Question

1-For random variable X~N(3,0.75²), what is P(X < 2.5)? Find the nearest answer.

2-For random variable X~N(3,0.75), what is the probability that X takes on a value within two standard deviations on either side of the mean?

3-For standard normal random variable Z, what is P(Z<0)? Answer to two decimal places.

Answer 1

1. X~N(3,0.752) ;

Z = (X-3.0)/0.752 follows a standard normal

P(X < 2.5) = P(Z < Z-score of 2.5)

Z-score for 2.5 = (2.5 - Mean)/Standard deviation = (2.5 - 3) / 0.752 = -0.5/0.752 = -0.66489 - 0.67

From Standard normal tables,

P(Z < Z-score of 2.5) = P(Z<-0.67) = 0.2514

P(X < 2.5) = P(Z < Z-score of 2.5) = 0.2514

P(X < 2.5) = 0.2514

2.  X~N(3,0.75), probability that X takes on a value within two standard deviations on either side of the mean

Mean = 3

Standard deviation = 0.75

X2 : Mean + 2 standard deviation = 3 + 2 x 0.75 = 3+1.5 = 4.5

Z-score for X2 = (X2 - mean) / standard deviation = (4.5-3)/0.75 = 1.5/0.75 = 2

X1 :

Mean - 2 standard deviation = 3 - 2 x 0.75 = 3-1.5 = 1.5

Z-score for X1 = (X1 - mean) / standard deviation = (1.5-3)/0.75 = -1.5/0.75 =- 2

probability that X takes on a value within two standard deviations on either side of the mean = P(X1 < Z< X2)

P(X1 < X< X2) = P(X<X2) - P(X<X1)

P(X<X2) = P(Z < Z-score of X2) = P(Z < 2)

From Standard normal tables , P(Z<2) = 0.9772

P(X<X2) = 0.9772

P(X<X1) = P(Z < Z-score of X1) = P(Z < -2)

From Standard normal tables , P(Z<-2) = 0.0228

P(X < X1) = 0.228

P(X1 < X< X2) = P(X<X2) - P(X<X1) = 0.9772 - 0.0228 = 0.9544

probability that X takes on a value within two standard deviations on either side of the mean = 0.9544

(Also,  empirical rule states that approximately 68%, 95% and 99% of values fall with 1,2 and 3 standard deviations of the mean By that rule Probability that X which follows a normal takes on a value with two standard deviations on either side of the mean = 0.95)

3.

For standard normal random variable Z , P(Z<0)

From standard normal tables,

P(Z< 0 ) = 0.5

Standard normal random variable Z, P(Z<0) =0.5

For Standard normal Mean = 0 and standard deviation =1.

It's well know fact that , For any normal distribution , P(X>Mean) = P(X<Mean) = 0.5

Therefore P(Z>0) = 0.5