IB Mathematics Applications & Interpretation Question Type 1: Using Properties of Normal Distribution to Find Probabilities Given Some Value Exercises

Calculate $P(X>5)$ for $X\sim N(2,4)$.

Let $X\sim N(2,{\sigma }^2)$ and $P(X>a)=0.3$. Find $1-P(X<4-a)$.

For $X\sim N(8,{\sigma }^2)$, if 95% of the data lies between 6.5 and 9.5, find the value of ${\sigma }^2$.

A test score is modeled by a normal distribution $X\sim N(100,{\sigma }^2)$. Given that 80% of students score between 90 and 110, find the value of ${\sigma }^2$.

Calculate $P(-1\le X\le 2)$ for $X\sim N(0,9)$.

For $X\sim N(3,16)$, show that $P(X<-1)=P(X>7)$ and compute this probability.

Let $X\sim N(2,{\sigma }^2)$ and $P(X>a)=0.3$. Express $a$ in terms of $\sigma$.

Calculate $P(X<1.96)$ for $X\sim N(0,1)$.

For $X\sim N(0,{\sigma }^2)$, if 95% of the data lies between $-3$ and $3$, find $\sigma$.

For $X\sim N(10,{\sigma }^2)$, the central $90\%$ of the data lies between 8 and 12. Find the value of $\sigma$.

Let $X\sim N(5,{\sigma }^2)$. Given that $99\%$ of the data lies between $3$ and $7$, find the value of $\sigma$.

For $X\sim N(20,{\sigma }^2)$, the interval $[10,30]$ contains $95\%$ of the values. Find ${\sigma }^2$.