Question

16 Let ? have a normal distribution with ?(?) = 0.2 and ??(?) = 0.08. a. Find ?(? < 0.36) e. Find ? such that ?(? ≤ ?) = 0.05. b. Find ?(? > 0.16) f. Find ? such that ?(? ≥ ?) = 0.10. c. Find ?(? ≥ 0.60) g. Find ? such that ?(? > ?) = 0.80. d. Find 0.28 < ? ≤ 0.40) h. Find ? such that ?(? < ?) = 0.95.

Answer 1

?(?) = µ = 0.2

??(?) = σ = 0.08.

a. Find P(? < 0.36) =

= P( (X̅-μ)/(σ/√n) < (0.36-0.2)/(0.08/√1) )

= P(z < 2)

Using excel function:

= NORM.S.DIST(2, 1)

= 0.9772

b. Find ?(? > 0.16)

= P( (X-µ)/σ > (0.16-0.2)/0.08)

= P(z > -0.5)

= 1 - P(z < -0.5)

Using excel function:

= 1 - NORM.S.DIST(-0.5, 1)

= 0.6915

c. Find ?(? ≥ 0.60)

= P( (X-µ)/σ > (0.6-0.2)/0.08)

= P(z > 5)

= 1 - P(z < 5)

Using excel function:

= 1 - NORM.S.DIST(5, 1)

= 0

d. Find P(0.28 < ? ≤ 0.40)

= P( (0.28-0.2)/0.08 < (X-µ)/σ < (0.4-0.2)/0.08 )

= P(1 < z < 2.5)

= P(z < 2.5) - P(z < 1)

Using excel function:

= NORM.S.DIST(2.5, 1) - NORM.S.DIST(1, 1)

= 0.1524

e. Find ? such that ?(? ≤ ?) = 0.05.

P(X < b) = 0.05  

Z score at p = 0.05 using excel = NORM.S.INV(0.05) = -1.6449

Value of X = µ + z*σ = 0.2 + (-1.6449)*0.08 = 0.0684

f. Find ? such that ?(? ≥ ?) = 0.10.

P(X > b) = 0.1  

= 1 - P(X < b) = 0.1  

= P(x < b) = 0.9  

Z score at p = 0.9 using excel = NORM.S.INV(0.9) = 1.2816

Value of X = µ + z*σ = 0.2 + (1.2816)*0.08 = 0.3025

g. Find ? such that ?(? > ?) = 0.80.

P(x > b) = 0.8  

= 1 - P(x < b) = 0.8  

= P(x < b) = 0.2  

Z score at p = 0.2 using excel = NORM.S.INV(0.2) =   -0.8416

Value of X = µ + z*σ = 0.2 + (-0.8416)*0.08 = 0.1327

h. Find ? such that ?(? < ?) = 0.95.

P(x < b) = 0.95  

Z score at p = 0.95 using excel = NORM.S.INV(0.95) =   1.6449

Value of X = µ + z*σ = 0.2 + (1.6449)*0.08 = 0.3316

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