Chapter Ten McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. One-Sample Tests of Hypothesis

What is a Hypothesis? A statement about the value of a population parameter. Example: The mean monthly income for a systems analyst is $5K.

Hypothesis testing Process used to determine if the hypothesis is a reasonable statement The hypothesis is either accepted or rejected Decision is based on sample data & probability theory Process involves taking a sample & calculating the sample mean. Then we look at how far the sample mean is from the hypothesized mean. If it is too far, we reject it; else, we accept it. Eg. If the mean salary of a sample of systems analysts is between 3-7K, accept hyp. 7K (say) 3K (say) $5K

Hypothesis Testing – Formal Steps Null: Innocent Alternate: Guilty Error level you are willing to tolerate; eg. 5% Identify method to weigh the evidence If evidence is stronger than error level (eg. 95%) reject null hypothesis. Listen to lawyers on both sides & decide based on above criteria

Practice time! Do Self-Review 10-1 Page (2 tails) -2.56

Alternative Hypothesis H 1 : A statement that is accepted if the sample data provide evidence that the null hypothesis is false Null Hypothesis H 0: A statement about the value of a population parameter Step One: State the null and alternate hypotheses Eg. H 0 :The mean income of women financial planners is $65,000 H 1 : The mean income of … …. is not equal to …

Three possibilities H 0 :  = 0 H 1 :  = 0 H 0 :  < 0 H 1 :  > 0 H 0 :  > 0 H 1 :  < 0 Step One : State the null and alternate hypotheses The null hypothesis always contains equality. H 0 :The mean income of women financial planners is $65,000, ie. μ = $65,000. H 1 : The mean income of.. is not equal to… ie. μ ≠ $ H 0 :The mean income of women financial planners is ≤ $65,000 H 1 : The mean income of.. is > $ H 0 :The mean income of women financial planners is ≥ $65,000 H 1 : The mean income of.. is < $ Generally H 0 represents what is currently believed. H 1 represents a researcher’s claim. H 1 is accepted if H 0 is shown to be false

Level of Significance is the probability ( α ) of rejecting the null hypothesis when it is actually true. α = P (Reject H 0 | H 0 is true) = Type I error We are deciding upfront how much type 1 error we are willing to tolerate. H 0 : The suspect is innocent. H 1 : The suspect is guilty. If we set the Level of Significance (α) at 0.05 (5%), it implies that we are willing to convict with only 95% of evidence pointing to guilt (ie. Even though the suspect is innocent). If we set the Level of Significance for the testing at 0.01 (1%), it implies that we could mistakenly convict an innocent person only 1% of the time. Step Two: Select a Level of Significance.

Researcher Null Accepts Rejects Hypothesis H o H o H o is true H o is false Correct decision Type I error  Type II Error  Correct decision Two types of errors in hypothesis testing α = P (Reject H 0 | H 0 is true) = Type I error β = P (Accept H 0 | H 0 is false) = Type II error

Decide if you want to use z or t as the statistic. (No need to calculate anything yet!) Step 3: Identify test statistic

Step Four: Formulate the decision rule. Find the Critical Value(s) corresponding to α from the z or t table. Mark the rejection/ acceptance regions. 2 tails testing 1 tail testing H 0 :  = 0 H 1 :  = 0 H 0 :  < 0 H 1 :  > 0

Step Five: Make a decision. Now, you compute the z or t statistic. Check if it falls inside the Rejection or Acceptance region If it falls inside the Rejection region, reject H 0. If it falls inside the Acceptance region, do not reject H 0.

p-Value The probability of observing a sample value as extreme as, or more extreme than the calculated test statistic value. Decision Rule: If the p-Value is smaller than the significance level, , H 0 is rejected. Cut-off Z Calculated Z p-value

Practice Self-Review 10-2, Page 291 (1 tail) (from table for α=0.01) (1 tail on the right) 0.01 Z=2.33 Z=1.81

Practice Self-Review 10-3, Page (1 tail) - Z= -.87 Acceptance Region