Answer:
a. P(X<68) = 0.9082
b. P(61 < X< 70) = 0.8185
c. the gestation period that corresponds to the top 10% of gestation periods = 67.846
d. the gestation period that corresponds to the 25th percentile = 61.977
Explanation:
Given that:
population mean [tex]\mu[/tex] = 64
standard deviation [tex]\sigma[/tex] = 3
a. What proportion of kittens have a gestation period of less than 68 days?
here the sample mean x = 68
The standard normal distribution for the z score is
[tex]z = \dfrac{x -\mu}{\sigma}[/tex]
[tex]z = \dfrac{68 -64}{3}[/tex]
[tex]z = \dfrac{4}{3}[/tex]
z = 1.33
The proportion of the kittens having a gestation period of less than 68 days is:
P(X<68) = P(Z< 1.33)
Using the z - tables
P(X<68) = 0.9082
b. What proportion of kittens have a gestation period between 61 and 70 days?
here ; sample mean x₁ = 61 and x₂ = 70
the standard normal distribution for the z score is:
[tex]z_1 = \dfrac{61 -64}{3}[/tex]
[tex]z_1 = \dfrac{-3}{3}[/tex]
[tex]z_1 =-1[/tex]
[tex]z_2 = \dfrac{70-64}{3}[/tex]
[tex]z_2= \dfrac{6}{3}[/tex]
[tex]z_2= 2[/tex]
So, the proportion of kittens having a gestation period between 61 and 70 days is:
P(61 < X< 70) = P(-1 < Z < 2)
P(61 < X< 70) = P(Z < 2) - P( Z< -1)
From z tables
P(61 < X< 70) = 0.9772 - 0.1587
P(61 < X< 70) = 0.8185
c. What gestation period corresponds to the top 10% of gestation periods?
i.e
P(X >[tex]x_o[/tex] ) = 0.1
P(X < [tex]x_o[/tex] ) = 1 - 0.1
P(X >[tex]x_o[/tex] ) = 0.9
[tex]P(Z < \dfrac{x - \mu}{\sigma}) =0.9[/tex]
Using the Excel Function : =NORMINV (0.9)
[tex]P(Z < \dfrac{x - \mu}{\sigma}) =1.282[/tex]
⇒ [tex]\dfrac{x - \mu}{\sigma}=1.282[/tex]
[tex]{x - \mu}=1.282 \times \sigma[/tex]
[tex]x =1.282 \times \sigma + \mu[/tex]
given that:
[tex]\mu = 64 \\ \sigma =3[/tex]
x = 1.282 × 3 + 64
x = 67.846
The gestation period that corresponds to the top 10% of gestation periods = 67.846
d. What gestation period corresponds to the 25th percentile?
At 25 percentile, using the EXCEL FUNCTION = NORMINV(0.25;64;3)
the gestation period that corresponds to the 25th percentile = 61.977
