Test hypothesis & p-value
- 1. BIOSTATSTICS Diagnosing Test-Statistic & General Procedure of Testing a Hypothesis Presented & Compiled By: Raeesa Mukhtar M.Phil (Pharmaceutics) 2019-21 University of Sargodha, Sargodha, Pakistan
- 2. Test statistic • The test statistic is some random value that may be computed from the data of the sample. As a rule, there are many possible values that the test statistic may assume, the particular value observed depending on the particular sample drawn. • The test statistic serves as a decision maker, since the decision to reject or not to reject the null hypothesis depends on the magnitude of the test statistic.
- 3. General Formula for Test Statistic • The following is a general formula for a test statistic that will be applicable Test statistic = sample Population mean – hypothesized mean standard deviation/√sample size
- 4. Decision: • All possible values that the test statistic can assume are points on the horizontal axis of the graph of the distribution of the test statistic and are divided into two groups; one group constitutes what is known as the rejection region and the other group makes up the non rejection region. • The values of the test statistic forming the rejection region are those values that are less likely to occur if the null hypothesis is true, while the values making up the acceptance region are more likely to occur if the null hypothesis is true. • The decision rule tells us to reject the null hypothesis if the value of the test statistic that we compute from our sample is one of the values in the rejection region and to not reject the null hypothesis if the computed value of the test statistic is one of the values in the non-rejection region.
- 5. Significance Level: The decision as to which values go into the rejection region and which ones go into the non-rejection region is made on the basis of the desired level of significance, designated by ‘’α’’ . The level of significance, α , specifies the area under the curve of the distribution of the test statistic that is above the values on the horizontal axis constituting the rejection region.
- 6. Statistical decision • The statistical decision consists of rejecting or of not rejecting the null hypothesis. • It is rejected if the computed value of the test statistic falls in the rejection region, and • it is not rejected if the computed value of the test statistic falls in the non-rejection region. Conclusion If H0 is rejected, we conclude that is HA true. If H0 is not rejected, we conclude that HA may be true.
- 7. p-values. • The p value is a number that tells us how unusual our sample results are, given that the null hypothesis is true. A p value indicating that the sample results are not likely to have occurred, if the null hypothesis is true, provides justification for doubting the truth of the null hypothesis.
- 8. Purpose of Hypothesis Testing • The purpose of hypothesis testing is to assist administrators and clinicians in making decisions. • The administrative or clinical decision usually depends on the statistical decision. • If the null hypothesis is rejected, the administrative or clinical decision usually reflects this, in that the decision is compatible with the alternative hypothesis. The reverse is usually true if the null hypothesis is not rejected. • The administrative or clinical decision, however, may take other forms, such as a decision to gather more data.
- 9. EXAMPLE • The goal of a study by Klingler et al. was to determine how symptom recognition and perception influence clinical presentation as a function of race. They characterized symptoms and care-seeking behavior in African-American patients with chest pain seen in the emergency department. One of the presenting vital signs was systolic blood pressure. Among 157 African-American men, the mean systolic blood pressure was 146 mm Hg with a standard deviation of 27. We wish to know if, on the basis of these data, is there any evidence to support the claim at α = 0.05 that the mean systolic blood pressure for a population of African-American men is greater than 140.
- 10. Solution: • Hypotheses H0 : µ ≤ 140 HA : µ > 140 Decision rule. Since α = 0.05 The critical value of the test statistic is 1.65. The rejection and non-rejection regions are shown in Figure. Reject H0 if Z ≥ 1.65 computed .
- 12. Calculation of test statistic Statistical decision. Reject H0 since 2.78 > 1.65 Conclusion. Conclude that the mean systolic blood pressure for the sampled population is greater than 140
- 13. p values • Instead of saying that an observed value of the test statistic is significant or is not significant, most writers in the research literature prefer to report the exact probability of getting a value as extreme as or more extreme than that observed if the null hypothesis is true. • In the present instance the p value for this test is 1-0.9973 =0.0027 • These writers would give the computed value of the test statistic along with the statement p value = 0.0027
- 14. • p value for a test may be defined also as the smallest value of α for which the null hypothesis can be rejected. Since, in Example, our p value is 0.0027, we know that we could have chosen an α value as small as 0.0027 and still have rejected the null hypothesis. If we had chosen an α smaller than 0.0027, we would not have been able to reject the null hypothesis. A general rule worth remembering, then, is this: if the p value is less than or equal to α, we reject the null hypothesis; if the p value is greater than α, we do not reject the null hypothesis.
- 15. General Procedure of Testing a Hypothesis There are several steps in testing of hypothesis, which lead to a conclusion to accept or not to accept the hypothesis. These steps are common for all types of tests of significance. These general steps lead us to the final decision about the null hypothesis. Step 1: Write two statements, which are appropriate concerning value of the parameter i.e. to state null and alternative hypotheses. Step 2: State whether the test is a one-tailed or a two-tailed test. Step 3: Choose the level of significance. Usually 1% or 5% level of significance is chosen. Step 4: State an appropriate test-statistic to be used.
- 16. Step 5: Calculate the value using the test-statistic mentioned in Step 4. Step 6: State the decision rule for the acceptance of null hypothesis. The decision rule is to accept the null- hypothesis if calculated value is less than table value at a given level of significance otherwise do not accept the null hypothesis Health scientists usually interpret the result in terms of p-value (observed level of significance). If the observed p-value is less than the stated p-value (given level of significance), then the null hypothesis is not accepted. Step 7: Draw the inference about the parameter on the basis of the above steps.