Mathematics of a Lady Tasting TeaMathematics 15: Lecture 19

Dan Sloughter

Furman University

November 2, 2006

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 1 / 9

Ronald Aylmer Fisher

I 1890 - 1962

I Two classic books on statistics: Statistical Methods for ResearchWorkers, first published in 1925, and The Design of Experiments, firstpublished in 1935

I Equally famous as a geneticist (for example, the text The GeneticalTheory of Natural Selection)

I Refused a prestigious position in London to pursue statisticalproblems in agriculture at Rothamsted, where he developed, amongmany other fundamental notions of modern statistics, the theory ofrandomized experimental design

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 2 / 9

Ronald Aylmer Fisher

I 1890 - 1962

I Two classic books on statistics: Statistical Methods for ResearchWorkers, first published in 1925, and The Design of Experiments, firstpublished in 1935

I Equally famous as a geneticist (for example, the text The GeneticalTheory of Natural Selection)

I Refused a prestigious position in London to pursue statisticalproblems in agriculture at Rothamsted, where he developed, amongmany other fundamental notions of modern statistics, the theory ofrandomized experimental design

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 2 / 9

Ronald Aylmer Fisher

I 1890 - 1962

I Two classic books on statistics: Statistical Methods for ResearchWorkers, first published in 1925, and The Design of Experiments, firstpublished in 1935

I Equally famous as a geneticist (for example, the text The GeneticalTheory of Natural Selection)

I Refused a prestigious position in London to pursue statisticalproblems in agriculture at Rothamsted, where he developed, amongmany other fundamental notions of modern statistics, the theory ofrandomized experimental design

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 2 / 9

Ronald Aylmer Fisher

I 1890 - 1962

I Two classic books on statistics: Statistical Methods for ResearchWorkers, first published in 1925, and The Design of Experiments, firstpublished in 1935

I Equally famous as a geneticist (for example, the text The GeneticalTheory of Natural Selection)

I Refused a prestigious position in London to pursue statisticalproblems in agriculture at Rothamsted, where he developed, amongmany other fundamental notions of modern statistics, the theory ofrandomized experimental design

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 2 / 9

The experiment

I Given a cup of tea with milk, a lady claims she can discriminate as towhether milk or tea was first added to the cup.

I To test her claim, eight cups of tea are prepared, four of which havethe milk added first and four of which have the tea added first.

I Question: How many cups does she have to correctly identify toconvince us of her ability?

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 3 / 9

The experiment

I Given a cup of tea with milk, a lady claims she can discriminate as towhether milk or tea was first added to the cup.

I To test her claim, eight cups of tea are prepared, four of which havethe milk added first and four of which have the tea added first.

I Question: How many cups does she have to correctly identify toconvince us of her ability?

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 3 / 9

The experiment

I Given a cup of tea with milk, a lady claims she can discriminate as towhether milk or tea was first added to the cup.

I To test her claim, eight cups of tea are prepared, four of which havethe milk added first and four of which have the tea added first.

I Question: How many cups does she have to correctly identify toconvince us of her ability?

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 3 / 9

Choosing subsets

I There are 8× 7× 6× 5 = 1680 ways to choose a first cup, a secondcup, a third cup, and a fourth cup, in order.

I There are 4× 3× 2× 1 = 24 ways to order four cups.

I So the number of ways to choose 4 cups out of 8 is

1680

24= 70.

I Note: the lady performs the experiment by selecting 4 cups, say, theones she claims to have had the tea poured first.

I For example, the probability that she would correctly identify all 4cups is 1

70 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 4 / 9

Choosing subsets

I There are 8× 7× 6× 5 = 1680 ways to choose a first cup, a secondcup, a third cup, and a fourth cup, in order.

I There are 4× 3× 2× 1 = 24 ways to order four cups.

I So the number of ways to choose 4 cups out of 8 is

1680

24= 70.

I Note: the lady performs the experiment by selecting 4 cups, say, theones she claims to have had the tea poured first.

I For example, the probability that she would correctly identify all 4cups is 1

70 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 4 / 9

Choosing subsets

I There are 8× 7× 6× 5 = 1680 ways to choose a first cup, a secondcup, a third cup, and a fourth cup, in order.

I There are 4× 3× 2× 1 = 24 ways to order four cups.

I So the number of ways to choose 4 cups out of 8 is

1680

24= 70.

I Note: the lady performs the experiment by selecting 4 cups, say, theones she claims to have had the tea poured first.

I For example, the probability that she would correctly identify all 4cups is 1

70 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 4 / 9

Choosing subsets

I There are 8× 7× 6× 5 = 1680 ways to choose a first cup, a secondcup, a third cup, and a fourth cup, in order.

I There are 4× 3× 2× 1 = 24 ways to order four cups.

I So the number of ways to choose 4 cups out of 8 is

1680

24= 70.

I Note: the lady performs the experiment by selecting 4 cups, say, theones she claims to have had the tea poured first.

I For example, the probability that she would correctly identify all 4cups is 1

70 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 4 / 9

Choosing subsets

I There are 8× 7× 6× 5 = 1680 ways to choose a first cup, a secondcup, a third cup, and a fourth cup, in order.

I There are 4× 3× 2× 1 = 24 ways to order four cups.

I So the number of ways to choose 4 cups out of 8 is

1680

24= 70.

I Note: the lady performs the experiment by selecting 4 cups, say, theones she claims to have had the tea poured first.

I For example, the probability that she would correctly identify all 4cups is 1

70 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 4 / 9

Choosing 3

I To get exactly 3 right, and, hence, 1 wrong, she would first have tochoose 3 from the 4 correct ones.

I She can do this 4× 3× 2 = 24 ways with order.I Since 3 cups can be ordered in 3× 2 = 6 ways, there are 4 ways for her

to choose the 3 correctly.

I Since she can now choose the 1 incorrect cup 4 ways, there are a totalof 4× 4 = 16 ways for her to choose exactly 3 right and 1 wrong.

I Hence the probability that she chooses exactly 3 correctly is 1670 = 8

35 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 5 / 9

Choosing 3

I To get exactly 3 right, and, hence, 1 wrong, she would first have tochoose 3 from the 4 correct ones.

I She can do this 4× 3× 2 = 24 ways with order.

I Since 3 cups can be ordered in 3× 2 = 6 ways, there are 4 ways for herto choose the 3 correctly.

I Since she can now choose the 1 incorrect cup 4 ways, there are a totalof 4× 4 = 16 ways for her to choose exactly 3 right and 1 wrong.

I Hence the probability that she chooses exactly 3 correctly is 1670 = 8

35 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 5 / 9

Choosing 3

I To get exactly 3 right, and, hence, 1 wrong, she would first have tochoose 3 from the 4 correct ones.

I She can do this 4× 3× 2 = 24 ways with order.I Since 3 cups can be ordered in 3× 2 = 6 ways, there are 4 ways for her

to choose the 3 correctly.

I Since she can now choose the 1 incorrect cup 4 ways, there are a totalof 4× 4 = 16 ways for her to choose exactly 3 right and 1 wrong.

I Hence the probability that she chooses exactly 3 correctly is 1670 = 8

35 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 5 / 9

Choosing 3

I To get exactly 3 right, and, hence, 1 wrong, she would first have tochoose 3 from the 4 correct ones.

I She can do this 4× 3× 2 = 24 ways with order.I Since 3 cups can be ordered in 3× 2 = 6 ways, there are 4 ways for her

to choose the 3 correctly.

I Since she can now choose the 1 incorrect cup 4 ways, there are a totalof 4× 4 = 16 ways for her to choose exactly 3 right and 1 wrong.

I Hence the probability that she chooses exactly 3 correctly is 1670 = 8

35 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 5 / 9

Choosing 3

I To get exactly 3 right, and, hence, 1 wrong, she would first have tochoose 3 from the 4 correct ones.

I She can do this 4× 3× 2 = 24 ways with order.I Since 3 cups can be ordered in 3× 2 = 6 ways, there are 4 ways for her

to choose the 3 correctly.

I Since she can now choose the 1 incorrect cup 4 ways, there are a totalof 4× 4 = 16 ways for her to choose exactly 3 right and 1 wrong.

I Hence the probability that she chooses exactly 3 correctly is 1670 = 8

35 .

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 5 / 9

Statistical significance

I Suppose the lady correctly identifies all 4 cups.

I Conclusion

I Either she has no ability, and has chosen the correct 4 cups purely bychance, or

I she has the discriminatory ability she claims.

I Since choosing correctly is highly unlikely in the first case (one chancein seventy), we decide for the second.

I Note: if she got 3 correct and 1 wrong, this would be evidence for herability, but not persuasive evidence since the chance of getting 3 ormore correct is 17

70 = 0.2429.

I Note: typically, a result is considered statistically significant if theprobability of its occurrence is less than 0.05, that is, less than 1 outof 20.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 6 / 9

Statistical significance

I Suppose the lady correctly identifies all 4 cups.I Conclusion

I Either she has no ability, and has chosen the correct 4 cups purely bychance, or

I she has the discriminatory ability she claims.

I Since choosing correctly is highly unlikely in the first case (one chancein seventy), we decide for the second.

I Note: if she got 3 correct and 1 wrong, this would be evidence for herability, but not persuasive evidence since the chance of getting 3 ormore correct is 17

70 = 0.2429.

I Note: typically, a result is considered statistically significant if theprobability of its occurrence is less than 0.05, that is, less than 1 outof 20.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 6 / 9

Statistical significance

I Suppose the lady correctly identifies all 4 cups.I Conclusion

I Either she has no ability, and has chosen the correct 4 cups purely bychance, or

I she has the discriminatory ability she claims.

I Since choosing correctly is highly unlikely in the first case (one chancein seventy), we decide for the second.

I Note: if she got 3 correct and 1 wrong, this would be evidence for herability, but not persuasive evidence since the chance of getting 3 ormore correct is 17

70 = 0.2429.

I Note: typically, a result is considered statistically significant if theprobability of its occurrence is less than 0.05, that is, less than 1 outof 20.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 6 / 9

Statistical significance

I Suppose the lady correctly identifies all 4 cups.I Conclusion

I Either she has no ability, and has chosen the correct 4 cups purely bychance, or

I she has the discriminatory ability she claims.

I Since choosing correctly is highly unlikely in the first case (one chancein seventy), we decide for the second.

I Note: if she got 3 correct and 1 wrong, this would be evidence for herability, but not persuasive evidence since the chance of getting 3 ormore correct is 17

70 = 0.2429.

I Note: typically, a result is considered statistically significant if theprobability of its occurrence is less than 0.05, that is, less than 1 outof 20.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 6 / 9

Statistical significance

I Suppose the lady correctly identifies all 4 cups.I Conclusion

I Either she has no ability, and has chosen the correct 4 cups purely bychance, or

I she has the discriminatory ability she claims.

I Since choosing correctly is highly unlikely in the first case (one chancein seventy), we decide for the second.

I Note: if she got 3 correct and 1 wrong, this would be evidence for herability, but not persuasive evidence since the chance of getting 3 ormore correct is 17

70 = 0.2429.

I Note: typically, a result is considered statistically significant if theprobability of its occurrence is less than 0.05, that is, less than 1 outof 20.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 6 / 9

Statistical significance

I Suppose the lady correctly identifies all 4 cups.I Conclusion

I Either she has no ability, and has chosen the correct 4 cups purely bychance, or

I she has the discriminatory ability she claims.

I Since choosing correctly is highly unlikely in the first case (one chancein seventy), we decide for the second.

I Note: if she got 3 correct and 1 wrong, this would be evidence for herability, but not persuasive evidence since the chance of getting 3 ormore correct is 17

70 = 0.2429.

I Note: typically, a result is considered statistically significant if theprobability of its occurrence is less than 0.05, that is, less than 1 outof 20.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 6 / 9

Statistical significance

I Suppose the lady correctly identifies all 4 cups.I Conclusion

I Either she has no ability, and has chosen the correct 4 cups purely bychance, or

I she has the discriminatory ability she claims.

I Since choosing correctly is highly unlikely in the first case (one chancein seventy), we decide for the second.

I Note: if she got 3 correct and 1 wrong, this would be evidence for herability, but not persuasive evidence since the chance of getting 3 ormore correct is 17

70 = 0.2429.

I Note: typically, a result is considered statistically significant if theprobability of its occurrence is less than 0.05, that is, less than 1 outof 20.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 6 / 9

The null hypothesis

I The null hypothesis is a specific description of a possible state ofnature.

I In this example, the null hypothesis is the hypothesis that the ladyhas no special ability to discriminate between the cups of tea.

I Note: we can never prove the null hypothesis, but the data mayprovide evidence to reject it.

I Note: in most situations, rejecting the null hypothesis is what wehope to do.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 7 / 9

The null hypothesis

I The null hypothesis is a specific description of a possible state ofnature.

I In this example, the null hypothesis is the hypothesis that the ladyhas no special ability to discriminate between the cups of tea.

I Note: we can never prove the null hypothesis, but the data mayprovide evidence to reject it.

I Note: in most situations, rejecting the null hypothesis is what wehope to do.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 7 / 9

The null hypothesis

I The null hypothesis is a specific description of a possible state ofnature.

I In this example, the null hypothesis is the hypothesis that the ladyhas no special ability to discriminate between the cups of tea.

I Note: we can never prove the null hypothesis, but the data mayprovide evidence to reject it.

I Note: in most situations, rejecting the null hypothesis is what wehope to do.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 7 / 9

The null hypothesis

I The null hypothesis is a specific description of a possible state ofnature.

I In this example, the null hypothesis is the hypothesis that the ladyhas no special ability to discriminate between the cups of tea.

I Note: we can never prove the null hypothesis, but the data mayprovide evidence to reject it.

I Note: in most situations, rejecting the null hypothesis is what wehope to do.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 7 / 9

Randomization

I It is randomization which allows us to make the probabilitycalculations which reveal whether the data are significant or not.

I Randomization takes care of all the possible causes for which wecannot control.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 8 / 9

Randomization

I It is randomization which allows us to make the probabilitycalculations which reveal whether the data are significant or not.

I Randomization takes care of all the possible causes for which wecannot control.

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 8 / 9

Problems

1. Suppose the lady samples 10 cups of tea, among which 5 had the teapoured first and 5 had the milk poured first.

a. What is the probability she correctly identifies all five cups which had thetea poured first?

b. What is the probability she correctly identifies exactly four of the cupswhich had the tea poured first?

c. What is the probability she correctly identifies four or more of the cupswhich had the tea poured first?

d. Would we reject the null hypothesis if she correctly identified exactly fourof the cups which had the tea poured first?

Dan Sloughter (Furman University) Mathematics of a Lady Tasting Tea November 2, 2006 9 / 9