Respuesta :
Answer:
(a) The value of P (X < 21 | μ = 23 and σ = 3) is 0.2514.
(b) The value of P (X ≥ 66 | μ = 50 and σ = 9) is 0.0427.
(c) The value of P (X > 47 | μ = 50 and σ = 5) is 0.7258.
(d) The value of P (17 < X < 24 | μ = 21 and σ = 3) is 0.7495.
(e) The value of P (X ≥ 95 | μ = 80 and σ = 1.82) is 0.
Step-by-step explanation:
The random variable X is Normally distributed.
(a)
The mean and standard deviation are:
[tex]\mu=23\\\sigma=3[/tex]
Compute the value of P (X < 21) as follows:
[tex]P(X<21)=P(\frac{X-\mu}{\sigma}<\frac{21-23}{3})[/tex]
[tex]=P(Z<-0.67)\\=1-P(Z<0.67)\\=1-0.74857\\=0.25143\\\approx0.2514[/tex]
Thus, the value of P (X < 21 | μ = 23 and σ = 3) is 0.2514.
(b)
The mean and standard deviation are:
[tex]\mu=50\\\sigma=9[/tex]
Compute the value of P (X ≥ 66) as follows:
Use continuity correction.
P (X ≥ 66) = P (X > 66 - 0.5)
= P (X > 65.5)
[tex]=P(\frac{X-\mu}{\sigma}>\frac{65.5-50}{9})[/tex]
[tex]=P(Z>1.72)\\=1-P(Z<1.72)\\=1-0.9573\\=0.0427[/tex]
Thus, the value of P (X ≥ 66 | μ = 50 and σ = 9) is 0.0427.
(c)
The mean and standard deviation are:
[tex]\mu=50\\\sigma=5[/tex]
Compute the value of P (X > 47) as follows:
[tex]P(X>47)=P(\frac{X-\mu}{\sigma}>\frac{47-50}{5})[/tex]
[tex]=P(Z>-0.60)\\=P(Z<0.60)\\=0.7258[/tex]
Thus, the value of P (X > 47 | μ = 50 and σ = 5) is 0.7258.
(d)
The mean and standard deviation are:
[tex]\mu=21\\\sigma=3[/tex]
Compute the value of P (17 < X < 24) as follows:
[tex]P(17<X<24)=P(\frac{17-21}{3}<\frac{X-\mu}{\sigma}<\frac{24-21}{3})[/tex]
[tex]=P(-1.33<Z<1)\\=P(Z<1)-P(Z<-1.33)\\=0.8413-0.0918\\=0.7495[/tex]
Thus, the value of P (17 < X < 24 | μ = 21 and σ = 3) is 0.7495.
(e)
The mean and standard deviation are:
[tex]\mu=80\\\sigma=1.82[/tex]
Compute the value of P (X ≥ 95) as follows:
Use continuity correction:
P (X ≥ 95) = P (X > 95 - 0.5)
= P (X > 94.5)
[tex]=P(\frac{X-\mu}{\sigma}>\frac{94.5-80}{1.82})[/tex]
[tex]=P(Z>7.97)\\=1-P(Z<7.97)\\=1-(\approx1)\\=0[/tex]
Thus, the value of P (X ≥ 95 | μ = 80 and σ = 1.82) is 0.
