Answer:
The probability that less than 800 students who said they still had their original major is 0.50 or 50%.
Step-by-step explanation:
Let the random variable <em>X</em> be described as the number of third-year college students if they still had their original major.
The probability of the random variable <em>X</em> is, P (X) = <em>p</em> = 0.50.
The sample selected consisted of <em>n</em> = 1600 third-year college students.
The random variable <em>X </em>thus follows Binomial distribution with parameters n = 1600 and p = 0.50.
![X\sim Bin(1600, 0.50)](https://tex.z-dn.net/?f=X%5Csim%20Bin%281600%2C%200.50%29)
As the sample size is large, i.e.<em>n</em> > 30, and the probability of success is closer to 0.50, Normal approximation can be used to approximate the binomial distribution.
The mean of <em>X</em> is:
![\mu_{x}=np=1600\times0.50=800\\](https://tex.z-dn.net/?f=%5Cmu_%7Bx%7D%3Dnp%3D1600%5Ctimes0.50%3D800%5C%5C)
The standard deviation of <em>X</em> is:
![\sigma_{x}=\sqrt{np(1-p}=\sqrt{1600\times0.50(1-0.50)}=20](https://tex.z-dn.net/?f=%5Csigma_%7Bx%7D%3D%5Csqrt%7Bnp%281-p%7D%3D%5Csqrt%7B1600%5Ctimes0.50%281-0.50%29%7D%3D20)
It is provided that Picky Polls got less than 800 students who said they still had their original major.
Then the probability of this event is:
![P(X](https://tex.z-dn.net/?f=P%28X%3C800%29%3DP%28%5Cfrac%7BX-%5Cmu_%7Bx%7D%7D%7B%5Csigma_%7Bx%7D%7D%20%3C%5Cfrac%7B800-800%7D%7B20%7D%20%29%5C%5C%3DP%28Z%3C0%29%5C%5C%3D0.50)
**Use the <em>z</em>-table for the probability.
Thus, the probability that less than 800 students who said they still had their original major is 0.50.