Question Type 1: Using properties of normal distribution to find probabilities given some value Exercises

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

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

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

Let $X \sim N(5,\sigma^2)$ and 99% of the data lies between 3 and 7. Find $\sigma$.

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

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 $\sigma$.

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

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

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

A test score $X \sim N(100,\sigma^2)$. If 80% of students score between 90 and 110, find $\sigma^2$.

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