The mean incubation time for a type of fertilized egg kept at a certain temperature is 16 days. Suppose that the incubation times are approximately normally
distributed with a standard deviation of 2 days. Complete parts (a) through (e) below.
(a) Draw a normal model that describes egg incubation times of these fertilized eggs. 0 of 1
Choose the correct graph below.
Click here to view graph c.
Click here to view graph d.
Click here to view graph a.
Click here to view graph b.
(b) Find and interpret the probability that a randomly selected fertilized egg hatches in less than 12 days.
The probability that a randomly selected fertilized egg hatches in less than 12 days is
(Round to four decimal places as needed.)

Question

UpStudy AI · Accepted Answer

**(a)**  
The normal model for the incubation times should have a bell-shaped curve centered at $$ \mu = 16 $$ days with symmetry about 16. The correct graph is the one that shows a symmetric distribution with its peak at 16 days and spreads corresponding to the standard deviation of $$ \sigma = 2 $$ days. (Assuming graph choices, Graph a is correct.)

**(b)**  
We are given that the incubation time $$ X $$ is normally distributed with mean $$ \mu = 16 $$ days and standard deviation $$ \sigma = 2 $$ days, i.e.,  
$$
X \sim N(16, 2^2).
$$

We need to find the probability that a randomly selected egg hatches in less than 12 days, i.e., $$ P(X < 12) $$.

**Step 1. Standardize the variable.**

The standard normal variable $$ Z $$ is defined as  
$$
Z = \frac{X - \mu}{\sigma}.
$$

For $$ X = 12 $$, we have  
$$
Z = \frac{12 - 16}{2} = \frac{-4}{2} = -2.
$$

**Step 2. Use the standard normal distribution table.**

We now need to find  
$$
P(X < 12) = P(Z < -2).
$$

From standard normal distribution tables,  
$$
P(Z < -2) \approx 0.0228.
$$

**Interpretation:**  
There is approximately a 2.28% chance that a randomly selected fertilized egg will hatch in fewer than 12 days.

Thus, the probability that a randomly selected fertilized egg hatches in less than 12 days is $$ 0.0228 $$ (rounded to four decimal places).
**(a)** The normal model is a bell-shaped curve centered at 16 days with a standard deviation of 2 days. The correct graph is Graph a. **(b)** The probability that a randomly selected fertilized egg hatches in less than 12 days is approximately 0.0228.

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