Question

The gestation period for cats has an approximate mean of 64 days and a standard deviation of 3 days. The distribution of the gestation period is approximately Normal.

a. What proportion of kittens have a gestation period of less than 68 days?
b. What proportion of kittens have a gestation period between 61 and 70 days?
c. What gestation period corresponds to the top 10% of gestation periods?
d. What gestation period corresponds to the 25th percentile?

Answer 1

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

Step-by-step explanation:

Given that:

population mean
`\mu` = 64

standard deviation
`\sigma` = 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

`z = (x -\mu)/(\sigma)`

`z = (68 -64)/(3)`

`z = (4)/(3)`

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:

`z_1 = (61 -64)/(3)`

`z_1 = (-3)/(3)`

`z_1 =-1`

`z_2 = (70-64)/(3)`

`z_2= (6)/(3)`

`z_2= 2`

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 >
`x_o` ) = 0.1

P(X <
`x_o` ) = 1 - 0.1

P(X >
`x_o` ) = 0.9

`P(Z < (x - \mu)/(\sigma)) =0.9`

Using the Excel Function : =NORMINV (0.9)

`P(Z < (x - \mu)/(\sigma)) =1.282`

⇒
`(x - \mu)/(\sigma)=1.282`

`{x - \mu}=1.282 * \sigma`

`x =1.282 * \sigma + \mu`

given that:

`\mu = 64 \\ \sigma =3`

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