Lecture 10-Hypothesis Testing Flashcards
Statistical decisions
Very often in practice, we are called upon to make decisions about populations on the basis of sample information
*Such decisions are Statistical Decisions.
*In attempting to reach such decisions it may become useful or necessary to make assumptions
*These assumptions are Statistical Hypotheses.
*It seems logical, therefore, that a framework exists within which we can TEST such hypotheses
Example of decision errors
You are a juror in a court case
*You must begin with the assumption that the defendant is innocent until proven guilty
*Your first position is that of innocence. Evidence is presented to you. Based on that evidence, you decide between two options: either remain at your original position (innocence) or move to a different position (guilty)
You will decide guilt only if you are convinced, beyond a critical minimum (beyond reasonable doubt) to move beyond your stance of innocence
*Sometimes (unfortunately) the courts get it wrong. What are the types of errors that can be made? (1) an innocent man is found guilty
(2) a guilty man is found innocent
Found Guilty, but really Innocent:
–A Type I Error is the error that results from rejecting the null hypothesis when indeed the null hypothesis is true.
*Found Not Guilty, but really Guilty:
–A Type II Error is the error that results from not rejecting the Null Hypothesis when indeed the null hypothesis is no longer true (i.e. when the alternative Hypothesis is true).
Significance levels of a hypothesis
is the maximum probability with which we would be willing to commit a Type I error
*This probability, often denoted by ‘alpha’ is generally specified before samples are drawn, so the results obtained don’t influence the researcher’s choice.
*If the significance level is 5%, then there are about 5 chances in 100 that we would reject the hypothesis, when it should be accepted.
*In other words, we are about 95% confident that we have made the right decision!
Steps of hypothesis testing
*(1) you have an original position
*(2) you have an alternative position
*(3) you have some critical region that defines your movement from one position to the next
*(4) you are presented evidence
*(5) you compare that evidence with your critical region
*(6) you come to a conclusion
The Original Question : The Null Hypothesis
The null hypothesis is denoted by Ho
*This is the hypothesis being tested (in our courtroom example, the assumption of innocence)
*It is always phrased in neutral, non-confrontational language that speaks to “no change” and “no difference”
*Hence the only sign used in a null hypothesis is the “=” sign
*must be rejected or not rejected based on the evidence presented.
The Alternative Position : The Alternative Hypothesis
The alternative hypothesis is denoted by H1 or Ha
*This is the alternative to the null hypothesis when H0 is rejected (in our courtroom example – guilty)
*This is phrased so as to include language such as “different from”, “greater than” or ‘less than”.
*There are therefore three choices for an alternative hypothesis:
*Less than “<”
*Greater than “>”
*Not equal to “≠”
One tailed and Two Tailed Hypothesis Tests
There are therefore three choices for H1 :
–Less than “<”
–Greater than “>”
–Not equal to “≠”
*If H1 is “not equal to”, we have what is called a “two-tailed test”
*If H1 is either “less than” or “greater than”, we have what is called a “one-tailed test”
*If H1 is “less than”, we have a “left-tailed test”
*If H1 is “greater than”, we have a “right-tailed test
The Critical Region
defines the conditions under which the Null Hypothesis will be rejected.
*It is based on three things:
–The distribution of the underlying population
–The form of the alternative hypothesis: “less than”, “greater than” or “not equal to”
–The chosen significance level
The Form of the Alternative Hypothesis
A two-tailed test has rejection
regions in both tails
* A Left-tailed test has the rejection
region in the left tail
* A Right-tailed test has the rejection
region in the right tail
* Note the presence of versus /2 in terms of the
definition of the critical region!
The chosen significance level a
The significance level is the probability of making a Type 1 Error (mistakingly rejecting the Null when the Null is true).
*Is usually given. If not given, a popular choice is 5%
Calculating the critical region
Assuming normality, how can we calculate the critical regions for one-tailed and two-tailed tests?
*For a left-tailed test, the CR is defined as Z< -Z
*For a right-tailed test, the CR is defined as Z > Z
*For a two-tailed test, the CR is defined as Z< -Z/2 and Z > Z/2
The Evidence : The Test Statistic
The Test Statistic is what is used as our evidence. We calculate the Test Statistic, and compare it to the Critical Region previously discussed.
*We can distinguish among the following sets of Test Statistics:
–Testing Means
–Testing Proportions
–Testing Differences Between Means
–Testing Differences Between Proportions
Testing Means
X bar is the value of the sample mean
u is the value of the population mean under the ( side a sign ) null hypothesis is the population standard deviation
n is the sample size
Testing Proportions
p hat is the sample proportion
p is the population proportion under the null
q is 1-p
n is the sample size
The Decision Rule/Criteria
The decision criteria or decision rule is the logic by which a decision is made in a test of hypothesis
*Golden Rule: If the Test Statistic Falls Within the Critical Region, you REJECT the Null Hypothesis
*Clearly, if the Test Statistic falls outside of the Critical Region, you DO NOT REJECT the Null Hypothesis
