M140 Introducing Statistics

• This tutorial will begin at 7:30pm for about 70 minutes. Correlations & • I will be recording this presentation so please let me know if you have any questions or concerns about this. • Mics will be muted until the end of the tutorial, Confidence when I will also stop recording. • Feel free to use the Chat Box at any time! • I will email slides out after the tutorial and post Intervals them on my OU blog.

• Things you might need today: Dr Jason Verrall • M140 Book 4 [email protected] • Calculator 07311 188 800 • Note-taking equipment • Drink of your choice

Tutorials are enhanced by your interaction Please vote in the polls and ask questions! M140 Introducing Statistics Correlations & Confidence Intervals

• Scatter Plots & Correlations • r values • Correlation & Causation • Confidence Intervals Computer Book has • Prediction Intervals detailed instructions!

• Plus Minitab examples But you don’t need Minitab running today

2 Scatter Plots

Scatter Plots & Correlations 1) Which is the positive relationship? A Variable 2 Variable

Variable 1 B Variable 2 Variable

Variable 1 3 Scatter Plots

Scatter Plots & Correlations 1) Which is the positive relationship? A Variable 2 Variable

Variable 1 B Variable 2 Variable

Variable 1 4 Scatter Plots Scatter Plots & Correlations

Scatterplots are notoriously tricky to interpret unless very clear, particularly the strength of any relationship

5 Scatter Plots Scatter Plots & Correlations

Scatterplots are notoriously tricky to interpret unless very clear, particularly the strength of any relationship

How can we improve this?

6 Scatter Plots Scatter Plots & Correlations

We can use a measure of correlation to Scatterplots are notoriously tricky to objectively assess the direction and interpret unless very clear, particularly strength of any relationship between the strength of any relationship variables

How can we improve this?

7 Scatter Plots Scatter Plots & Correlations

We can use a measure of correlation to Scatterplots are notoriously tricky to objectively assess the direction and interpret unless very clear, particularly strength of any relationship between the strength of any relationship variables

How can we improve this? Pearson product- moment correlation coefficient

8 Correlation Coefficient Book 4 pp.98-107 Pearson’s Correlation Coefficient, r

9 Correlation Coefficient Book 4 pp.98-107 Pearson’s Correlation Coefficient, r

Covariance between X & Y variables

10 Correlation Coefficient Book 4 pp.98-107 Pearson’s Correlation Coefficient, r

Covariance between X & Y variables

Standard deviation of each variable

11 Correlation Coefficient Book 4 pp.98-107 Pearson’s Correlation Coefficient, r

Covariance between X Difference between & Y variables each point and the variable mean

Standard deviation of each variable

12 Correlation Coefficient Book 4 pp.98-107 Pearson’s Correlation Coefficient, r

Covariance between X Difference between & Y variables each point and the variable mean

Standard deviation of X Standard deviation of & Y (n is the same for each variable both variables and cancels out)

13 Correlation Coefficient Book 4 pp.98-107 Pearson’s Correlation Coefficient, r

Covariance between X Difference between r is always between & Y variables each point and the -1 and 1 variable mean

Standard deviation of X Standard deviation of & Y (n is the same for each variable both variables and cancels out)

14 Correlation Coefficient Book 4 pp.98-107 Worked Example of r pp. 104-105

15 Correlation Coefficient Book 4 pp.98-107 Worked Example of r pp. 104-105 Quantity Value S x 508 S y 355.6 S x2 37008.1 S y2 18600.62 S xy 26030.33

16 Correlation Coefficient Book 4 pp.98-107 Worked Example of r pp. 104-105 Quantity Value S x 508 S y 355.6 S x2 37008.1 S y2 18600.62 S xy 26030.33

17 Correlation Coefficient Book 4 pp.98-107 Worked Example of r pp. 104-105 Quantity Value S x 508 S y 355.6 S x2 37008.1 S y2 18600.62 S xy 26030.33

Quantity Value S (푥 − 푥̅) 141.814 S (푦 − 푦) 536.140 S(푥 − 푥̅)(푦 − 푦) 223.930

18 Correlation Coefficient Book 4 pp.98-107 Worked Example of r 2) What is r?

Quantity Value S (푥 − 푥̅) 141.814 S (푦 − 푦) 536.140 S(푥 − 푥̅)(푦 − 푦) 223.930

19 Correlation Coefficient Book 4 pp.98-107 Worked Example of r 2) What is r?

223.93 223.93 푟 = = 141.814 × 536.14 275.74

= ퟎ. ퟖퟏퟐ

Quantity Value S (푥 − 푥̅) 141.814 S (푦 − 푦) 536.140 S(푥 − 푥̅)(푦 − 푦) 223.930

20 Correlation Coefficient Book 4 pp.98-107 Worked Example of r 2) What is r?

223.93 223.93 푟 = = 141.814 × 536.14 275.74

= ퟎ. ퟖퟏퟐ

Quantity Value S (푥 − 푥̅) 141.814 S (푦 − 푦) 536.140 S(푥 − 푥̅)(푦 − 푦) 223.930

21 Correlation Coefficient Samples of r

22 Correlation Coefficient Samples of r https://en.wikipedia.org/wiki/Pearson_correlation_coefficient#/media/File:Correlation_examples2.svg 23 Correlation Coefficient Samples of r

24 Correlation Coefficient Samples of r

25 Correlation Coefficient Samples of r

26 Correlation Coefficient Samples of r

Pearson Correlation is best for linear relationships but can help with other relationships

27 Correlation Coefficient

3) For which graph will Samples of r correlation be informative about strength of relationship? A

B

28 Correlation Coefficient

3) For which graph will Samples of r correlation be informative about strength of relationship? A

Non-linear relationship and correlation unlikely to be helpful

B

Relationship is ‘more’ linear and correlation more likely to be informative (although not definitive)

29 Minitab Comp Bk pp. 81-91 Correlations in Minitab

30 Minitab Comp Bk pp. 81-91 Correlations in Minitab

Stat → Basic Statistics → Correlation..

1. Open Correlation dialogue box

31 Minitab Comp Bk pp. Correlations in Minitab

Stat → Basic Statistics → Correlation..

1. Open Correlation dialogue box

32 Minitab Comp Bk pp. 81-91 Correlations in Minitab

Stat → Basic Statistics → Correlation..

1. Open Correlation dialogue box 2. Select the variable pair to be analysed

33 Minitab Comp Bk pp. 81-91 Correlations in Minitab

Stat → Basic Statistics → Correlation..

1. Open Correlation dialogue box 2. Select the variable pair to be analysed

34 Minitab Comp Bk pp. 81-91 Correlations in Minitab

Stat → Basic Statistics → Correlation..

1. Open Correlation dialogue box 2. Select the variable pair to be analysed 3. Select Pearson in Options

35 Minitab Comp Bk pp. 81-91 Correlations in Minitab

Stat → Basic Statistics → Correlation..

1. Open Correlation dialogue box 2. Select the variable pair to be analysed 3. Select Pearson in Options

36 Minitab Comp Bk pp. 81-91 Correlations in Minitab

Stat → Basic Statistics → Correlation..

1. Open Correlation dialogue box 2. Select the variable pair to be analysed 3. Select Pearson in Options 4. Review output

37 Minitab Comp Bk pp. 81-91 Correlations in Minitab

Stat → Basic Statistics → Correlation..

1. Open Correlation dialogue box 2. Select the variable pair to be analysed 3. Select Pearson in Options 4. Review output

38 Correlation & Causation Does the Nose Cause the Tail?

39 Correlation & Causation Does the Nose Cause the Tail?

40 Correlation & Causation Does the Nose Cause the Tail?

A strong relationship does not indicate that A causes B

41 Correlation & Causation Does the Nose Cause the Tail?

A strong relationship does It only indicates that A and not indicate that A causes B are associated B

42 Correlation & Causation Does the Nose Cause the Tail?

Further analysis is A strong relationship does necessary to confirm It only indicates that A and not indicate that A causes causation, including B are associated B considering the mechanism of action

43 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

44 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

We use statistics such as the mean and median to summarise a group of data Sample mean = 푥̅

45 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals Population mean = 휇

We use statistics such as the mean and median to summarise a group of data Sample mean = 푥̅

However, we might want to calculate properties about a population of data which we don’t have

46 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals Population mean = 휇 Point estimates We use statistics such as the mean and median to summarise a group of data Sample mean = 푥̅

However, we might want to calculate properties about a population of data which we don’t have

47 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals Population mean = 휇 Point estimates We use statistics such as the mean and median to summarise a group of data Sample mean = 푥̅

However, we might want to calculate properties about a population of data which we don’t have

We can use 푥̅ to estimate 휇

48 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals Population mean = 휇 Point estimates We use statistics such as the mean and median to summarise a group of data Sample mean = 푥̅

However, we might want to calculate properties about a population of data which we don’t have

How can the uncertainty about this We can use 푥̅ to estimate 휇 estimate be quantified?

49 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals Population mean = 휇 Point estimates We use statistics such as the mean and median to summarise a group of data Sample mean = 푥̅

However, we might want to calculate properties about a population of data which we don’t have

How can the Confidence Intervals! uncertainty about this We can use 푥̅ to estimate 휇 estimate be quantified?

50 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

Population mean = 20 Sample mean = 15

51 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

Population mean = 20 Sample mean = 15

As we have only calculated an estimate, there must be a way of quantifying the uncertainty about this estimate, or how confident we are in our estimate

52 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

95% confidence

53 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

95% confidence

54 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

95% confidence

z=-1.96 z=1.96

55 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

95% confidence We can get confidence about the location of our point estimates by using a method similar to a Z-test – effectively reversing the test to tell us the values of our estimate that are the bounds on our critical region

z=-1.96 z=1.96

56 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals

95% confidence We can get confidence about the location of our point estimates by using a method similar to a Z-test – effectively reversing the test to tell us the values of our estimate that are the bounds on our critical region

−1.96<푧<1.96 →푎<휇<푏

z=-1.96 z=1.96

57 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

58 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

퐻: μ = 퐴 & 퐻: μ ≠ 퐴

푥̅ = 50.72,푠 = 17.2,푛 = 56

What values of A would be rejected in a test at the 5% significance level, and what values would we fail to reject?

59 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

퐻: μ = 퐴 & 퐻: μ ≠ 퐴

푥̅ = 50.72,푠 = 17.2,푛 = 56

What values of A would be rejected in a test at the 5% significance level, and what values would we fail to reject?

60 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

퐻: μ = 퐴 & 퐻: μ ≠ 퐴 Normal distribution so.. 푥̅ − 퐴 50.7 − 퐴 50.7 − 퐴 푥̅ = 50.72,푠 = 17.2,푛 = 56 푧 = = = s 17.2 2.3 푛 What values of A would be rejected in a test at the 5% 56 significance level, and what values would we fail to reject?

61 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

퐻: μ = 퐴 & 퐻: μ ≠ 퐴 Normal distribution so.. 푥̅ − 퐴 50.7 − 퐴 50.7 − 퐴 푥̅ = 50.72,푠 = 17.2,푛 = 56 푧 = = = s 17.2 2.3 푛 What values of A would be rejected in a test at the 5% 56 significance level, and what values would we fail to reject? For 95% significance, we would fail to reject at −1.96 < 푧 < 1.96, which gives us a clue:

62 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

퐻: μ = 퐴 & 퐻: μ ≠ 퐴 Normal distribution so.. 푥̅ − 퐴 50.7 − 퐴 50.7 − 퐴 푥̅ = 50.72,푠 = 17.2,푛 = 56 푧 = = = s 17.2 2.3 푛 What values of A would be rejected in a test at the 5% 56 significance level, and what values would we fail to reject? For 95% significance, we would fail to reject at −1.96 < 푧 < 1.96, which gives us a clue:

50.7 − 퐴 < 1.96 ⇒ 50.7 − 1.96 × 2.3 < 퐴 2.3

63 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

퐻: μ = 퐴 & 퐻: μ ≠ 퐴 Normal distribution so.. 푥̅ − 퐴 50.7 − 퐴 50.7 − 퐴 푥̅ = 50.72,푠 = 17.2,푛 = 56 푧 = = = s 17.2 2.3 푛 What values of A would be rejected in a test at the 5% 56 significance level, and what values would we fail to reject? For 95% significance, we would fail to reject at −1.96 < 푧 < 1.96, which gives us a clue:

50.7 − 퐴 < 1.96 ⇒ 50.7 − 1.96 × 2.3 < 퐴 2.3

64 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

퐻: μ = 퐴 & 퐻: μ ≠ 퐴 Normal distribution so.. 푥̅ − 퐴 50.7 − 퐴 50.7 − 퐴 푥̅ = 50.72,푠 = 17.2,푛 = 56 푧 = = = s 17.2 2.3 푛 What values of A would be rejected in a test at the 5% 56 significance level, and what values would we fail to reject? For 95% significance, we would fail to reject at −1.96 < 푧 < 1.96, which gives us a clue:

50.7 − 퐴 < 1.96 ⇒ 50.7 − 1.96 × 2.3 < 퐴 2.3

Using symmetry of Normal PDF…

65 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

66 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

46.19 < 퐴 < 55.21

67 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

46.19 < 퐴 < 55.21

95% confidence interval for A is (46.19, 55.21)

68 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

CI for a population mean

푥̅ − 퐴 푠 푠 푧 = s → 푥̅ − 푧 × < 퐴 < 푥̅ + 푧 × 46.19 < 퐴 < 55.21 푛 푛 푛

95% confidence interval for A is (46.19, 55.21)

69 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

CI for a population mean

푥̅ − 퐴 푠 푠 푧 = s → 푥̅ − 푧 × < 퐴 < 푥̅ + 푧 × 46.19 < 퐴 < 55.21 푛 푛 푛

95% confidence interval for A is (46.19, 55.21)

70 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals – Worked Example

CI for a population mean

푥̅ − 퐴 푠 푠 푧 = s → 푥̅ − 푧 × < 퐴 < 푥̅ + 푧 × 46.19 < 퐴 < 55.21 푛 푛 푛

As for Z-tests, make sure n>25!

95% confidence interval for A is (46.19, 55.21)

71 Confidence Intervals Book 4 pp. 129-138

4) What is the 99% Confidence Intervals confidence interval to 1dp?

72 Confidence Intervals Book 4 pp. 129-138

4) What is the 99% Confidence Intervals confidence interval to 1dp?

12.8 48.3 ± 2.58 × = 48.3 ± 3.669 81

73 Confidence Intervals Book 4 pp. 129-138 CIs for Point Estimates - Context

About 95% of the possible random samples we could select will give rise to a 95% confidence interval that does include the population mean.

74 Confidence Intervals Book 4 pp. 129-138 CIs for Point Estimates - Context

About 95% of the possible random samples we could select will give rise to a 95% confidence interval that does include the population mean.

About 5% of possible random samples that might be selected will give rise to a 95% confidence interval that does not include the population mean.

75 Confidence Intervals Book 4 pp. 129-138 CIs for Point Estimates - Context

About 95% of the possible random samples we could select will give rise to a 95% confidence interval that does include the population mean.

About 5% of possible random samples that might be selected will give rise to a 95% confidence interval that does not include the population mean.

A 95% CI includes the population mean 95% of the time!

76 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals In Reverse

77 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals In Reverse

We can infer a point estimate from a confidence interval by identifying the midpoint

78 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals In Reverse

We can infer a point estimate from a confidence interval by identifying the midpoint

95% CI for A: (48.6, 55.2)

.. Midpoint:

79 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals In Reverse

We can infer a point estimate from a confidence interval by identifying the midpoint

95% CI for A: (48.6, 55.2)

.. Midpoint:

Point estimate

80 Minitab Computer Book pp. 84-85 CIs in Minitab

81 Minitab Computer Book pp. 84-85 CIs in Minitab

Stat → Basic Statistics → 1-Sample Z..

82 Minitab Computer Book pp. 84-85 CIs in Minitab

Stat → Basic Statistics → 1-Sample Z..

83 Minitab Computer Book pp. 84-85 CIs in Minitab

Stat → Basic Statistics → 1-Sample Z..

84 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals For 2 Samples

85 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals For 2 Samples

86 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals For 2 Samples

87 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals For 2 Samples

88 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals For 2 Samples

18.2 20.6 퐸푆퐸 = + = 2.478 100 150

89 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals For 2 Samples

18.2 20.6 퐸푆퐸 = + = 2.478 100 150

90 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals For 2 Samples

18.2 20.6 퐸푆퐸 = + = 2.478 100 150 95% 퐶퐼: 58.4 − 52.1 ± 1.96 × 퐸푆퐸

= 6.3 ± 4.857

91 Confidence Intervals Book 4 pp. 129-138 Confidence Intervals For 2 Samples

18.2 20.6 퐸푆퐸 = + = 2.478 100 150 95% 퐶퐼: 58.4 − 52.1 ± 1.96 × 퐸푆퐸

= 6.3 ± 4.857

95% CI for 휇 − 휇: (1.44, 11.16)

92 Confidence Intervals Book 4 pp. 139-150 Confidence Intervals From Fitted Lines

93 Confidence Intervals Book 4 pp. 139-150 Confidence Intervals From Fitted Lines Confidence Intervals Book 4 pp. 139-150 Confidence Intervals From Fitted Lines

95% confidence interval limit

95% confidence interval limit Confidence Intervals Book 4 pp. 139-150 Confidence Intervals From Fitted Lines

95% confidence interval limit

95% confidence interval limit

96 Confidence Intervals Book 4 pp. 139-150 Confidence Intervals From Fitted Lines

95% confidence interval limit

95% confidence interval limit

CI for the mean response

97 Confidence Intervals Book 4 pp. 139-150 Confidence Intervals for the Mean Response

Upper bound

Lower bound

Narrowest point ⇒ 푥̅ = 8

98 Confidence Intervals Book 4 pp. 139-150 Confidence Intervals for the Mean Response

푦 = 푎 + 푏푥 휇̂ Upper bound

Lower bound

99 Prediction Intervals Book 4 pp. 147-150 Prediction Intervals

100 Prediction Intervals Book 4 pp. 147-150 Prediction Intervals

101 Prediction Intervals Book 4 pp. 147-150 Prediction Intervals

Confidence intervals give us a range of values within which our point estimate will lie

102 Prediction Intervals Book 4 pp. 147-150 Prediction Intervals

Confidence intervals give us a range of values within which our point estimate will lie

Sometimes we also want a range of values within which individual observations will lie, based on the properties of our sample

103 Prediction Intervals Book 4 pp. 147-150 Prediction Intervals

Confidence intervals give us a range of values within which our point estimate will lie

Sometimes we also want a range of values within which individual observations will lie, based on the properties of our sample

Prediction intervals!

104 Prediction Intervals Book 4 pp. 147-150 Prediction Intervals

Confidence intervals give us a range of values within which our point estimate will lie

Sometimes we also want a range of values within which individual observations will lie, based on the properties of our sample

Prediction intervals!

105 Prediction Intervals Book 4 pp. 147-150 Prediction Intervals

Confidence intervals give us a range of values within which our point estimate will lie Upper CI bound

Sometimes we also want a range of values within which individual Lower CI bound observations will lie, based on the properties of our sample

Prediction intervals!

106 Prediction Intervals Book 4 pp. 147-150 Prediction Intervals

Confidence intervals give us a range of values within which our point estimate will lie Upper PI bound

Sometimes we also want a range of values within which individual observations will lie, based on the properties of our sample Lower PI bound Prediction intervals!

107 Prediction Intervals Book 4 pp. Prediction Intervals

108 Prediction Intervals Book 4 pp. Prediction Intervals

For a school with a pass rate in English of 35%, the 95% confidence interval for the mean response is 35.9% to 46.4%. Two different students quote the corresponding prediction interval.

109 Prediction Intervals Book 4 pp. Prediction Intervals

For a school with a pass rate in English of 35%, the 95% confidence interval for the mean response is 35.9% to 46.4%. Two different students quote the corresponding prediction interval. Ali says the interval is 37.8% to 44.5%, and Charlie says the interval is 17.3% to 55.0%. Explain why both Ali and Charlie must be mistaken.

110 Prediction Intervals Book 4 pp. Prediction Intervals

For a school with a pass rate in English of 35%, the 95% confidence interval for the mean response is 35.9% to 46.4%. Two different students quote the corresponding prediction interval. Ali says the interval is 37.8% to 44.5%, and Charlie says the interval is 17.3% to 55.0%. Explain why both Ali and Charlie must be mistaken.

Pass rate: 35%

95% CI: (35.9%, 46.4%)

Ali’s PI: (37.8%, 44.5%)

Charlie’s PI: (17.3%, 55%)

111 Prediction Intervals Book 4 pp. Prediction Intervals

For a school with a pass rate in English of 35%, the 95% confidence interval for the mean response is 35.9% to 46.4%. Two different students quote the corresponding prediction interval. Ali says the interval is 37.8% to 44.5%, and Charlie says the interval is 17.3% to 55.0%. Explain why both Ali and Charlie must be mistaken.

Pass rate: 35%

95% CI: (35.9%, 46.4%) Midpoint = 41.15%

Ali’s PI: (37.8%, 44.5%) Midpoint = 41.15%

Charlie’s PI: (17.3%, 55%) Midpoint = 36.15%

112 Prediction Intervals Book 4 pp. Prediction Intervals

For a school with a pass rate in English of 35%, the 95% confidence interval for the mean response is 35.9% to 46.4%. Two different students quote the corresponding prediction interval. Ali says the interval is 37.8% to 44.5%, and Charlie says the interval is 17.3% to 55.0%. Explain why both Ali and Charlie must be mistaken.

Pass rate: 35%

95% CI: (35.9%, 46.4%) Midpoint = 41.15% Range = 10.5

Ali’s PI: (37.8%, 44.5%) Midpoint = 41.15% Range = 6.7

Charlie’s PI: (17.3%, 55%) Midpoint = 36.15% Range = 37.7

113 Prediction Intervals Book 4 pp. Prediction Intervals

For a school with a pass rate in English of 35%, the 95% confidence interval for the mean response is 35.9% to 46.4%. Two different students quote the corresponding prediction interval. Ali says the interval is 37.8% to 44.5%, and Charlie says the interval is 17.3% to 55.0%. Explain why both Ali and Charlie must be mistaken.

Pass rate: 35%

95% CI: (35.9%, 46.4%) Midpoint = 41.15% Range = 10.5

Ali’s PI: (37.8%, 44.5%) Midpoint = 41.15% Range = 6.7 Ali’s PI range < 95% CI

Charlie’s PI: (17.3%, 55%) Midpoint = 36.15% Range = 37.7

114 Prediction Intervals Book 4 pp. Prediction Intervals

For a school with a pass rate in English of 35%, the 95% confidence interval for the mean response is 35.9% to 46.4%. Two different students quote the corresponding prediction interval. Ali says the interval is 37.8% to 44.5%, and Charlie says the interval is 17.3% to 55.0%. Explain why both Ali and Charlie must be mistaken.

Pass rate: 35%

95% CI: (35.9%, 46.4%) Midpoint = 41.15% Range = 10.5

Ali’s PI: (37.8%, 44.5%) Midpoint = 41.15% Range = 6.7 Ali’s PI range < 95% CI

Charlie’s PI: (17.3%, 55%) Midpoint = 36.15% Range = 37.7 Charlie’s PI midpoint < 95% CI midpoint

115 Minitab ks4subjects.mwx Computer Book pp. 89-90 CI & PI for the Mean Response

Stat → Regression → Regression → Fit..

What is the mean pass rate in Maths when the English pass rate is 50%?

116 Minitab ks4subjects.mwx Computer Book pp. 89-90 CI & PI for the Mean Response

Stat → Regression → Regression → Fit..

What is the mean pass rate in Maths when the English pass rate is 50%?

117 Minitab ks4subjects.mwx Computer Book pp. 89-90 CI & PI for the Mean Response

Stat → Regression → Regression → Fit..

What is the mean pass rate in Maths when the English pass rate is 50%?

118 Minitab ks4subjects.mwx Computer Book pp. 89-90 CI & PI for the Mean Response

Stat → Regression → Regression → Fit..

What is the mean pass rate in Maths when the English pass rate is 50%?

Having fit a line to some data, we can generate both the CI and PI at 95%

119 Minitab ks4subjects.mwx Computer Book pp. 89-90 CI & PI for the Mean Response

Stat → Regression → Regression → Predict..

What is the mean pass rate in Maths when the English pass rate is 50%?

120 Minitab ks4subjects.mwx Computer Book pp. 89-90 CI & PI for the Mean Response

Stat → Regression → Regression → Predict..

What is the mean pass rate in Maths when the English pass rate is 50%?

121 Minitab ks4subjects.mwx Plotting CIs & PIs

Stat → Regression → Regression → Fitted Line Plot..

122 Minitab ks4subjects.mwx Plotting CIs & PIs

Stat → Regression → Regression → Fitted Line Plot..

123 Minitab ks4subjects.mwx Plotting CIs & PIs

Stat → Regression → Regression → Fitted Line Plot..

124 Minitab ks4subjects.mwx Plotting CIs & PIs

Stat → Regression → Regression → Fitted Line Plot..

125 M140 Introducing Statistics Correlations & Confidence Intervals

• Scatter Plots & Correlations • r values • Correlation & Causation • Confidence Intervals Computer Book has • Prediction Intervals detailed instructions!

• Plus Minitab examples

126 Thank you! Any questions?

• OU Resources • M140 Course Books & Screencasts • https://learn2.open.ac.uk/course/view.php?id=208584&area=resources • M140 Student Forums • Downloadable full text e-books: • Error Analysis For Biologists: Statistics You Can Understand by Gierlinski, Marek • https://pmt-eu.hosted.exlibrisgroup.com/permalink/f/gvehrt/TN_cdi_askewsholts_vlebooks_9781119106906 • A Modern Introduction To Probability And Statistics : Understanding Why And How by Dekking, Michel • https://pmt-eu.hosted.exlibrisgroup.com/permalink/f/h21g24/44OPN_ALMA_DS5172530560002316 • Medical Statistics For Beginners by HK, Ramakrishna • https://pmt-eu.hosted.exlibrisgroup.com/permalink/f/gvehrt/TN_cdi_askewsholts_vlebooks_9789811019234 • Online Resources • Wikipedia Contact me: • CrossValidated • https://stats.stackexchange.com/ [email protected] • Minitab channel on YouTube: 07311 188 800 • https://www.youtube.com/user/MinitabInc My blog • Minitab help (https://learn1.open.ac.uk/mod/oublog/view. • https://support.minitab.com/en-us/minitab/19/ php?user=554822)

Recording will be available from M140-20J Online Tutorial Room https://learn2.open.ac.uk/mod/connecthosted/view.php?id=1644077&group=274133