We are dealing with a normally distributed variable representing incubation times. Let’s solve both parts step by step.

Given:

• Mean $\mu = 2121 \, \text{days}$
• Standard deviation $\sigma = 11 \, \text{days}$

We will use the standard normal distribution (Z-distribution) to solve these problems.

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Part (a): 13th Percentile (P13)

To find the 13th percentile, we need the Z-score associated with the 13th percentile in a standard normal distribution.

1. Find the Z-score: Using a Z-table or inverse Z-scores for percentiles, the Z-score for 13th percentile (P13) is approximately: $Z \approx -1.13$ This means that the 13th percentile lies 1.13 standard deviations below the mean.

2. Convert Z-score to incubation time: Use the formula to convert from Z-score to the raw value $X$: $X = \mu + Z \cdot \sigma$ Substituting the known values: $X = 2121 + (-1.13) \cdot 11$ $X = 2121 - 12.43 \approx 2109 \, \text{days}$

So, the 13th percentile is approximately 2109 days.

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Part (b): Incubation times for the middle 97%

For the middle 97%, the goal is to determine the cutoff values that leave 1.5% on each side of the distribution (since $100\% - 97\% = 3\%$, divided equally into tails).

1. Find Z-scores: Using a Z-table:

   • The lower Z-score for 1.5th percentile is approximately $Z = -2.17$.
   • The upper Z-score for 98.5th percentile is approximately $Z = 2.17$.

2. Convert Z-scores to incubation times:

   • Lower bound: $X = 2121 + (-2.17) \cdot 11 = 2121 - 23.87 \approx 2097 \, \text{days}$
   • Upper bound: $X = 2121 + (2.17) \cdot 11 = 2121 + 23.87 \approx 2145 \, \text{days}$

So, the middle 97% of incubation times are approximately 2097 to 2145 days.

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Final Answers:

(a) The 13th percentile for incubation times is 2109 days.
(b) The incubation times that make up the middle 97% are 2097 to 2145 days.

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Do you want to explore the calculations further or have questions?
Here are five related questions to expand your understanding:

1. How can we interpret a Z-score in practical terms?
2. What percentile corresponds to a Z-score of 0?
3. How would the 13th percentile change if the mean incubation time were higher?
4. What is the probability that an egg will hatch in fewer than 2100 days?
5. How do changes in standard deviation affect the spread of a normal distribution?

Tip: Always sketch a bell curve when working with normal distributions—it helps visualize percentiles and probabilities effectively!