Answer: About 191 students scored between a 60 and an 80.
Step-by-step explanation:
Given : A set of 200 test scores are normally distributed with a mean of 70 and a standard deviation of 5.
i.e. [tex]\mu=70[/tex] and [tex]\sigma=5[/tex]
let x be the random variable that denotes the test scores.
Then, the probability that the students scored between a 60 and an 80 :
[tex]P(60<x<80)=P(\dfrac{60-70}{5}<\dfrac{x-\mu}{\sigma}<\dfrac{80-70}{5})\\\\=P(-2<z<2)\ \ [\because z=\dfrac{x-\mu}{\sigma}]\\\\=P(z<2)-P(z<-2)\ \ [\because\ P(z_1<Z<z_2)=P(Z<z_2)-P(Z<z_1)]\\\\=P(z<2)-(1-P(z<2))\ \ [\because\ P(Z<-z)=1-P(Z<z)]\\\\=2P(z<2)-1\\\\=2(0.9772)- 1 \ \ [\text{By z-table}]\\\\=0.9544[/tex]
The number of students scored between a 60 and an 80 = 0.9544 x 200
= 190.88 ≈ 191
Hence , about 191 students scored between a 60 and an 80.
