Respuesta :
Answer:
(a) 7.6%
(b) 46.2% 42.4%
Step-by-step explanation:
(a)According to the definition of Continuous probability distribution
[tex]f(x) = \frac{d}{dx}F(x)[/tex]
[tex]f(x) = \frac{F(x + h) - F(x - h)}{(x + h) - (x - h)}[/tex]
[tex]f(69) = \frac{F(69 + 0.2) - F(69 - 0.2)}{0.4}[/tex]
⇒ 0.19 × 0.4 = F(69.2) - F(68.8)
⇒ F(69.2) - F(68.8) = 0.076
⇒ 7.6%
(b) Given F(69) = 0.5
[tex]f(x) = \frac{d}{dx}F(x)[/tex]
[tex]f(x) = \frac{F(x) - F(x - h)}{x - (x - h)}[/tex]
[tex]f(69) = \frac{F(69) - F(69 - 0.2)}{0.2}[/tex]
⇒ 0.19 × 0.2 = F(69) - F(68.8)
⇒ F(68.8) = 0.5 - 0.038 = 0.462
⇒ 46.2%
[tex]f(x) = \frac{d}{dx}F(x)[/tex]
[tex]f(x) = \frac{F(x) - F(x - h)}{x - (x - h)}[/tex]
[tex]f(69) = \frac{F(69) - F(69 - 0.4)}{0.4}[/tex]
⇒ 0.19 × 0.4 = F(69) - F(68.8)
⇒ F(68.8) = 0.5 - 0.076 = 0.424
⇒ 42.4%
Part(a): Therefore, there is 7.6% of American men between 68.8 and 69.2 inches.
Part(b): The required values are,
[tex]f(68.8)=46.2\%[/tex]
[tex]f(68.8)=42.4\%[/tex]
Probability density function:
The Probability Density Function(PDF) defines the probability function representing the density of a continuous random variable lying between a specific range of values.
Part(a):
Calculate the percentage of American men are between 68.8 and 69.2 inches tall.
[tex]f(68.8\le f(x)\le69.2)=f(69.2)-f(68.8)\\=[F(69)+(dx)f(69)]-[F(69)-(dx)f(69)]\\=[0.5+(0.2\times 0.19)]-[0.5-(0.2\times 0.19)]\\=0.076\\=7.6\%[/tex]
Part(b):
Calculating [tex]F(68.8)[/tex]
[tex]f(68.8)=[F(69)-(dx)f(69)]\\=[0.5-(0.2\times 0.19)]\\=0.462\\=46.2\%[/tex]
Now, calculating [tex]f(68.6)[/tex]
[tex]f(68.6)=[F(69)-(dx)f(69)]\\=[0.5-(0.4\times 0.19)]\\=0.424\\=42.4\%[/tex]
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