Respuesta :
Answer:
0.8554 is the probability that the proportion of students living in dormitories is between 0.70 and 0.85
Step-by-step explanation:
We are given the following in the question:
Proportion of students who live in dormitories = 80%
[tex]p= 0.80[/tex]
Sample size, n = 80
Thus, the number of students follow a binomial distribution,
We use normal approximation to the binomial distribution to find the probability.
Parameters of Normal distribution:
[tex]\mu = np = 80\times 0.80 = 64\\\sigma = \sqrt{np(1-p)} = \sqrt{80(0.80)(1-0.80)} = 3.58[/tex]
We have to find the probability that the sample proportion (the proportion living in the dormitories) is between 0.70 and 0.85.
[tex]P(0.70\leq p \leq 0.85)\\\\=P(0.70(80)\leq x \leq0.85(80))\\\\=P(56 \leq x \leq 68) \\\\= P(\displaystyle\frac{56 - 64}{3.58} \leq z \leq \displaystyle\frac{68-64}{3.58})\\\\ = P(-2.2346 \leq z \leq 1.1173)\\\\= P(z \leq 1.1173) - P(z < -2.2346)\\\\= 0.8681 - 0.0127 = 0.8554[/tex]
Thus, 0.8554 is the probability that the proportion of students living in dormitories is between 0.70 and 0.85
