Consistency of the Weather

Nicole Baratelle, Cara Barskey, Youjin Kwon

Topic• To determine the accuracy of

the weather• Used the normal and actual

temperature for specific days to see if difference in mean temperature

• Analyzed mean difference • Information from the National

Oceanic & Atmospheric Administration (NOAA) – National Weather Service

Procedure • Mean Difference 1-Sample T-Test/

Confidence Interval

– SRS of 50 Days in 2008 (numbered, software)

– Found average temperature on each day (avg. observed & avg. normal)

– Mean Difference ( observed-normal )

– Analyzed accuracy of weather by performing 1-Sample T-test and Confidence Interval for mean difference

Procedure (cont.) • Mean Difference 2-Sample T-

Test/ Confidence Interval – Two SRS’s Summer Days

(May-Oct.) & Winter Days (Nov.-April)

– For each day, found observed and normal avg. temperature

– Mean Difference ( observed-normal ) for each sample

– Compared Accuracy of Summer and Winter months using 2-sample T-test for mean difference

• Monthly weather predictions made in Farmers’ Almanac since 1818 with 80%-85% accuracy– Predictions made 2yrs in

advance• NOAA’s National Weather Service

predicting weather since 1870– Weekly weather predictions – Able to calculate daily “normal”– Daily Almanac gives temperature,

precipitation, snow depth and the normals

– Use Doppler Weather Radar to predict weather patterns

Background

Data

Collection 3

observed expected <new>

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

48.5 48

42.5 40

75.0 72

80.5 78

78.0 78

45.0 35

42.0 49

80.5 78

63.5 48

78.0 76

71.0 74

69.0 71

76.5 78

43.0 43

38.0 34

77.5 77

40.5 47

84.0 78

38.5 43

71.0 65

44.0 32

80.5 78

66.5 67

44.5 42

69.5 77

48.0 39

74.5 74

50.0 52

28.5 33

55.0 62

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

48.5 48

42.5 40

75.0 72

80.5 78

78.0 78

45.0 35

42.0 49

80.5 78

63.5 48

78.0 76

71.0 74

69.0 71

76.5 78

43.0 43

38.0 34

77.5 77

40.5 47

84.0 78

38.5 43

71.0 65

44.0 32

80.5 78

66.5 67

44.5 42

69.5 77

48.0 39

74.5 74

50.0 52

28.5 33

55.0 62

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

48.0 39

74.5 74

50.0 52

28.5 33

55.0 62

31.0 40

28.0 32

80.0 72

52.5 56

79.5 75

56.0 60

59.0 50

42.5 53

65.5 54

61.5 67

41.5 44

72.5 75

77.5 78

75.5 77

31.0 46

60.5 64

68.5 54

60.0 62

69.0 68

60.5 55

Yearly Average Temperatures

DataWinter Average Temperatures observed expected <new>

1

2

3

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5

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9

10

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12

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14

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16

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20

21

22

23

24

25

26

27

28

29

30

31

32

54.5 50

53.0 53

38.0 34

64.0 51

29.5 36

29.5 32

53.5 46

38.5 43

59.5 57

35.0 32

59.0 50

40.0 34

28.0 36

32.0 45

37.0 36

49.0 44

26.0 32

58.0 55

54.5 37

34.5 46

57.0 52

47.5 33

53.5 51

42.0 46

31.0 46

35.0 40

53.5 33

33.5 32

26.0 32

40.5 40

42.5 40

58.5 54

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

49.0 44

26.0 32

58.0 55

54.5 37

34.5 46

57.0 52

47.5 33

53.5 51

42.0 46

31.0 46

35.0 40

53.5 33

33.5 32

26.0 32

40.5 40

42.5 40

58.5 54

21.5 33

41.5 32

37.5 33

39.0 32

43.5 45

36.5 35

55.0 50

36.0 41

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2

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5

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10

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12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

54.5 50

53.0 53

38.0 34

64.0 51

29.5 36

29.5 32

53.5 46

38.5 43

59.5 57

35.0 32

59.0 50

40.0 34

28.0 36

32.0 45

37.0 36

49.0 44

26.0 32

58.0 55

54.5 37

34.5 46

57.0 52

47.5 33

53.5 51

42.0 46

31.0 46

35.0 40

53.5 33

33.5 32

26.0 32

40.5 40

42.5 40

58.5 54

DataSummer Average Temperatures Collection 5

observed expected <new>

1

2

3

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5

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15

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18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

76.5 78

43.5 53

80.0 72

86.0 71

52.5 56

53.0 58

63.5 63

72.0 64

74.5 73

80.5 78

74.0 72

67.0 73

63.0 67

43.5 53

80.5 78

54.0 65

80.5 78

75.0 75

82.5 75

81.0 75

49.5 56

67.0 58

62.0 59

72.0 67

68.0 58

70.0 61

72.5 77

48.0 55

85.0 78

79.0 78

56.0 60

76.5 78

78.5 78

61.5 67

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

82.5 75

81.0 75

49.5 56

67.0 58

62.0 59

72.0 67

68.0 58

70.0 61

72.5 77

48.0 55

85.0 78

79.0 78

56.0 60

76.5 78

78.5 78

61.5 67

70.0 69

65.0 66

76.5 78

57.5 59

78.5 78

56.5 65

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2

3

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5

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14

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16

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18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

76.5 78

43.5 53

80.0 72

86.0 71

52.5 56

53.0 58

63.5 63

72.0 64

74.5 73

80.5 78

74.0 72

67.0 73

63.0 67

43.5 53

80.5 78

54.0 65

80.5 78

75.0 75

82.5 75

81.0 75

49.5 56

67.0 58

62.0 59

72.0 67

68.0 58

70.0 61

72.5 77

48.0 55

85.0 78

79.0 78

56.0 60

76.5 78

78.5 78

61.5 67

Exploratory Analysis

2

4

6

8

10

observed

0 20 40 60 80 100

Collection 3 Histogram

2

4

6

8

10

12

expected

0 10 20 30 40 50 60 70 80 90

Collection 3 Histogram

Collection 3

observed

59.51

50

16.5442

2.3397

0

28

44

61

75

84

S1 = meanS2 = countS3 = stdDevS4 = stdErrorS5 = missing countS6 = minS7 = Q1S8 = medianS9 = Q3S10 = max

Collection 3

expected

59

50

15.5839

2.20389

0

32

46

61

75

78

S1 = meanS2 = countS3 = stdDevS4 = stdErrorS5 = missing countS6 = minS7 = Q1S8 = medianS9 = Q3S10 = max

Yearly Average Temperatures

Conclusion

•Statistical summaries are similar but not the same

•Slight difference between the expected and normal temperatures

1

2

3

4

5

6

7

observed

10 20 30 40 50 60 70

Collection 4 Histogram

1

2

3

4

5

6

7

8

expected

10 20 30 40 50 60

Collection 4 Histogram

Collection 4

observed

42.6

40

11.2576

1.77998

0

21.5

34.75

40.25

53.5

64

S1 = meanS2 = countS3 = stdDevS4 = stdErrorS5 = missing countS6 = minS7 = Q1S8 = medianS9 = Q3S10 = max

Collection 4

expected

41.225

40

8.04311

1.27173

0

32

33

40

48

57

S1 = meanS2 = countS3 = stdDevS4 = stdErrorS5 = missing countS6 = minS7 = Q1S8 = medianS9 = Q3S10 = max

1

2

3

4

5

6

observed

40 50 60 70 80 90

Collection 5 Histogram

2

4

6

8

10

12

expected

40 50 60 70 80

Collection 5 Histogram

Collection 5

observed

68.3125

40

11.845

1.87286

0

43.5

59.5

71

78.5

86

S1 = meanS2 = countS3 = stdDevS4 = stdErrorS5 = missing countS6 = minS7 = Q1S8 = medianS9 = Q3S10 = max

Collection 5

expected

68.05

40

8.57531

1.35588

0

53

59.5

68

77.5

78

S1 = meanS2 = countS3 = stdDevS4 = stdErrorS5 = missing countS6 = minS7 = Q1S8 = medianS9 = Q3S10 = max

Conclusion

•Winter and summer expected graphs are both higher temperatures as compared to the observed graphs

•Therefore, both winter and summer expected over predicted •No difference

Inferential Analysis

Assumptions1.SRS yes 2.Normal population 50 or

Mean Difference 1-Sample T-Test

30

dn 30

Mean Difference 1-Sample T-Test (cont.)

05.010140.1)608.8(

608.8

50

05.0

0:

0:

11

tP

nsx

t

n

H

H

normalobserved

d

d

dd

da

do

d

Conclusion: We reject in favor of because the P-value of 8.608 < . We have sufficient evidence that the absolute value of differences between observed and expected temperatures is greater than zero.

**Predicted and observed temperatures are different; therefore NOAA did not do great job predicting the temperatures in 2008.**

oH aH05.0

Mean Difference 1-Sample T-Test (cont.)

Mean Difference 1-Sample Confidence Interval

We are 95% confident that the absolute value of mean differences between observed and expected temperatures is between 3.8097 and 6.1303°F; Zero is not in the interval so there is a significant difference between observed and predicted temperatures.

)1303.6,8097.3()/(* ddd nstx

Mean Difference 2-Sample T-Test

Assumptions1. 2 independent SRS yes 2. 2 Normal populations

or 40

sn

30

wn3030

3040

Mean Difference 2-Sample T-Test (cont.)

Summer: May to OctoberWinter: November to April normalobserved

normalobserved

w

s

05.00564.0)198.72605267.1(

605267.1)()(

40

40

05.0

:

:

22

dftP

ns

ns

xxt

n

n

H

H

w

w

s

s

ws

s

s

wsa

wso

• Conclusion: We fail to reject because the p-value of 0.0564 > . We have sufficient evidence that the mean difference between observed and predicted temperatures in the summer is equal to the mean difference between observed and predicted temperatures in the winter.

**Incorrect predictions in summer months are equal to the incorrect predictions in winter

months.**

Mean Difference 2-Sample T-Test (cont.)

oH05.0

Mean Difference 2-Sample Confidence Interval

0.37172) (-3.447,)()(

)(22

* w

w

s

sws n

s

n

stxx

We are 95% confident that the difference between the mean difference between observed and predicted temperatures in the summer and the mean difference between observed and predicted temperatures in the winter is between -3.447 and 0.37172°F. We conclude that the mean difference between observed and predicted temperatures in the summer is equal to the mean difference between observed and predicted temperatures in the winter since the interval contains 0, which makes two mean differences equal to each other.

• Yearly Average Temperatures– Thought it would be off – Weather is never perfect– Climate changing

• Summer vs. Winter – Surprised that they are same mean difference– Winter temperatures seems to be more

changing

Personal Opinions/Conclusions

• Use this to know how much you can trust the weather forecast

• Relation to global warming– See if there is a pattern

of temperatures being warmer/cooler than normal

Application to population

• Used the average daily temperature– If outstanding high or low,

avg. thrown off

• Used data collected from a internet site – Legitimate site but can

never be certain

Bias/Error

Activity

Date (2008) Class Predictions1/6

7/8

5/4

9/10

10/11

1/22

7/24

4/9

12/1

7/16

3/29

3/20

5/6

11/10

4/3

6/7

6/20

5/23

1/15

8/9