The National Center for Health Statistics (NCHS) keeps records of many health-re

Question

The National Center for Health Statistics (NCHS) keeps records of many health-related aspects of people, including the birth weights of all babies born in the United States. The birth weight of a baby is related to its gestation period (the time between conception and birth). For a given gestation period, the birth weights can be approximated by a normal distribution.

The means and standard deviations of the birth weights of the birth weights for various gestation periods are shown in the table below. One of the goals of the NCHS is to reduce the percentage of babies born with low birth weights. However, the problem of low birth weights increased from the 1990's to the 2000's.

Gestation Period

Mean Birth Weight (#)

Standard Deviation (#)

under 28 wks

1.9

1.22

28-31 weeks

4.12

1.87

32-33 weeks

5.14

1.57

34-36 weeks

6.19

1.29

37-39 weeks

7.29

1.08

40 weeks

7.66

1.04

41 weeks

7.75

1.07

42 weeks and over

7.57

1.11

For the problem you choose:

Show all your work.
Include the name of the software you used, the function, and the inputs to it.
Express any probabilities rounded to 3-significant digits.
Explain what each of your results means.

3. For each of the following gestation periods, what is the probability that a baby will weigh between 6 and 9 pounds at birth?

(a) under 28 weeks

(b) 28 to 31 weeks

(c) 34 to 36 weeks

(d) 37 to 39 weeks

Gestation Period

Mean Birth Weight (#)

Standard Deviation (#)

under 28 wks

1.9

1.22

28-31 weeks

4.12

1.87

32-33 weeks

5.14

1.57

34-36 weeks

6.19

1.29

37-39 weeks

7.29

1.08

40 weeks

7.66

1.04

41 weeks

7.75

1.07

42 weeks and over

7.57

1.11

Explanation / Answer

Gestation Period

Mean Birth Weight (#)

Standard Deviation (#)

under 28 wks

1.9

1.22

28-31 weeks

4.12

1.87

32-33 weeks

5.14

1.57

34-36 weeks

6.19

1.29

37-39 weeks

7.29

1.08

40 weeks

7.66

1.04

41 weeks

7.75

1.07

42 weeks and over

7.57

1.11

Since the birth weights (x) follows normal distribution:

P(6 < x < 9 )= P([(6-mean)/std ]< (x-µ)/ < [(9-mean)/std ])

a)under 28 weeks:

P(6 < x < 9 )= P((6-1.9)/1.22 < z < (9-1.9)/1.22)

=P(3.3607 < z < 5.8197)

=P(z <5.8197) - P(z < 3.3607)

=1-0.999611 (using normdistin excel, “=normdist(9,1.9,1.22,TRUE)” , likewise for x=6)

=0.000389

b)28 to 31 weeks:

P(6 < x < 9 )= P((6-4.12)/1.87 < z < (9-4.12)/1.87)

=P(1.00534< z < 2.6096)

=P(z <2.6096) - P(z < 1.00534)

=0.995468-0.842635

=0.152833

c)34 to 36 weeks

P(6 < x < 9 )= P((6-6.19)/1.29 < z < (9-6.19)/1.29)

=P(-0.1472< z < 2.17829)

=P(z <2.17829) - P(z < -0.1472)

=0.985308-0.441453

=0.543855

d)37 to 39 weeks

P(6 < x < 9 )= P((6-7.29)/1.08 < z < (9-7.29)/1.08)

=P(-1.194< z < 1.5833)

=P(z <1.5833) - P(z < -1.194)

=0.943327-0.116152

=0.827175

Probability that a baby will weigh between 6 and 9 pounds at birth:

a)under 28 weeks = 0.000389

b)28 to 31weeks =0.152833

c)34 to 36 weeks=0.543855

d)37 to 39 weeks=0.82717

Gestation Period

Mean Birth Weight (#)

Standard Deviation (#)

under 28 wks

1.9

1.22

28-31 weeks

4.12

1.87

32-33 weeks

5.14

1.57

34-36 weeks

6.19

1.29

37-39 weeks

7.29

1.08

40 weeks

7.66

1.04

41 weeks

7.75

1.07

42 weeks and over

7.57

1.11

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