1. Measuring body fat is not simple. Muscle and bone are denser than fat so an estimate of bodydensitycanbeusedtoestimate theproportionoffatinthebody.Measuringsomeone's weightiseasybutvolumeismoredifficult.Onemethodrequires submergingthebodyunderwaterinatankandmeasuringtheincreaseinthewaterlevel.Mostpeoplewouldprefernottobe submerged underwater to get a measure of body fat so we would like to have an easier method.In order to develop such a method,researchers recorded age, weight, height,and10 body circumference measurements for 252 men.Each man's percentage of body fat wasaccurately estimated by an underwater weighing technique. We wish to predict body fat usingjust the easy-to-record measurements. For simplicity, four variables selected for the study arelistedinTable.

Use data: Project-2-data.txt from Blackboard Variable Description X1 chest circumference (cm) X2 percent body fat using Brozek's equation X3 weight (lbs) (`Heavy' if greater than 180, `Light' otherwise) X4 height (inches) (`Tall' if greater than 70, `Short' otherwise)

1. [4pts]Foreachvariable,describeitasquantitativeorqualitative.
2. [8pts]Developappropriatedescriptivestatisticstosummarizethedata.
3. [6pts]Drawahistogram foreachcontinuousvariable.Interpret.
4. [4pts]Drawascatterplotbetweentwocontinuousvariables.
5. [6pts]Drawabargraphforeachcategoricalvariable.
6. [6pts]Foreachcontinuousvariable,computethe95%confidenceintervalforapopulationmean.
7. [6pts]Foreachcontinuousvariable,computethe95%confidenceintervalforapopulationvariance.
8. [6pts]Foreachcontinuousvariable,performastatisticaltesttoseewhetheritspopulationmeaniszeroornot.
9. [10pts]Foreachcategoricalvariable, performat-testonx₂.(thatis,testwhetherthere issignificantmeandifferenceonx₂betweentwolevelsofeachcategoricalvariable)
10. [5pts]Calculatecorrelationsbetween continuousvariables.
11. [5pts]Performastatisticaltesttoseewhetherthecorrelationbetween twocontinuousvariablesissignificantornot.
12. [10pts]Testindependencebetween categoricalvariables.

[10pts]Fitasimplelinear regressionmodelrelatingx₂tox₁(thatis,x₂isaresponsevariable).Writedownthefittedmodel,andinterpret.

Project-2-data.txt:

x1	x2	x3	x4
93.1	12.6	Light	Short
93.6	6.9	Light	Tall
95.8	24.6	Light	Short
101.8	10.9	Heavy	Tall
97.3	27.8	Heavy	Tall
104.5	20.6	Heavy	Tall
105.1	19	Heavy	Short
99.6	12.8	Light	Tall
100.9	5.1	Heavy	Tall
99.6	12	Heavy	Tall
101.5	7.5	Heavy	Tall
103.6	8.5	Heavy	Tall
102	20.5	Heavy	Short
104.1	20.8	Heavy	Tall
101.3	21.7	Heavy	Short
99.1	20.5	Light	Short
101.9	28.1	Heavy	Tall
107.6	22.4	Heavy	Tall
106.8	16.1	Heavy	Short
106.2	16.5	Heavy	Tall
103.3	19	Light	Short
111.4	15.3	Heavy	Short
86	15.7	Light	Short
86.7	17.6	Light	Short
90.2	14.2	Light	Short
89.6	4.6	Light	Tall
88.6	8.5	Light	Short
97.4	22.4	Light	Short
93.5	4.7	Light	Short
97.4	9.4	Light	Short
100.5	12.3	Heavy	Tall
93.5	6.5	Light	Tall
93	13.4	Light	Tall
111.7	20.9	Heavy	Tall
117	31.1	Heavy	Tall
118.5	38.2	Heavy	Short
106.5	23.6	Heavy	Short
105.6	27.5	Heavy	Short
136.2	33.8	Heavy	Tall
114.8	31.3	Heavy	Short
128.3	33.1	Heavy	Short
106	31.7	Heavy	Short
113.3	30.4	Heavy	Short
106.6	30.8	Heavy	Tall
85.1	8.4	Light	Short
96.6	14.1	Light	Tall
88.2	11.2	Light	Short
89.8	6.4	Light	Tall
92.3	13.4	Light	Short
83.4	5	Light	Short
90.2	10.7	Light	Tall
89.2	7.4	Light	Short
89.7	8.7	Light	Short
93.3	7.1	Light	Tall
87.6	4.9	Light	Short
107.6	22.2	Heavy	Tall
100	20.1	Heavy	Short
111.5	27.1	Heavy	Short
115.4	30.4	Heavy	Tall
104.8	24	Light	Short
112.3	25.4	Heavy	Tall
102.9	28.8	Light	Short
107.6	29.6	Heavy	Tall
105.3	25.1	Light	Short
105.3	31	Heavy	Short
103	28.9	Heavy	Short
90	21.1	Light	Tall
95.4	14	Light	Tall
89.3	7.1	Light	Short
94.4	13.2	Light	Tall
97.6	23.7	Light	Tall
88.5	9.4	Light	Short
93.6	9.1	Light	Tall
87.7	13.7	Light	Short
93.4	12	Light	Short
91.6	18.3	Light	Short
91.6	9.2	Light	Short
102	21.7	Light	Short
96.4	21.1	Light	Tall
102.7	18.6	Light	Short
97.7	30.2	Light	Short
97.1	26	Light	Short
103.1	18.2	Heavy	Tall
101.8	26.2	Light	Short
101.4	26.1	Light	Short
98.9	25.8	Light	Short
97.5	15	Light	Short
104.3	22.6	Light	Short
97.3	8.8	Light	Tall
96.7	14.3	Light	Tall
99.7	20.2	Light	Short
101.9	18.1	Light	Short
97.2	9.2	Light	Tall
106.6	24.2	Heavy	Tall
99.6	9.6	Heavy	Tall
113.2	17.3	Heavy	Tall
99.1	10.1	Heavy	Tall
99.4	11.1	Light	Short
95.1	17.7	Light	Short
107.5	21.7	Heavy	Tall
106.5	20.8	Heavy	Tall
99.1	20.1	Light	Tall
96.7	19.8	Light	Tall
103.5	21.9	Heavy	Tall
104	24.7	Light	Short
93.1	17.8	Light	Short
105.2	19.1	Heavy	Tall
110	18.2	Heavy	Tall
110.1	17.2	Heavy	Tall
97.8	21	Light	Short
96.3	19.5	Light	Short
108	27.1	Heavy	Short
99.7	21.6	Light	Short
93.5	20.9	Light	Tall
100.7	25.9	Light	Tall
97	16.7	Light	Short
96	19.8	Light	Tall
99.2	14.1	Light	Tall
95.4	25.1	Heavy	Tall
101.8	17.9	Heavy	Tall
104.3	27	Heavy	Tall
99.2	24.6	Heavy	Tall
99.3	14.8	Light	Short
94	16	Light	Short
98.9	14	Light	Short
101	17.4	Light	Short
98.7	26.4	Light	Short
95.9	17.4	Light	Short
103.9	20.4	Heavy	Tall
96.2	15	Light	Short
97.8	18	Light	Tall
94.6	22.2	Light	Tall
103.6	23.1	Heavy	Tall
100.4	25.3	Light	Short
98.4	23.8	Light	Short
104.6	26.3	Heavy	Short
92.9	21.4	Light	Tall
97.8	28.4	Heavy	Tall
98.3	21.8	Light	Tall
104.7	20.1	Heavy	Tall
98.6	24.3	Light	Tall
99.5	18.1	Light	Short
102.7	22.7	Light	Short
92.1	9.9	Light	Tall
96.6	10.8	Heavy	Tall
92.7	14.4	Light	Tall
102	19	Heavy	Tall
110.9	28.6	Heavy	Short
92.3	6.1	Light	Tall
114.1	24.5	Heavy	Tall
92.9	9.9	Light	Short
108.3	19.1	Heavy	Tall
88.5	10.6	Light	Tall
94	16.5	Light	Short
101.1	20.5	Heavy	Tall
92.1	17.2	Light	Tall
105.6	30.1	Heavy	Short
98.5	10.5	Heavy	Tall
88.7	12.8	Light	Short
101.1	22	Light	Tall
94	9.9	Light	Tall
103.8	14.8	Heavy	Tall
98.9	13.3	Heavy	Short
89.2	15.2	Light	Tall
111.4	26.5	Heavy	Tall
107.5	19	Heavy	Tall
99.1	21.4	Light	Short
108.2	20	Heavy	Tall
114.9	34.7	Heavy	Short
99.1	16.5	Light	Short
92.2	4.1	Light	Short
90.8	1.9	Light	Short
100.5	20.2	Light	Tall
98.2	16.8	Light	Tall
115.3	24.6	Heavy	Tall
96.8	10.4	Light	Short
92.6	13.4	Light	Short
119.2	28.8	Heavy	Tall
102.7	22	Heavy	Short
109.5	16.8	Heavy	Tall
108.5	25.8	Heavy	Tall
79.3	0	Light	Short
95.5	11.9	Light	Short
92.3	12.4	Light	Short
98.9	17.4	Light	Tall
89.5	9.2	Light	Tall
117.5	23	Heavy	Tall
107.4	20.1	Heavy	Tall
109.2	20.2	Heavy	Tall
103.4	23.8	Heavy	Short
91.4	11.8	Light	Short
115.2	36.5	Heavy	Tall
104.9	16	Heavy	Tall
106.7	24	Heavy	Tall
92.2	22.3	Light	Tall
101.6	24.8	Light	Short
97.8	21.5	Light	Short
92	17.6	Light	Tall
94	7.3	Light	Tall
103.7	22.6	Light	Short
102.7	12.5	Light	Tall
91.1	21.7	Light	Short
107.2	27.7	Heavy	Tall
100.8	6.8	Heavy	Tall
121.6	33.4	Heavy	Short
105.6	16.6	Heavy	Tall
100.6	31.7	Light	Short
102.7	31.5	Heavy	Tall
99.8	10.1	Light	Tall
92.9	11.3	Light	Tall
91.2	7.8	Light	Short
115.6	26.4	Heavy	Tall
98.3	19.3	Light	Tall
103.7	18.5	Heavy	Tall
98.7	19.3	Light	Tall
119.8	45.1	Heavy	Short
92.8	13.8	Light	Short
93.3	8.2	Light	Short
106.8	23.9	Heavy	Tall
93.9	15.1	Light	Short
99	12.7	Light	Tall
119.9	25.3	Heavy	Tall
94.2	11.9	Light	Short
92.7	6.1	Light	Short
106.9	11.3	Light	Short
88.8	12.8	Light	Short
101.7	14.9	Light	Short
105.3	24.5	Heavy	Tall
104	15	Light	Short
98.6	16.9	Light	Short
99.6	11.1	Light	Short
103.4	16.1	Heavy	Tall
100.2	15.5	Light	Tall
94.9	25.9	Light	Short
97.2	25.5	Light	Short
104.7	18.4	Light	Short
104	24	Heavy	Tall
117.6	26.4	Heavy	Short
95.8	12.7	Light	Short
106.4	28.8	Heavy	Short
93	17	Light	Short
119.6	33.6	Heavy	Short
119.7	29.3	Heavy	Tall
115.8	31.4	Heavy	Tall
118.3	28.1	Heavy	Short
97.4	15.3	Light	Short
113.7	29.1	Heavy	Tall
89.2	11.5	Light	Short
108.5	32.3	Heavy	Short
111.1	28.3	Heavy	Short
108.3	25.3	Heavy	Tall
112.4	30.7	Heavy	Short