The distribution of birth weight infants at 40 weeks gestational is approximately normal with a mean of 3500 grams and a standard deviation of 430 grams. Given a sample of five newborns all with gestational age 40 weeks a what is the probability that their mean birth weight is less than 2500 grams? What is the probability that only one of the five newborns that has a birth weight less than 2500 grams? 

We are given a normal distribution for birth weight at 40 weeks gestation, where the **mean** (\(\mu\)) is **3500 grams** and the **standard deviation** (\(\sigma\)) is **430 grams**. Let's solve the problem in two parts.

### Part A: Probability that the **mean birth weight** of a sample of 5 newborns is less than 2500 grams

To solve this, we need to use the concept of the **sampling distribution of the sample mean**. For a normally distributed population, the distribution of the sample mean \(\bar{X}\) is also normal with the same mean, but the standard deviation is scaled by the square root of the sample size. This is known as the **standard error**.

The standard error of the mean (SEM) is given by:
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{430}{\sqrt{5}} \approx 192.32
\]

Now we want the probability that the mean of the sample is less than 2500 grams, i.e., \(P(\bar{X} < 2500)\).

We calculate the **Z-score** for 2500 grams using the formula:
\[
Z = \frac{\bar{X} - \mu}{\text{SEM}} = \frac{2500 - 3500}{192.32} \approx \frac{-1000}{192.32} \approx -5.2
\]

The Z-score is approximately \(-5.2\), which is extremely low. The probability of having a Z-score this low is nearly **0**. Using standard normal distribution tables, we can see that:

\[
P(\bar{X} < 2500) \approx 0
\]

Thus, the probability that the mean birth weight of 5 newborns is less than 2500 grams is essentially **0**.

---

### Part B: Probability that only **one** of the five newborns has a birth weight less than 2500 grams

For this part, we treat the problem as a **binomial probability problem**, where we are interested in the probability of exactly one newborn having a birth weight less than 2500 grams. 

First, we need to find the probability that a **single newborn** has a birth weight less than 2500 grams. This can be done using the **Z-score** for an individual birth weight:

\[
Z = \frac{2500 - 3500}{430} = \frac{-1000}{430} \approx -2.33
\]

Using a standard normal distribution table, the probability corresponding to a Z-score of \(-2.33\) is approximately:

\[
P(X < 2500) \approx 0.0099
\]

Thus, the probability of a single newborn having a birth weight less than 2500 grams is approximately **0.0099**.

Now, we use the **binomial probability formula** to calculate the probability of exactly 1 success (a birth weight less than 2500 grams) out of 5 newborns:

\[
P(X = 1) = \binom{5}{1} \cdot (0.0099)^1 \cdot (1 - 0.0099)^4
\]

Let’s calculate this probability:
\[
P(X = 1) = 5 \cdot (0.0099) \cdot (0.9901)^4
\]

Now let's compute the value to get the final answer.

The probability that exactly one of the five newborns has a birth weight less than 2500 grams is approximately **0.0476** or **4.76%**.

### Summary:
- **Part A:** The probability that the mean birth weight of a sample of 5 newborns is less than 2500 grams is essentially **0**.
- **Part B:** The probability that only one of the five newborns has a birth weight less than 2500 grams is approximately **4.76%**.

Would you like more details, or do you have any questions?

### Related questions:
1. What is the probability that more than one newborn weighs less than 2500 grams?
2. How would the probability change if the sample size increased to 10?
3. What is the probability that none of the five newborns weighs less than 2500 grams?
4. How does changing the standard deviation affect these probabilities?
5. What is the 95th percentile of birth weights at 40 weeks?

### Tip:
For very small probabilities in a binomial setting, approximations like the Poisson distribution can sometimes be useful.

The distribution of birth weight infants at 40 weeks gestational is approximately normal with a mean of 3500 grams and a standard deviation of 430 grams. Given a sample of five newborns all with gestational age 40 weeks, what is the probability that their mean birth weight is less than 2500 grams? What is the probability that only one of the five newborns has a birth weight less than 2500 grams?

Explore the normal distribution of birth weights at 40 weeks gestation with a mean of 3500 grams and standard deviation of 430 grams. Learn how to calculate the probability of the mean birth weight of five newborns being less than 2500 grams, as well as the probability that only one of five newborns has a birth weight below 2500 grams. Suitable for college-level or advanced high school statistics students.

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