Question

If a random variable has the standard normal distribution, what are the probabilities that it will take on a valuea) Less than 1.64b) Greater than – 0.47c) Greater than 0.76d) Less than –1.35e) Between 0.95 and 1.36

Answer 1

• (a)The probability that it will take on a value less than 1.64 is 0.9495.

,

• (b)The probability that it will take on a value greater than – 0.47 is 0.6808.

,

• (c)The probability that it will take on a value greater than 0.76 is 0.2236.

,

• (d)The probability that it will take on a value less than -1.35 is 0.0885

,

• (e)The probability that it will take on a value between 0.95 and 1.36​ is 0.0841.

Explanation:

Given that a random variable has the standard normal distribution, then, we use a normal distribution table or calculator to find these values:

(a) Less than 1.64

`P(X<1.64)=0.9495`

The probability that it will take on a value less than 1.64 is 0.9495.

(b)Greater than – 0.47

`\begin{gathered} P(X>-0.47)=1-P(X<-0.47) \\ =1-0.3192 \\ =0.6808 \end{gathered}`

The probability that it will take on a value greater than – 0.47 is 0.6808.

(c)Greater than 0.76

`\begin{gathered} P(X>0.76)=1-P(X<0.76) \\ =1-0.7764 \\ =0.2236 \end{gathered}`

The probability that it will take on a value greater than 0.76 is 0.2236.

(d)Less than –1.35

`P(X<-1.35)=0.0885`

The probability that it will take on a value less than -1.35 is 0.0885

(e) Between 0.95 and 1.36​

[tex]\begin{gathered} P\left(x<0.95\right)=0.8289 \\ P\left(x>1.36\right)=0.0869 \\ P\left(x<0.95\;or\;x>1.36\right)=0.8289+0.0869=0.9159 \\ P\left(0.95The probability that it will take on a value between 0.95 and 1.36​ is 0.0841.