We need to accept or reject a statement about some parameter of a population Statement is called the Hypothesis Random variable Example We are interested in the burn rate of a solid rocket propellant We may be interested in determining if the mean burn rate is 50 cm s Null hypothesis we want to test H 0 50 We say H1 50 Alternative hypothesis rejecting null hypothesis leads to accepting alternative hypothesis How do we determine the value of the population parameter specified by the null hypothesis Past experience Theory Evaluation want to know if engineering design specifications were met Hypothesis testing relies on information from a random sample If this is consistent with the null hypothesis then we conclude it is plausible If not then we conclude it is false Reject H 0 50 How does our sample mean compare with 50 H 50 1 A certain type of car engine emits a mean of 100 mg s of oxides of nitrogen NOx at 100 horsepower A modification to the engine has been proposed that may reduce NOx emissions and it will be put into production if the emission is less than 100 mg s A sample of 50 modified engines are built and tested that have a mean NOx emission of 92 mg s and a standard deviation of 21mg s What are the possible interpretations of this result 1 Population mean is actually greater than 100 sample mean is lower due to random variations Null hypothesis 2 Population mean is actually less than 100 experimental data is real alternative hypothesis A certain type of car engine emits a mean of 100 mg s of oxides of nitrogen NOx at 100 horsepower A modification to the engine has been proposed that may reduce NOx emissions and it will be put into production if the emission is less than 100mg A sample of 50 modified engines are built and tested that have a mean NOx emission of 92 mg s and a standard deviation of 21mg s Hypothesis Testing put the null hypothesis on trial H 0 100 H1 100 Hypothesis test measure the strength of disagreement between the sample and Ho to produce a number between 0 and 1 small Evidence against Ho P value large Evidence for Ho If P value is small enough we can reject the null hypothesis A certain type of car engine emits a mean of 100 mg s of oxides of nitrogen NOx at 100 horsepower A modification to the engine has been proposed that may reduce NOx emissions and it will be put into production if the emission is less than 100mg A sample of 50 modified engines are built and tested that have a mean NOx emission of 92 mg s and a standard deviation of 21mg s 1 We compute the distribution of x under the assumption Ho true null distribution H 0 100 Need mean and standard deviation Choose mean closest to the alternative hypothesis 100 Sx 21 2 97 Choose standard deviation of sample S sdom 50 50 A certain type of car engine emits a mean of 100 mg s of oxides of nitrogen NOx at 100 horsepower A modification to the engine has been proposed that may reduce NOx emissions and it will be put into production if the emission is less than 100mg A sample of 50 modified engines are built and tested that have a mean NOx emission of 92 mg s and a standard deviation of 21mg s 1 Compute P value probability of observing a 92 or less given this distribution 92 100 z 2 69 How do we do this Use z table 2 97 What is P z 2 69 P z 2 69 0 0036 P z 2 69 0 0036 The P value for this test is 0 0036 What does this tell us If Ho were true the probability of drawing a sample with a mean this far from Ho or greater is 0 36 P 0 0036 Ho is false Ho is true and we got lucky unlucky and picked a sample that only occurs 0 36 of the time A P value less than 0 05 is often sufficient to reject null hypothesis There are several types of Hypothesis Tests here we used a zscore as the test statistic z test All follow the same general steps 1 Define Ho and H1 2 Assume Ho to be true 3 Compute test statistic that will be used to assess the evidence against H0 such as the z score 4 Compute the P value probability assuming Ho to be true of the test statistic showing as great or greater disagreement with that actually observed 5 State a conclusion about the strength of evidence against Ho We can either reject the null hypothesis of find that it is plausible Let s try another example A scale is to be calibrated by weighing a 1000g test weight 60 times The 60 scale measurements have a mean of 1000 6g and a standard deviation of 2 g Find the P value for testing the hypothesis H 0 1000 H1 1000 What is our null hypothesis H 0 false What can we learn from this experiment H 0 plausible Let s follow the steps and assume H0 true We need a mean and standard deviation 1000 2 Sx 0 258 60 Now we need the z value 1000 6 1000 z 2 32 0 258 H 0 1000 1000 Two sided test P value Sum A1 A2 Area 2 999 4 z 2 32 Area 1 1000 6 z 2 32 z 1000 6 1000 2 32 0 258 What is the P value P 0 0204 What does this mean Recalibrate the scale We reject the null hypothesis Let s try to set up this problem In one experiment 45 steel balls lubricated with purified paraffin were subjected to a 40 kg load at 600 rpm for 60 minutes The average wear measured by the reduction in diameter was 673 2 m and the standard deviation was 14 9 m You are interested in determining if the specification for a lubricant that the mean wear be less than 675 m has been met Let s just set up the null and alternate hypothesis H 0 675 H1 675