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To solve these normal distribution problems, we need to use the mean (\(\mu = 6.9\) pounds) and the standard deviation (\(\sigma = 2.1\) pounds) provided. We can calculate probabilities using the standard normal distribution (Z-scores).

Let’s go through each question:

1. **How many would you expect to weigh between 4 and 9 pounds?**
   - Find the Z-scores for 4 and 9 pounds:
     \[
     Z = \frac{X - \mu}{\sigma}
     \]
     For 4 pounds:
     \[
     Z_1 = \frac{4 - 6.9}{2.1} = \frac{-2.9}{2.1} \approx -1.38
     \]
     For 9 pounds:
     \[
     Z_2 = \frac{9 - 6.9}{2.1} = \frac{2.1}{2.1} = 1
     \]
     Use a Z-table or calculator to find the probabilities for these Z-scores. The probability between Z-scores of -1.38 and 1 is approximately 0.8315.

     So, for 900 newborns, expect:
     \[
     0.8315 \times 900 = 748.35 \text{ (around 748 newborns)}
     \]

2. **How many would you expect to weigh less than 6 pounds?**
   - Find the Z-score for 6 pounds:
     \[
     Z = \frac{6 - 6.9}{2.1} = \frac{-0.9}{2.1} \approx -0.43
     \]
     The probability for \(Z = -0.43\) is approximately 0.3336.

     So, for 900 newborns:
     \[
     0.3336 \times 900 = 300.24 \text{ (around 300 newborns)}
     \]

3. **How many would you expect to weigh more than 5 pounds?**
   - Find the Z-score for 5 pounds:
     \[
     Z = \frac{5 - 6.9}{2.1} = \frac{-1.9}{2.1} \approx -0.90
     \]
     The probability for \(Z = -0.90\) is approximately 0.1841, but we want the probability of weighing *more* than 5 pounds, so:
     \[
     P(X > 5) = 1 - 0.1841 = 0.8159
     \]
     So, for 900 newborns:
     \[
     0.8159 \times 900 = 734.31 \text{ (around 734 newborns)}
     \]

4. **How many would you expect to weigh between 6.9 and 10 pounds?**
   - Find the Z-scores for 6.9 and 10 pounds:
     \[
     Z_1 = \frac{6.9 - 6.9}{2.1} = 0
     \]
     \[
     Z_2 = \frac{10 - 6.9}{2.1} \approx 1.48
     \]
     The probability between Z-scores of 0 and 1.48 is approximately 0.4306.

     So, for 900 newborns:
     \[
     0.4306 \times 900 = 387.54 \text{ (around 388 newborns)}
     \]

Let me know if you'd like more details on the process or have any questions!

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Here are 5 related questions to expand this topic:

1. How do you calculate probabilities from Z-scores using a Z-table?
2. What is the relationship between standard deviation and the spread of a normal distribution?
3. How do you interpret the area under the normal curve in real-world terms?
4. How would you calculate the probability of weights in a different range (e.g., between 3 and 7 pounds)?
5. Can you explain how changing the standard deviation affects the distribution?

**Tip:** Always remember to subtract from 1 when calculating probabilities for "more than" or "greater than" scenarios in normal distributions.

The weights for newborn babies are approximately normally distributed with a mean of 6.9 pounds and a standard deviation of 2.1 pounds. Consider a group of 900 newborn babies: 1) How many would you expect to weigh between 4 and 9 pounds? 2) How many would you expect to weigh less than 6 pounds? 3) How many would you expect to weigh more than 5 pounds? 4) How many would you expect to weigh between 6.9 and 10 pounds?

Solve normal distribution problems involving newborn baby weights with a mean of 6.9 pounds and a standard deviation of 2.1 pounds. Includes calculations for Z-scores and probabilities for different weight ranges. Suitable for college-level statistics or advanced high school students.

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