ANSWER:
(a) 68.27%
(b) 2.28%
(c) 0.13%
(d) 81.85%
STEP-BY-STEP EXPLANATION:
We have the following information:
[tex]\begin{gathered} m=70 \\ sd=5 \end{gathered}[/tex]
To calculate the probability we must calculate the z-value, which we do by means of the following formula:
[tex]Z=\frac{x-m}{sd}[/tex]
Then, with the value of Z and the help of the normal distribution table, we can calculate the probability.
The table is as follows:
Now, we calculate the probability in each case using the information above.
(a)
between 65 and 75:
[tex]\begin{gathered} P(65\le x\le70)=\frac{65-70}{5}\le x\le\frac{75-70}{5} \\ P(65\le x\le70)=-1\le x\le1 \\ P=0.8413-0.1587=0.6827 \end{gathered}[/tex]
68.27% between 65 and 75.
(b)
above 80:
[tex]\begin{gathered} P(x>80)=x>\frac{80-70}{5} \\ P(x>80)=x>2\rightarrow1-x<2 \\ P=1-0.9772=0.0228 \end{gathered}[/tex]
2.28% above 80.
(c)
below 55:
[tex]\begin{gathered} P(x<55)=x<\frac{55-70}{5} \\ P(x<55)=x<-3 \\ P=0.0013 \end{gathered}[/tex]
0.13% below 55.
(d)
between 65 and 80:
[tex]\begin{gathered} P(65\le x\le80)=\frac{65-70}{5}\le x\le\frac{80-70}{5} \\ P(65\le x\le80)=-1\le x\le2 \\ P=0.9772-0.1587=0.8185 \end{gathered}[/tex]
81.85% between 65 and 80.
