The P- value of the test is the associated with the critical region. Any smaller value for expands the critical region and the test fails to reject the null hypothesis when . The P -value is easy to compute after the test statistic is observed. 3 . 51 x 3 . 51 x 038 . 0 962 . 0 1 ) 08 . 2 08 . 2 ( 1 16 / 5 . 2 50 3 . 51 16 / 5 . 2 50 7 . 48 1 ) 3 . 51 7 . 48 ( 1 value - Z P Z P X P P Figure 9-6 P -value is area of shaded region when . 3 . 51 x
9-1 Hypothesis Testing 9-1.5 Connection between Hypothesis Tests and Confidence Intervals A close relationship exists between the test of a hypothesis for , and the confidence interval for . If [ l , u ] is a 100(1 ) confidence interval for the parameter , the test of size of the hypothesis H 0 : 0 H 1 : 0 will lead to rejection of H 0 if and only if 0 is not in the 100(1 ) CI [ l , u ].
9-1 Hypothesis Testing • CI provides a range of likely values for 𝜇 at a stated confidence level • Hypothesis testing is a framework for displaying the risk levels such as the p-value associated with a specific decision. 9-1.5 Connection between Hypothesis Tests and Confidence Intervals
Recall of the terms • Type-1 error • Type-2 error • Power of the test • Significance level • Size of a test • p-value • Confidence level
9-1 Hypothesis Testing 9-1.6 General Procedure for Hypothesis Tests 1. Identify the parameter of interest. 2. Formulate the null hypothesis, H 0 . 3. Specify an appropriate alternative hypothesis, H 1 . 4. Choose a significance level, . 5. Determine an appropriate test statistic. 6. State the rejection criteria for the statistic. 7. Compute necessary sample quantities for calculating the test statistic. 8. Draw appropriate conclusions.
9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.1 Hypothesis Tests on the Mean Consider the two-sided hypothesis test The test statistic is : H 0 : 0 H 1 : ≠ 0 n X Z / 0 0 (9-1)
9-2 Tests on the Mean of a Normal Distribution, Variance Known 9-2.1 Hypothesis Tests on the Mean Reject H 0 if the observed value of the test statistic z 0 is either : z 0 > z /2 or z 0 < - z /2 Fail to reject H 0 if the observed value of the test statistic z 0 is - z /2 < z 0 < z /2
9-2 Tests on the Mean of a Normal Distribution, Variance Known • The reference distribution for this test is the standard normal distribution. The test is usually called a z -test .
EXAMPLE 9-2 Propellant Burning Rate Air crew escape systems are powered by a solid propellant. The burning rate of this propellant is an important product characteristic. Specifications require that the mean burning rate must be 50 centimeters per second and the standard deviation is 2 centimeters per second. The significance level of 0.05 and a random sample of n 25 has a sample average burning rate of centimeters per second. Draw conclusions. The seven-step procedure is 1. Parameter of interest: The parameter of interest is , the mean burning rate.
- Summer '20
- Normal Distribution, Null hypothesis, Statistical hypothesis testing