Chapter 10 Section 1

1. Chapter 10Section 1 The Language of Hypothesis Testing

2. 1 2 3 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

3. 1 2 3 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

4. 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 hypothesistesting • This is one of the most common goals of statistics

5. 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. 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. Chapter 10 – Section 1 • An example of what we want to quantify • 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. Chapter 10 – Section 1 • How confident are we that the gas economy is definitely less than 29 miles per gallon? • 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. Chapter 10 – Section 1 • A hypothesistest 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 hypothesistest 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 levelofsignificance represents the confidence we have in our conclusion

10. Chapter 10 – Section 1 • How do we state our claim? • Our claim • Is the statement to be tested • Is called the nullhypothesis • Is written as H0 (and is read as “H-naught”)

11. 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 alternativehypothesis • Is written as H1 (and is read as “H-one”)

12. Chapter 10 – Section 1 • There are different types of null hypothesis / alternative hypothesis pairs, depending on the claim and the counter-claim • There are different types of null hypothesis / alternative hypothesis pairs, depending on the claim and the counter-claim • One type of H0 / H1 pair, called a two-tailedtest, tests whether the parameter is either equal to, versus not equal to, some value • H0: parameter = some value • H1: parameter ≠ some value

13. Chapter 10 – Section 1 • An example of a two-tailed test • An example of a two-tailed test • A bolt manufacturer claims that the diameter of the bolts average 10 mm • H0: Diameter = 10 • H1: Diameter ≠ 10 • An example of a two-tailed test • A bolt manufacturer claims that the diameter of the bolts average 10 mm • H0: Diameter = 10 • H1: 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 • H0: Diameter = 10 • H1: 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. Chapter 10 – Section 1 • Another type of pair, called a left-tailedtest, tests whether the parameter is either equal to, versus less than, some value • H0: parameter = some value • H1: parameter < some value

15. Chapter 10 – Section 1 • An example of a left-tailed test • An example of a left-tailed test • A car manufacturer claims that the mpg of a certain model car is at least 29.0 • H0: MPG = 29.0 • H1: 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 • H0: MPG = 29.0 • H1: 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 • H0: MPG = 29.0 • H1: 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. Chapter 10 – Section 1 • Another third type of pair, called a right-tailedtest, tests whether the parameter is either equal to, versus greater than, some value • H0: parameter = some value • H1: parameter > some value

17. Chapter 10 – Section 1 • An example of a right-tailed test • 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 • H0: Defect Rate = 0.001 • H1: 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 • H0: Defect Rate = 0.001 • H1: 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 • H0: Defect Rate = 0.001 • H1: 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. 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

19. Chapter 10 – Section 1 • A manufacturer claims that there are at least two scoops of cranberries in each box of cereal • 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. Chapter 10 – Section 1 • A manufacturer claims that there are exactly 500 mg of a medication in each tablet • 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. Chapter 10 – Section 1 • A manufacturer claims that there are at most 8 grams of fat per serving • 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. Chapter 10 – Section 1 • There are two possible results for a hypothesis test • There are two possible results for a hypothesis test • If we believe that the null hypothesis could be true, this is called notrejectingthenullhypothesis • 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 notrejectingthenullhypothesis • 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 rejectingthenullhypothesis • Note that this is “we are pretty sure that … is”

23. 1 2 3 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

24. 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. 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. 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 TypeIerror • When we reject (and state that the null hypothesis is false) but the null hypothesis is actually true … this is called a TypeIerror • When we do not reject (and state that the null hypothesis could be true) but the null hypothesis is actually false … this called a TypeIIerror • When we reject (and state that the null hypothesis is false) but the null hypothesis is actually true … this is called a TypeIerror • When we do not reject (and state that the null hypothesis could be true) but the null hypothesis is actually false … this called a TypeIIerror • In general, Type I errors are considered the more serious of the two

27. Chapter 10 – Section 1 • A very good analogy for Type I and Type II errors is in comparing it to a criminal trial • 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 • H0: the defendant is innocent

28. Chapter 10 – Section 1 • If the defendant is not innocent, then • The defendant is guilty • The alternative hypothesis is that the defendant is guilty • H1: the defendant is guilty • If the defendant is not innocent, then • The defendant is guilty • The alternative hypothesis is that the defendant is guilty • H1: the defendant is guilty • The summary of the set-up • H0: the defendant is innocent • H1: the defendant is guilty

29. Chapter 10 – Section 1 • Our possible conclusions • Our possible conclusions • Reject the null hypothesis • Go with the alternative hypothesis • H1: the defendant is guilty • We vote “guilty” • Our possible conclusions • Reject the null hypothesis • Go with the alternative hypothesis • H1: the defendant is guilty • We vote “guilty” • Do not reject the null hypothesis • Go with the null hypothesis • H0: the defendant is innocent • We vote “not guilty” (which is not the same as voting innocent!)

30. 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. Chapter 10 – Section 1 • Which error do we try to control? • 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. 1 2 3 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

33. 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. 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 H0 versus the alternative hypothesis H1 • If there is sufficient evidence to conclude that H0 is false, we reject the null hypothesis • If there is insufficient evidence to conclude that H0 is false, we do not reject the null hypothesis