Question

The mean incubation time of fertilized eggs is 23 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day. Find the incubation time that separates the bottom 2.5​% from the rest of the incubation times.

Answer 1

The incubation time that separates the bottom 2.5​% from the rest of the incubation times is 21.04 days.

Explanation:

We are given that the mean incubation time of fertilized eggs is 23 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day.

Let X = the incubation times

So, X ~ N(
`\mu=23,\sigma^(2) = 1^(2)`)

The z score probability distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = population mean incubation time = 23 days

`\sigma` = standard deviation = 1 day

Now, we have to find the incubation time that separates the bottom 2.5​% from the rest of the incubation times.

So, Probability that the incubation time separate the bottom 2.5​% is given by;

P(X > x) = 0.025

P(
`(X-\mu)/(\sigma)` >
`(x-23)/(1)` ) = 0.025

P(Z >
`(x-23)/(1)` ) = 0.025

So, the critical value of x in z table which separate the bottom 2.5% is given as -1.96, which means;

`(x-23)/(1)` = -1.96

`x` = 23 - 1.96 = 21.04

Therefore, the incubation time that separates the bottom 2.5​% from the rest of the incubation times is 21.04 days.

Answer 2

Answer: the incubation time that separates the bottom 2.5​% from the rest of the incubation times is 24.96 days.

Explanation:

Suppose the incubation times are approximately normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = incubation times.

µ = mean time

σ = standard deviation

From the information given,

µ = 23 days

σ = 1 day

The probability value for the incubation time that separates the bottom 2.5​% from the rest of the incubation times would be (1 - 2.5/100) = (1 - 0.025) = 0.975

Looking at the normal distribution table, the z score corresponding to the probability value is 1.96

Therefore,

1.96 = (x - 23)/1

x = 1.96 + 23

x = 24.96