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 X be described as the number of third-year college students if they still had their original major.
The probability of the random variable X is, P (X) = p = 0.50.
The sample selected consisted of n = 1600 third-year college students.
The random variable X thus follows Binomial distribution with parameters n = 1600 and p = 0.50.
[tex]X\sim Bin(1600, 0.50)[/tex]
As the sample size is large, i.e.n > 30, and the probability of success is closer to 0.50, Normal approximation can be used to approximate the binomial distribution.
The mean of X is:
[tex]\mu_{x}=np=1600\times0.50=800\\[/tex]
The standard deviation of X is:
[tex]\sigma_{x}=\sqrt{np(1-p}=\sqrt{1600\times0.50(1-0.50)}=20[/tex]
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:
[tex]P(X<800)=P(\frac{X-\mu_{x}}{\sigma_{x}} <\frac{800-800}{20} )\\=P(Z<0)\\=0.50[/tex]
**Use the z-table for the probability.
Thus, the probability that less than 800 students who said they still had their original major is 0.50.
