k Likely sample means Assume the conjecture is true!

Figure 7-1 Central Limit Theorem: Distribution of Sample Means z = –1.96 x = 20.2k or z = 1.96 x = 39.8k or Sample data: z = 2.62 x = 43.1k or µ x = 30k Likely sample means Assume the conjecture is true!

COMPONENTS OF A FORMAL HYPOTHESIS TEST

TWO-TAILED, LEFT-TAILED, RIGHT-TAILED TESTS

Problem 1 Test µ = 0 against µ > 0, assuming normally and using the sample [multiples of 0.01 radians in some revolution of a satellite] 1, -1, 1, 3, -8, 6, 0 (deviations of azimuth) Choose α = 5%.

Problem 2 In one of his classical experiments Buffon obtained 2048 heads in tossing a coin 4000 times. Was the coin fair?

Problem 3 In one of his classical experiments K Pearson obtained 6019 heads in trials. Was the coin fair?

Problem 5 Assuming normality and known variance б 2 = 4, test the hypotheses µ = 30 using a sample of size 4 with mean X = 28.5 and choosing α = 5%.

Problem 7 Assuming normality and known variance б 2 = 4, test the hypotheses µ = 30 using a sample of size 10 with mean X = What is the rejection region in case of a two sided test with α = 5%.

Problem 9 A firm sells oil in cans containing 1000 g oil per can and is interested to know whether the mean weight differs significantly from 1000 g at the 5% level, in which case the filling machine has to be adjusted. Set up a hypotheses and an alternative and perform the test, assuming normality and using a sample of 20 fillings with mean 996 g and Standard Deviation 5g.

Problem 11 If simultaneous measurements of electric voltage by two different types of voltmeter yield the differences (in volts) 0.8, 0.2, -0.3, 0.1, 0.0, 0.5, 0.7 and 0.2 Can we assert at the 5% level that there is no significant difference in the calibration of the two types of instruments? Assume normality.