In order to test any hypothesis (even non statistical hypotheses) you need three elements: 1. The Hypotheses (Both the hypothesis that you are testing and the alternative hypothesis, which is the opposite of the hypothesis) 2. An unbiased test statistic- A measure that you will use to evaluate the hypotheses. 3. A rejection rule- the rule that you will use to ultimately decide if a hypothesis should be rejected.
Hypothesis Testing Rules In order to test a hypothesis, you must first find the test statistic and rejection rule that is appropriate for evaluating your hypotheses (i.e., we could not evaluate OJ’s innocence based upon his LSAT score). All hypothesis tests will always end in 1 of 2 ways: 1. You conclude that that you must reject the null hypothesis hypothesis (This is the same as concluding that you have proven the alternative true); Or 2. You conclude that you do not have enough evidence to reject the null hypothesis (This is the same as concluding that you do not have enough evidence to prove that the alternative is true).
Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. The null hypothesis, hypothesis , denoted H , is a statement of the basic proposition
5 6 7.
**If ** If using the critical value approach, use to determine the critical value. (Find the rejection rule that is appropriate for for your null hypothesis) Collect the sample data and compute the value of the test statistic. **If ** If using the p-value approach, find the p-value using t he test statistic. Use the value of the test statistic and the reje ction rule to determine whether to reject H 0. State the conclusion in layman’s terms.
Follow our hypothesis testing steps on slide 29. (The (The emergency service is meeting the response H : Step 1: 0 goal; no appropriate follow-up action is necessary. ) H : The emergency service is not meeting the a response goal; appropriate follow-up action is necessary . - mean filling weight for the machine. Step 2: =.05 So far we have only learned about conducting hypothesis tests for large samples with σ known. In order to find the appropriate test statistic and rejection rule to test our type of null hypothesis, we must verify that this is a large sample case with σ known.
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being tested. The statement is not rejected unless there is convincing sample evidence that it is false. The alternative or research hypothesis, hypothesis , denoted H , is an alternative (to
Step 3: 3: Step 4: Reject H0 if z > z = 1.645 (α =0.05) =0.05) α
a
the null hypothesis) statement that will be accepted only if there is convincing sample evidence that it is true. Developing Null and Alternative Hypotheses If you wish to find evidence to contradict a claim, the claim should be stated as the null hypothesis. If you wish to prove a claim to be true, then you should state the claim as the alternative hypothesis. Claims that test whether the mean is equal to a specific value must be stated as the null hypothesis. SEE SLIDE 10 Type I Errors Since hypothesis tests are based on sample data, we must allow for the possibility of errors. A Type I error is rejecting H when it is true. 0
The person conducting the hypothesis test specifies the maximum allowable probability of making a Type I error, denoted by a and called the level of significance. Type I and Type II Errors A Type II error is accepting H when it is false.
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Generally, we cannot control for the probability of making a Type II error, denoted by . Statistician avoids the risk of making a Type II error by using the phrase “do not not reject H ” instead of “accept H ”. SEE SLIDE 24 0
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Step 5: Step 6: Is z> 1.645? Since 2.47 > 1.645, we reject H0 in favor of H a. Step 7: Conclusion: We are 95% confident that Metro EMS is not meeting the response goal of 12 minutes; appropriate action should be taken to improve service. Large Sample Tests about Mean (n 30) and unknown OR Small Sample Tests about Mean, Normal pop. and unknown Steps for Computing ta Computing t upper (lower) tail area of . 1.
2. 3.
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Steps for Computing t /2 1.
Steps for Computing z z value for an upper tail area of 1.
2.
In order to determine the z value with an upper tail area of , we need the area beneath the normal curve to the left of the z value of interest. Compute this area= 1- . Go to the area section of the standard normal table and find the area
2. 3. 4.
closest to the area computed in 1. The corresponding z value is z ***To find the z value for a lower tail area of ***To of , look up (or closest value) directly on chart and the corresponding z value is z .
Steps for Computing z 1 2
3
/2
(The critical value for a 2 tailed test)
4.
Go to t chart and find the t value with the upper (lower) tail area (computed in step 2), and the degrees of freedom (computed in step 2). This is t /2 .
Computing t /2 1.
Compute
In order to determine the z value with an upper tail area of /2, we need the area beneath the normal curve to the left of the z value of
interest. Compute this area= 1- . Go to the area section of the standard normal table and find the area
Steps of Hypothesis Testing
2. 3.
SEE SLIDE 37 Compute . =1 – (Confidence Coefficient). **Sometimes is given, in this case we do not have to compute anything. We can just use what is given. Compute the upper (lower) tail area : /2. Compute the degrees of freedom= n-1.
Compute /2.
closest to the area computed in 2. The corresponding z value is z /2. 1.
Compute the upper (lower) tail area . =1 – (Confidence Coefficient). **Sometimes is given, in this case we do not have to compute anything. We can just use what is given. Compute the degrees of freedom= n-1. Go to t chart and find the t value with the upper (lower) tail area (computed in step 1), and the degrees of freedom (computed in step 2). This is t
Determine the null and alternative hypotheses. Specify the level of significance . Select the test statistic that will be used to test the hypothesis. State the rejection rule for H 0 .
. =1
–
(Confidence Coefficient).
**Sometimes is given, in this case we do not have to compute anything. We can just use what is given.
2. 3.
Compute the upper (lower) tail area : /2. Compute the degrees of freedom= n-1.
4.
Go to t chart and find the t value with the upper (lower) tail area (computed in step 2), and the degrees of freedom (computed in step 2). This is t /2 .
