X ~ N (-7, 4^2). Probabilities must be expressed to 4 decimal places and values

Question

X ~ N (-7, 4^2). Probabilities must be expressed to 4 decimal places and values of x to 2 decimal places. (1 - 9) Find these probabilities. P(X lessthanorequalto 0) P(-9 lessthanorequalto x lessthanorequalto -2) P(X greaterthanorequalto -12) P(-16 lessthanorequalto x lessthanorequalto 2) P(x lessthanorequalto -13 or x greaterthanorequalto -8) Find x. P(x lessthanorequalto x) = -75 P(x greaterthanorequalto x) = 45 P(-x lessthanorequalto x + 7 lessthanorequalto) = 35 P(x + 7 lessthanorequalto -x or x + 7 greaterthanorequalto x) = 25 X is normal. P(x lessthanorequalto 3) = 65 and P(x > 7) = 25 Find mu and sigma to 2 decimal places.

Explanation / Answer

1-      P(X<=0)= 0.9599 using excel function =NORMDIST(0,-7,4,TRUE)

2-      P(-9<=X<-2)=0.5858 using excel function =NORMDIST(-2,-7,4,TRUE)-NORMDIST(-9,-7,4,TRUE)

3-      P(X>=-12)=0.8944 using excel function =1-NORMDIST(-12,-7,4,TRUE)

4-      P(-16<=X<=2)=0.9756 using excel function =NORMDIST(2,-7,4,TRUE)-NORMDIST(-16,-7,4,TRUE)

5-      P(X<=-13 or X>=-8)=0.6655 using excel function =NORMDIST(-13,-7,4,TRUE)+1-NORMDIST(-8,-7,4,TRUE)

6-      P(X<=x)=0.75 gives x= using excel function =NORMINV(0.75,-7,4)

7-      P(X>=x)=0.45 or P(X<x)=0.55 gives x=-6.50 using excel function =NORMINV(0.55,-7,4)

8-      As normal distribution is symmetric so we have

P(X+7<=-x)=(1-0.35)/2 = 0.325

Or P(X<-x-7)=0.325

So, -x-7=-8.82 using excel function =NORMINV(0.325,-7,4)

Hence, x= 8.82-7=1.82

9-      As normal distribution is symmetric so we have

P(X+7<=-x)=0.25/2 = 0.125

Or P(X<-x-7)=0.125

So, -x-7=-11.60 using excel function =NORMINV(0.125,-7,4)

Hence, x= 11.60-7=4.60

10- P(X<=3)=0.65 gives P((X-µ)/)<=(3-µ)/) =0.65 or P(Z<z0)=0.65 where z0=(3-µ)/

From excel function =normsinv(0.65) we have z0=0.39

P(X>7)=0.25 gives P((X-µ)/)>(7-µ)/) =0.25 or P(Z>z1)=0.25 where z1=(7-µ)/

P(Z<z1)=1-0.25=0.75

From excel function =normsinv(0.75) we have z1=0.67

Z0=0.39 gives (3-µ)/=0.39 or 3-µ=0.39

z1=0.67 gives (7-µ)/=0.67 or 7-µ=0.67

so, 3-(7-0.67)=0.39

we have -4+0.67 =0.39

Hence, =4/0.28 = 14.28

So, µ=3-0.39*14.28=-2.57

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