Question

Assume that the duration of human pregnancies can be described by a Normal model with mean 268 days and standard deviation 16 days. ​a) What percentage of pregnancies should last between 274 and 285 ​days? ​b) At least how many days should the longest 15​% of all pregnancies​ last? ​c) Suppose a certain obstetrician is currently providing prenatal care to 51 pregnant women. Let y overbar represent the mean length of their pregnancies. According to the Central Limit​ Theorem, what's the distribution of this sample​ mean, y overbar​? Specify the​ model, mean, and standard deviation. ​d) What's the probability that the mean duration of these​ patients' pregnancies will be less than 258 ​days? ​a) The percentage of pregnancies that should last between 274 and 285 days is nothing​%. ​(Round to two decimal places as​ needed.) ​b) The longest 15​% of all

pregnancies should last at least nothing days. ​(Round to one decimal place as​ needed.) ​c) Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) within your choice. A. A Binomial model with nothing trials and a probability of success of nothing ​(Type integers or decimals rounded to two decimal places as​ needed.) B. A Normal model with mean nothing and standard deviation nothing ​(Type integers or decimals rounded to two decimal places as​ needed.) C. There is no model that fits this distribution. ​d) The probability that the mean duration of these​ patients' pregnancies will be less than 258 days is nothing.

Answer 1

µ = 268, σ = 16

a) The percentage of pregnancies that should last between 274 and 285 days, P(274 < X < 285) =

= P( (274-268)/16 < (X-µ)/σ < (285-268)/16 )

= P(0.375 < z < 1.0625)

= P(z < 1.0625) - P(z < 0.375)

Using excel function:

= NORM.S.DIST(1.0625, 1) - NORM.S.DIST(0.375, 1)

= 0.2098 = 20.98%

b)

P(x > a) = 0.15

= 1 - P(x < a) = 0.15

= P(x < a) = 0.85

Z score at p = 0.85 using excel = NORM.S.INV(0.85) = 1.0364

Value of X = µ + z*σ = 268 + (1.0364)*16 = 284.6

c) Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) within your choice. A.

µₓ = 268

σₓ = σ/√n = 16/√51 = 2.2404

Answer: B. A Normal model with mean 268 and standard deviation 2.24

d) Probability that the mean duration of these​ patients' pregnancies will be less than 258 days, P(X̅ < 258) =

= P( (X̅-μ)/(σ/√n) < (258-268)/(16/√51) )

= P(z < -4.4634)

Using excel function:

= NORM.S.DIST(-4.4634, 1)

= 0.00