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