The length of human pregnancies from conception to birth var

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 263 days and standard deviation 16 days.

(a) What proportion of pregnancies last less than 270 days (about 9 months)?

(b) What proportion of pregnancies last between 240 and 270 days (roughly between 8 months and 9 months)?

(c) How long do the longest 20% of pregnancies last? (Please use 2 decimal places.)

The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75.

(d) What are the quartiles of the standard Normal distribution? (Use 2 decimal places.)

Q1=

Q3=

(e) What are the quartiles of the distribution of lengths of human pregnancies? Please use 2 decimal places.

Q1=

Q3=

Solution

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    270      
u = mean =    263      
          
s = standard deviation =    16      
          
Thus,          
          
z = (x - u) / s =    0.4375      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   0.4375   ) =    0.669125612 [answer]

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b)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    240      
x2 = upper bound =    270      
u = mean =    263      
          
s = standard deviation =    16      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1.4375      
z2 = upper z score = (x2 - u) / s =    0.4375      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.075287986      
P(z < z2) =    0.669125612      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.593837626   [answer]

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c)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.8      
          
Then, using table or technology,          
          
z =    0.841621234      
          
As x = u + z * s          
          
where          
          
u = mean =    263      
z = the critical z score =    0.841621234      
s = standard deviation =    16      
          
Then          
          
x = critical value =    276.4659397   [answer]

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d)

The quartiles of the z distrbution are, using table or technology, using left tailed areas of 0.25 and 0.75, repspectively,

z(Q1) = -0.67448975 [answer]

z(Q3) = 0.67448975 [answer]

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e)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.25      
          
Then, using table or technology,          
          
z =    -0.67448975      
          
As x = u + z * s,          
          
where          
          
u = mean =    263      
z = the critical z score =    -0.67448975      
s = standard deviation =    16      
          
Then          
          
x = critical value =    252.208164   [ANSWER, Q1]

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.75      
          
Then, using table or technology,          
          
z =    0.67448975      
          
As x = u + z * s,          
          
where          
          
u = mean =    263      
z = the critical z score =    0.67448975      
s = standard deviation =    16      
          
Then          
          
x = critical value =    273.791836   [answer, Q2]