Answer:
1. 758 newborn babies
2. 918 newborn babies
3. 756 newborn babies
4. 587 newborn babies
Step-by-step explanation:
Let's start by defining the following event :
W : ''The weight of a newborn baby''
W ~ N(μ,σ) Where N is the normal distribution, μ is the mean and σ is the standard deviation
W ~ N(6.2,2)
To calculate probability, we need to turn this variable into a N(0,1) by doing the following :
First we subtract the mean to W and then we divide by the standard deviation
[(W-μ) / σ] ~ N(0,1)
1. ![P(5](https://tex.z-dn.net/?f=P%285%3CW%3C8%29)
![P(\frac{5-(6.2)}{2}](https://tex.z-dn.net/?f=P%28%5Cfrac%7B5-%286.2%29%7D%7B2%7D%3C%5Cfrac%7BW-%286.2%29%7D%7B2%7D%3C%5Cfrac%7B8-%286.2%29%7D%7B2%7D%29%3D)
![P(-0.6](https://tex.z-dn.net/?f=P%28-0.6%3CZ%3C0.9%29)
Where Z ~ N(0,1)
is the area below the normal curve N(0,1) between the values -0.6 and 0.9
P(-0.6<Z<0.9) = Φ(0.9) - Φ(-0.6)
Where Φ is the cumulative distribution for N(0,1)
P(-0.6<Z<0.9) = Φ(0.9) - Φ(-0.6) = 0.8159 - 0.2743 = 0.5416
Then the probability of the variable W to be between 5 and 8 pounds is 0.5416
To find the number of newborn babies expected to weigh between 5 and 8 pounds we multiply the group of 1400 and the probability
![(1400).(0.5416)=758.24](https://tex.z-dn.net/?f=%281400%29.%280.5416%29%3D758.24)
758.24 ≅ 758
Then 758 newborn babies are expected to weigh between 5 and 8 pounds
2.
![P(W](https://tex.z-dn.net/?f=P%28W%3C7%29%3DP%28Z%3C%5Cfrac%7B7-6.2%7D%7B2%7D%29%3DP%28Z%3C0.4%29)
P(Z<0.4) = Φ(0.4) = 0.6554
The expected number of newborn babies is
![(1400).(0.6554)=917.56](https://tex.z-dn.net/?f=%281400%29.%280.6554%29%3D917.56)
917.56 ≅ 918
918 newborn babies are expected to weigh less than 7 pounds
3.
![P(W>6)=1-P(W](https://tex.z-dn.net/?f=P%28W%3E6%29%3D1-P%28W%3C6%29%3D1-P%28Z%3C%5Cfrac%7B6-6.2%7D%7B2%7D%29%3D1-P%28Z%3C-0.1%29)
1-P(Z<-0.1) = 1 - Φ(-0.1) = 1 - 0.4602 = 0.5398
The expected number of babies is
![(1400).(0.5398)=755.72](https://tex.z-dn.net/?f=%281400%29.%280.5398%29%3D755.72)
755.72 ≅ 756
The expected number of babies to weigh more than 6 pounds is 756
4.
![P(6.2](https://tex.z-dn.net/?f=P%286.2%3CW%3C9%29%3DP%28%5Cfrac%7B6.2-6.2%7D%7B2%7D%3CZ%3C%5Cfrac%7B9-6.2%7D%7B2%7D%29%3DP%280%3CZ%3C1.4%29)
P(0<Z<1.4) = Φ(1.4) - Φ(0) = 0.9192 - 0.500 = 0.4192
The expected number of babies is
![(1400).(0.4192)= 586.88](https://tex.z-dn.net/?f=%281400%29.%280.4192%29%3D%20586.88)
586.88 ≅ 587
587 newborn babies are expected to weigh between 6.2 and 9 pounds