Question

X∼(μ=4.5,σ=4) find P(X<10.2)

X∼(μ=4.5,σ=4) find P(X>-2.8)

For X∼(μ=4.5,σ=4) find P(6.7<X<15.9)

For X∼(μ=4.5,σ=4) find P(-4.9<X<-0.2)

For X∼(μ=4.4,σ=4) find the 2-th percentile.

For X∼(μ=17.4,σ=2.9) find the 82-th percentile.

For X∼(μ=48.1,σ=3.4) find the 13-th percentile.

For X∼(μ=17.4,σ=2.9) find the 49-th percentile.

For X∼(μ=48.1,σ=3.4) find P(X>39)

For X∼(μ=48.1,σ=3.4) find P(X<39)

Answer 1

Solution :

Given that ,

mean = = 4.5

standard deviation = = 4

P(x < 10.2) = P[(x - ) / < (10.2 - 4.5) / 4]

= P(z < 1.425)

= 0.9229

P(x < 10.2) = 0.9229

mean = = 4.5

standard deviation = = 4

P(x > -2.28) = 1 - P(x < -2.28)

= 1 - P((x - ) / < (-2.28 - 4.5) / 4)

= 1 - P(z < -1.695)

= 1 - 0.045   

= 0.955

P(x > -2.28) = 0.955

mean = = 4.5

standard deviation = = 4

P(6.7 < x < 15.9) = P[(6.7 - 4.5)/ 4) < (x - ) /  < (15.9 - 4.5) / 4) ]

= P(0.55 < z < 2.85)

= P(z < 2.85) - P(z < 0.55)

= 0.9978 - 0.7088

= 0.289

P(6.7 < x < 15.9) = 0.289

mean = = 4.5

standard deviation = = 4

P(-4.9 < x < -0.2) = P[(-4.9 - 4.5)/ 4) < (x - ) /  < (-0.2 - 4.5) / 4) ]

= P(-2.35 < z < -1.175)

= P(z < -1.175) - P(z < -2.35)

= 0.12 - 0.0094

= 0.1106

P(-4.9 < x < -0.2) = 0.1106

mean = = 4.4

standard deviation = = 4

Using standard normal table,

P(Z < z) = 2%

P(Z < -2.05) = 0.02

z = -2.05

Using z-score formula,

x = z * +

x = -2.05 * 4 + 4.4 = -3.8

The 2-th percentile = -3.8

mean = = 17.4

standard deviation = = 2.9

Using standard normal table,

P(Z < z) = 82%

P(Z < 0.92) = 0.82

z = 0.92

Using z-score formula,

x = z * +

x = 0.92 * 2.9 + 17.4 = 20.07

The 82-th percentile = 20.07

mean = = 48.1

standard deviation = = 3.4

Using standard normal table,

P(Z < z) =13%

P(Z < -1.13) = 0.13

z = -1.13

Using z-score formula,

x = z * +

x = -1.13 * 3.4 + 48.1 = 44.26

The 13-th percentile = 44.26

mean = = 17.4

standard deviation = = 2.9

Using standard normal table,

P(Z < z) = 49%

P(Z < -0.025) = 0.49

z = -0.025

Using z-score formula,

x = z * +

x = -0.025 * 2.9 + 17.4 = 17.33

The 49-th percentile = 17.33

mean = = 4.5

standard deviation = = 4

P(x > 39) = 1 - P(x < 39)

= 1 - P((x - ) / < (39 - 48.1) / 3.4)

= 1 - P(z < -2.6765)

= 1 - 0.0037

= 0.9963

P(x > 39) = 0.9963

mean = = 4.5

standard deviation = = 4

P(x < 39) = P[(x - ) / < (39 - 48.1) / 3.4]

= P(z < -2.6765)

= 0.0037

P(x < 39) = 0.0037