Answer:
a. 0.1587
b. 0.8849
c. 0.1814
Step-by-step explanation:
a. Given that [tex]\mu=28, \ \ \sigma=2.4[/tex]
-The probability a randomly selected score is greater than 30.4 is calculated as:
[tex]P(X>30.4)=1-P(X<30.4)\\\\z=\frac{\bar X-\mu}{\sigma}\\\\=\frac{30.4-28}{2.4}=1\\\\\therefore P(X>30.4)=1-P(z<1)\\\\=1-0.84134\\\\=0.1587[/tex]
Hence, the probability of a score greater than 30.4 is 0.1587
b. Given that [tex]\mu=28, \ \ \ \sigma=2.4[/tex]
The probability a randomly selected score is less than 32.8 is calculated as:
[tex]P(X<32.8)=P(z<\frac{\bar X-\mu}{\sigma})\\\\z=\frac{\bar X-\mu}{\sigma}\\\\=\frac{32.8-28}{2.4}=1.2\\\\P(X<32.8)=P(z<1.2)\\\\=0.88493[/tex]
Hence, the probability that a randomly selected score is less than 32.8 is 0.8849
c. The probability that a score is between 25.6 and 32.8 is calculated as follows:
[tex]P(25.6<X<32.8)=P(\frac{\bar X-\mu}{\sigma}<z<\frac{\bar X-\mu}{\sigma})\\\\=P(\frac{25.6-28}{2.4}<z<\frac{32.8-28}{2.4})\\\\=P(-1<z<2.0)\\\\=0.15866+(1-0.97725)\\\\=0.1814[/tex]
Hence, the probability is 0.1814
