Presented by Math 70 Statisticians: Libby Jones Nicole Miritello Carla Giugliano
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Transcript of Presented by Math 70 Statisticians: Libby Jones Nicole Miritello Carla Giugliano
Variables taken into Variables taken into consideration:consideration:
Weight (lbs)
Height (inches)
GenderAge
Vision
What we hope to learn from our data:What we hope to learn from our data:• Is the relationship between height and weight different across the sexes?
• Does adding age as an independent variable change the relationship between height and weight?
• Can we prove, statistically that male height is different from female height in high schoolers? Is weight statistically different?
•Is female weight more variable than male weight? Is male height more variable than female height?
• Is there a statistical difference between male and female mean vision scores?
Normality of Height
height
53 78
0
.522039
Normality of Weight
weight
83 350
0
.130854
Distribution of Age
15 16 17 18F M F M F M F M98 94 156 147 59 69 42 60
n = 725
Scatter Plot of Weight vs. Height
height
53 78
83
350
Regression of Weight vs. Height
MalesMalesn = 370
ˆ y =4.89h−174.19
t-statistic for h = 9.55p-value = 0.00
95% Confidence Interval: (3.88, 5.89)
Regression of Weight vs. Height
FemalesFemalesn = 355
ˆ y =5.24h−190.28
t-statistic for h = 8.24p-value = 0.00
95% Confidence Interval: (3.99, 6.49)
Regression of Weight vs. Heightwith a Dummy Variable for SEX
sex = 1 if malesex = 1 if malesex = 0 if femalesex = 0 if female
ˆ y =5.03h−6.79sex−177.12
ˆ y =5.03h−183.91ˆ y =5.03h−177.12
t-statistic for h = 12.58p-value = 0.00
95% Confidence Interval: (4.25, 5.82)
<= males
<= females
t-statistic for sex = -2.34p-value = 0.02
95% Confidence Interval: (-12.49, -1.10)
Regression of Weight vs. Heightwith a Dummy Variable for sex in the slope
sex = 1 if malesex = 1 if malesex = 0 if femalesex = 0 if female
ˆ y =5.10h−.10h⋅sex−181.25
ˆ y =5.00h−181.25ˆ y =5.10h−181.25
t-statistic for h = 12.33p-value = 0.00
95% Confidence Interval: (4.28, 5.91)
<= males
<= females
t-statistic for h*sex = -2.36p-value = 0.02
95% Confidence Interval: (-.19, -.02)
Taking into account gender, Taking into account gender, we now predict (weight)we now predict (weight)with a 95% Confidence Interval of:with a 95% Confidence Interval of:
ˆ y
(148.07, (148.07, 150.52)150.52)
(148.07, (148.07, 150.52)150.52)
Graph of Weight vs. Height and yhatGraph of Weight vs. Height and yhat
height
weight Fitted values
53 78
83
350
Testing the mean weightweight for femalesfemales in high school:
H0 : μ =140
HA : μ >140vs.
t = 1.7024P > t =
0.04
∴ Reject the Null
Note: the sample mean is 143.39
Testing the mean weightweight for malesmales in high school:
H0 : μ =160
HA : μ ≠160vs.
t = -2.62P > |t| =
0.01
∴ Reject the Null
Note: the sample mean is 154.96
Testing the mean heightheight for femalesfemales in high school:
H0 : μ =6 ′ ′ 5
HA : μ ≠6 ′ ′ 5 vs.
t = -8.52P > |t|=
0.00
∴ Reject the Null
Note: the sample height is 63.70
Testing the mean heightheight for malesmales in high school:
H0 : μ =6 ′ ′ 6
HA : μ >6 ′ ′ 6 vs.
t = 7.69P > t =
0.00
∴ Reject the Null
Note: the sample height is 67.35
Regression of Weight vs. Heightwith a Dummy Variable for AGE
AgeAge11 = 15 yr olds = 15 yr olds Age Age22 = 16 yr olds Age = 16 yr olds Age33 = 17 yr olds Age = 17 yr olds Age44 = = 18 yr olds18 yr olds
ˆ y =4.85h−1.35age2 +3.37age3 +1.40age4 −171.78
ˆ y =4.85h−171.78
ˆ y =4.85h−173.13ˆ y =4.85h−168.41
ˆ y =4.85h−170.38
<= AgeAge11
<= AgeAge22
<= AgeAge33
<= AgeAge44
t-stat for h= 9.15, Age2=-.30, Age3=.63, Age4=.25p-value for h= 0.00, Age2=0.76, Age3=0.53, Age4=0.80
MalesMales
Regression of Weight vs. Heightwith a Dummy Variable for AGE
AgeAge11 = 15 yr olds = 15 yr olds Age Age22 = 16 yr olds Age = 16 yr olds Age33 = 17 yr olds Age = 17 yr olds Age44 = = 18 yr olds18 yr olds
<= AgeAge11
FemalesFemales
<= AgeAge22
<= AgeAge33
<= AgeAge44
t-stat for h= 8.13, Age2=1.22, Age3= 0.71, Age4= 1.63p-value for h= 0.00, Age2= 0.23, Age3= 0.48, Age4= 0.10
H0 : β1 =β2
HA : β1 ≠β2vs.
t forage22 = 0.04P > |t|= 0.97 ∴ Accept the Null
ˆ y =β0 +β1(h+age2)+(β2 −β1)age2 +β3age3 +β4age4
For females:For females:
ˆ y =5.19h+5.39age2 +4.01age3 +10.37age4 −191.31
H0 : β1 =β1*
∴ Accept the Null
t =ˆ β 1 −ˆ β 1
*
s1
(xi −x )2∑+
1(xi
* −x * )2∑
Test:Test:
−t.05,(m+n)−4 ≤t ≤t.05,(m+n)−4
t = -.4316
−1.64≤t ≤1.64
ˆ y =5.19h+5.39age2 +4.01age3 +10.37age4 −191.31
ˆ y =4.85h−1.35age2 +3.37age3 +1.40age4 −171.78 <= males
<= females
Where: beta1 is for males beta1* is for females
Regression of Weight vs. Height, Sex, AgeAgeAge11 = 15 yr olds = 15 yr olds Age Age22 = 16 yr olds Age = 16 yr olds Age33 = 17 yr olds Age = 17 yr olds Age44 = = 18 yr olds18 yr olds
ˆ y =4.93h+2.06age2 +3.89age3 +5.45age4 −6.67sex−173.14
<= AgeAge11
<= AgeAge22
<= AgeAge33
<= AgeAge44
t-stat: h= 12.15Age2=0.66Age3=1.00Age4=1.30Sex=-2.29
sex = 1 if male, sex = 0 if femalesex = 1 if male, sex = 0 if female
malemale femalefemale
ˆ y =4.93h−179.81
ˆ y =4.93h−177.75
ˆ y =4.93h−175.92
ˆ y =4.93h−174.36
ˆ y =4.93h−173.14
ˆ y =4.93h−171.08
ˆ y =4.93h−169.25
ˆ y =4.93h−167.69
p-value: h= 0.00Age2=0.51Age3=0.32Age4=0.19Sex=0.02
Taking into account age, we Taking into account age, we now predict yhat with a 95% now predict yhat with a 95% Confidence Interval of:Confidence Interval of:
(148.06, (148.06, 150.53)150.53)
(148.06, (148.06, 150.53)150.53)
Graph of Weight vs. Height, Age, Graph of Weight vs. Height, Age, Sex and yhatSex and yhat
height
weight Fitted values
53 78
83
350
Testing VarianceVariance in weightweight across gender:
vs.
F(354,369) ~0.79<1.03<1.24
∴ Accept the Null
H0 :σ w,m2 =σ w, f
2 HA:σ w,m2 ≠σ w, f
2
F=sw, f2
sw,m2 =1.03
Testing differences in Testing differences in meanmean weightweight across sexes:across sexes:
vs.
t = -4.182P > |t| =
0.000
∴ Reject the Null
H0 :μw, f =μw, m HA:μw, f < μw, m
Testing VarianceVariance in heightheight across gender:
vs.
F(354,369) ~0.84>0.73
∴ Reject the Null
H0 :σ h, m2 =σ h, f
2 HA:σ h,m2 >σh, f
2
F=sh, f2
sh, m2 =0.73
Since variances are not equal, we cannot test for the equality of mean height across the sexes.
ANOVAANOVATesting whether weight is dependent on age or not
H0 : μw,15 =μw,16 =μw,17 =μw,18
F-statistic: 3.94Probability > F: 0.01
Reject the Null∴
Testing VarianceVariance in visionvision across gender:
vs.
F(354,369) ~(.813, 1.229)2.0172 > 1.229
∴ Reject the Null
H0 :σ vis,m2 =σ vis, f
2HA:σ vis,m
2 ≠σ vis, f2
F=svis, f2
svis,m2 =2.0172
Since variances are not equal, we cannot check for equality of mean vision across the sexes.
vs.
F(97,41) ~(.0610<.862<1.733)
∴ Accept the Null
H0 :σ152 =σ18
2HA:σ15
2 ≠σ182
F=s152
s182 =.86190
Testing VarianceVariance in visionvision for 15 and 18 yr
olds:FemaleFemaless
Testing differences in Testing differences in meanmean visionvision for 15 and 18 year olds:for 15 and 18 year olds:
H0 : μ15 =μ18 HA : μ15 ≠μ18vs.
t = -0.64P > |t| =
0.522
∴ Accept the Null
FemalesFemales
vs.
F(93,59) ~(0.636<1.553<1.612)
∴ Accept the Null
H0 :σ152 =σ18
2HA:σ15
2 ≠σ182
F=s152
s182 =1.5525
Testing VarianceVariance in visionvision for 15 and 18 yr
olds:MalesMales
Testing differences in Testing differences in meanmean visionvision for 15 and 18 year olds:for 15 and 18 year olds:
H0 : μ15 =μ18 HA : μ15 ≠μ18vs.
t = 0.42P > |t| = 0.67
∴ Accept the Null
MalesMales
Possible Errors:Possible Errors:• R2 0.20 for all regressions
– Weight dependent on other factors– Diet,exercise, genetics, abnormal health conditions, muscle
to fat ratio, etc.
• Age variable approximates mean age from grade level
• Weight and height data may be overestimates due to method of collection
• Almost half of data is for 16 year old students
• Rounding errors in height and weight measurements
• Scale only measured up to 300 lbs
Conclusions:Conclusions:• Sex is statistically significant in determining the relationship
between height and weight• Age, as an independent variable, is statistically significant in
determining the relationship between height and weight for both males and females
• Mean female weight is less than mean male weight at the 95% level of significance
• At the 95% level of significance, variance of weight in females does not differ from that of males
• Male height is more variable than that of females at the 95%
level of significance • Because variance in vision is not equal between males and
females, we could not compare male and female mean vision scores by an unpaired t-test