1. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 1 of 34 Chapter 10 Section 1 The Language of Hypothesis Testing

2. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 2 of 34 Chapter 10 – Section 1 ●Learning objectives  Determine the null and alternative hypotheses from a claim  Understand Type I and Type II errors  State conclusions to hypothesis tests 1 2 3

3. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 3 of 34 Chapter 10 – Section 1 ●Learning objectives  Determine the null and alternative hypotheses from a claim  Understand Type I and Type II errors  State conclusions to hypothesis tests 1 2 3

4. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 4 of 34 Chapter 10 – Section 1 ●The environment of our problem is that we want to test whether a particular claim is believable, or not ●The process that we use is called hypothesis testing ●This is one of the most common goals of statistics

5. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 5 of 34 Chapter 10 – Section 1 ●Hypothesis testing involves two steps  Step 1 – to state what we think is true  Step 2 – to quantify how confident we are in our claim ●The first step is relatively easy ●The second step is why we need statistics

6. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 6 of 34 Chapter 10 – Section 1 ●We are usually told what the claim is, what the goal of the test is ●Now similar to estimation in the previous chapter, we will again use the material in Chapter 8 on the sample mean to quantify how confident we are in our claim

7. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 7 of 34 Chapter 10 – Section 1 ●An example of what we want to quantify  A car manufacturer claims that a certain model of car achieves 29 miles per gallon ●An example of what we want to quantify  A car manufacturer claims that a certain model of car achieves 29 miles per gallon  We test some number of cars ●An example of what we want to quantify  A car manufacturer claims that a certain model of car achieves 29 miles per gallon  We test some number of cars  We calculate the sample mean … it is 27 ●An example of what we want to quantify  A car manufacturer claims that a certain model of car achieves 29 miles per gallon  We test some number of cars  We calculate the sample mean … it is 27  Is 27 miles per gallon consistent with the manufacturer’s claim? How confident are we that the manufacturer has significantly overstated the miles per gallon achievable?

8. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 8 of 34 Chapter 10 – Section 1 ●How confident are we that the gas economy is definitely less than 29 miles per gallon? ●We would like to make either a statement “We’re pretty sure that the mileage is less than 29 mpg” ●How confident are we that the gas economy is definitely less than 29 miles per gallon? ●We would like to make either a statement “We’re pretty sure that the mileage is less than 29 mpg” or “It’s believable that the mileage is equal to 29 mpg”

9. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 9 of 34 Chapter 10 – Section 1 ●A hypothesis test for an unknown parameter is a test of a specific claim  Compare this to a confidence interval which gives an interval of numbers, not a “believe it” or “don’t believe it” answer ●A hypothesis test for an unknown parameter is a test of a specific claim  Compare this to a confidence interval which gives an interval of numbers, not a “believe it” or “don’t believe it” answer ●The level of significance represents the confidence we have in our conclusion

10. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 10 of 34 Chapter 10 – Section 1 ●How do we state our claim? ●Our claim  Is the statement to be tested  Is called the null hypothesis  Is written as H 0 (and is read as “H-naught”)

11. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 11 of 34 Chapter 10 – Section 1 ●How do we state our counter-claim? ●Our counter-claim  Is the opposite of the statement to be tested  Is called the alternative hypothesis  Is written as H 1 (and is read as “H-one”)

12. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 12 of 34 Chapter 10 – Section 1 ●There are different types of null hypothesis / alternative hypothesis pairs, depending on the claim and the counter-claim ●One type of H 0 / H 1 pair, called a two-tailed test, tests whether the parameter is either equal to, versus not equal to, some value  H 0 : parameter = some value  H 1 : parameter ≠ some value

13. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 13 of 34 Chapter 10 – Section 1 ●An example of a two-tailed test ●A bolt manufacturer claims that the diameter of the bolts average 10 mm  H 0 : Diameter = 10  H 1 : Diameter ≠ 10 ●An example of a two-tailed test ●A bolt manufacturer claims that the diameter of the bolts average 10 mm  H 0 : Diameter = 10  H 1 : Diameter ≠ 10 ●An alternative hypothesis of “≠ 10” is appropriate since  A sample diameter that is too high is a problem  A sample diameter that is too low is also a problem ●An example of a two-tailed test ●A bolt manufacturer claims that the diameter of the bolts average 10 mm  H 0 : Diameter = 10  H 1 : Diameter ≠ 10 ●An alternative hypothesis of “≠ 10” is appropriate since  A sample diameter that is too high is a problem  A sample diameter that is too low is also a problem ●Thus this is a two-tailed test

14. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 14 of 34 Chapter 10 – Section 1 ●Another type of pair, called a left-tailed test, tests whether the parameter is either equal to, versus less than, some value  H 0 : parameter = some value  H 1 : parameter < some value

15. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 15 of 34 Chapter 10 – Section 1 ●An example of a left-tailed test ●A car manufacturer claims that the mpg of a certain model car is at least 29.0  H 0 : MPG = 29.0  H 1 : MPG < 29.0 ●An example of a left-tailed test ●A car manufacturer claims that the mpg of a certain model car is at least 29.0  H 0 : MPG = 29.0  H 1 : MPG < 29.0 ●An alternative hypothesis of “< 29” is appropriate since  A mpg that is too low is a problem  A mpg that is too high is not a problem ●An example of a left-tailed test ●A car manufacturer claims that the mpg of a certain model car is at least 29.0  H 0 : MPG = 29.0  H 1 : MPG < 29.0 ●An alternative hypothesis of “< 29” is appropriate since  A mpg that is too low is a problem  A mpg that is too high is not a problem ●Thus this is a left-tailed test

16. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 16 of 34 Chapter 10 – Section 1 ●Another third type of pair, called a right-tailed test, tests whether the parameter is either equal to, versus greater than, some value  H 0 : parameter = some value  H 1 : parameter > some value

17. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 17 of 34 Chapter 10 – Section 1 ●An example of a right-tailed test ●A bolt manufacturer claims that the defective rate of their product is at most 1 part in 1,000  H 0 : Defect Rate = 0.001  H 1 : Defect Rate > 0.001 ●An example of a right-tailed test ●A bolt manufacturer claims that the defective rate of their product is at most 1 part in 1,000  H 0 : Defect Rate = 0.001  H 1 : Defect Rate > 0.001 ●An alternative hypothesis of “> 0.001” is appropriate since  A defect rate that is too low is not a problem  A defect rate that is too high is a problem ●An example of a right-tailed test ●A bolt manufacturer claims that the defective rate of their product is at most 1 part in 1,000  H 0 : Defect Rate = 0.001  H 1 : Defect Rate > 0.001 ●An alternative hypothesis of “> 0.001” is appropriate since  A defect rate that is too low is not a problem  A defect rate that is too high is a problem ●Thus this is a right-tailed test

18. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 18 of 34 Chapter 10 – Section 1 ●A comparison of the three types of tests ●The null hypothesis  We believe that this is true ●A comparison of the three types of tests ●The null hypothesis  We believe that this is true ●The alternative hypothesis Type of testSample value that is too low Sample value that is too high Two-tailed testA problem Left-tailed testA problemNot a problem Right-tailed testNot a problemA problem

19. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 19 of 34 Chapter 10 – Section 1 ●A manufacturer claims that there are at least two scoops of cranberries in each box of cereal ●What would be a problem?  The parameter to be tested is the number of scoops of cranberries in each box of cereal  If the sample mean is too low, that is a problem  If the sample mean is too high, that is not a problem ●A manufacturer claims that there are at least two scoops of cranberries in each box of cereal ●What would be a problem?  The parameter to be tested is the number of scoops of cranberries in each box of cereal  If the sample mean is too low, that is a problem  If the sample mean is too high, that is not a problem ●This is a left-tailed test  The “bad case” is when there are too few

20. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 20 of 34 Chapter 10 – Section 1 ●A manufacturer claims that there are exactly 500 mg of a medication in each tablet ●What would be a problem?  The parameter to be tested is the amount of a medication in each tablet  If the sample mean is too low, that is a problem  If the sample mean is too high, that is a problem too ●A manufacturer claims that there are exactly 500 mg of a medication in each tablet ●What would be a problem?  The parameter to be tested is the amount of a medication in each tablet  If the sample mean is too low, that is a problem  If the sample mean is too high, that is a problem too ●This is a two-tailed test  A “bad case” is when there are too few  A “bad case” is also where there are too many

21. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 21 of 34 Chapter 10 – Section 1 ●A manufacturer claims that there are at most 8 grams of fat per serving ●What would be a problem?  The parameter to be tested is the number of grams of fat in each serving  If the sample mean is too low, that is not a problem  If the sample mean is too high, that is a problem ●A manufacturer claims that there are at most 8 grams of fat per serving ●What would be a problem?  The parameter to be tested is the number of grams of fat in each serving  If the sample mean is too low, that is not a problem  If the sample mean is too high, that is a problem ●This is a right-tailed test  The “bad case” is when there are too many

22. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 22 of 34 Chapter 10 – Section 1 ●There are two possible results for a hypothesis test ●If we believe that the null hypothesis could be true, this is called not rejecting the null hypothesis  Note that this is only “we believe … could be” ●There are two possible results for a hypothesis test ●If we believe that the null hypothesis could be true, this is called not rejecting the null hypothesis  Note that this is only “we believe … could be” ●If we are pretty sure that the null hypothesis is not true, so that the alternative hypothesis is true, this is called rejecting the null hypothesis  Note that this is “we are pretty sure that … is”

23. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 23 of 34 Chapter 10 – Section 1 ●Learning objectives  Determine the null and alternative hypotheses from a claim  Understand Type I and Type II errors  State conclusions to hypothesis tests 1 2 3

24. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 24 of 34 Chapter 10 – Section 1 ●In comparing our conclusion (not reject or reject the null hypothesis) with reality, we could either be right or we could be wrong  When we reject (and state that the null hypothesis is false) but the null hypothesis is actually true  When we not reject (and state that the null hypothesis could be true) but the null hypothesis is actually false ●These would be undesirable errors

25. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 25 of 34 Chapter 10 – Section 1 ●A summary of the errors is ●We see that there are four possibilities … in two of which we are correct and in two of which we are incorrect

26. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 26 of 34 Chapter 10 – Section 1 ●When we reject (and state that the null hypothesis is false) but the null hypothesis is actually true … this is called a Type I error ●When we do not reject (and state that the null hypothesis could be true) but the null hypothesis is actually false … this called a Type II error ●When we reject (and state that the null hypothesis is false) but the null hypothesis is actually true … this is called a Type I error ●When we do not reject (and state that the null hypothesis could be true) but the null hypothesis is actually false … this called a Type II error ●In general, Type I errors are considered the more serious of the two

27. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 27 of 34 Chapter 10 – Section 1 ●A very good analogy for Type I and Type II errors is in comparing it to a criminal trial ●In the US judicial system, the defendant “is innocent until proven guilty”  Thus the defendant is presumed to be innocent  The null hypothesis is that the defendant is innocent  H 0 : the defendant is innocent

28. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 28 of 34 Chapter 10 – Section 1 ●If the defendant is not innocent, then  The defendant is guilty  The alternative hypothesis is that the defendant is guilty  H 1 : the defendant is guilty ●If the defendant is not innocent, then  The defendant is guilty  The alternative hypothesis is that the defendant is guilty  H 1 : the defendant is guilty ●The summary of the set-up  H 0 : the defendant is innocent  H 1 : the defendant is guilty

29. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 29 of 34 Chapter 10 – Section 1 ●Our possible conclusions ●Reject the null hypothesis  Go with the alternative hypothesis  H 1 : the defendant is guilty  We vote “guilty” ●Our possible conclusions ●Reject the null hypothesis  Go with the alternative hypothesis  H 1 : the defendant is guilty  We vote “guilty” ●Do not reject the null hypothesis  Go with the null hypothesis  H 0 : the defendant is innocent  We vote “not guilty” (which is not the same as voting innocent!)

30. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 30 of 34 Chapter 10 – Section 1 ●A Type I error  Reject the null hypothesis  The null hypothesis was actually true  We voted “guilty” for an innocent defendant ●A Type I error  Reject the null hypothesis  The null hypothesis was actually true  We voted “guilty” for an innocent defendant ●A Type II error  Do not reject the null hypothesis  The alternative hypothesis was actually true  We voted “not guilty” for a guilty defendant

31. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 31 of 34 Chapter 10 – Section 1 ●Which error do we try to control? ●Type I error (sending an innocent person to jail)  The evidence was “beyond reasonable doubt”  We must be pretty sure  Very bad! We want to minimize this type of error ●Which error do we try to control? ●Type I error (sending an innocent person to jail)  The evidence was “beyond reasonable doubt”  We must be pretty sure  Very bad! We want to minimize this type of error ●A Type II error (letting a guilty person go)  The evidence wasn’t “beyond a reasonable doubt”  We weren’t sure enough  If this happens … well … it’s not as bad as a Type I error (according to the US system)

32. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 32 of 34 Chapter 10 – Section 1 ●Learning objectives  Determine the null and alternative hypotheses from a claim  Understand Type I and Type II errors  State conclusions to hypothesis tests 1 2 3

33. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 33 of 34 Chapter 10 – Section 1 ●“Innocent” versus “Not Guilty” ●This is an important concept ●Innocent is not the same as not guilty  Innocent – the person did not commit the crime  Not guilty – there is not enough evidence to convict … that the reality is unclear ●To not reject the null hypothesis – doesn’t mean that the null hypothesis is true – just that there isn’t enough evidence to reject

34. Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 10 Section 1 – Slide 34 of 34 Summary: Chapter 10 – Section 1 ●A hypothesis test tests whether a claim is believable or not, compared to the alternative ●We test the null hypothesis H 0 versus the alternative hypothesis H 1 ●If there is sufficient evidence to conclude that H 0 is false, we reject the null hypothesis ●If there is insufficient evidence to conclude that H 0 is false, we do not reject the null hypothesis