The lengths of human pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. What is the probability that a pregnancy last at least 300 days?




Question 3Answer

a.

0.0166

b.

0.4834

c.

0.0179

d.

0.9834

To find the probability that a pregnancy lasts at least 300 days, we'll use the standard normal distribution.

### Step 1: Calculate the z-score
The z-score measures how many standard deviations a value (in this case, 300 days) is from the mean.

The z-score formula is:

\[
z = \frac{X - \mu}{\sigma}
\]

where:
- \(X = 300\) days (the value we are interested in),
- \(\mu = 268\) days (the mean),
- \(\sigma = 15\) days (the standard deviation).

Substitute the values:

\[
z = \frac{300 - 268}{15} = \frac{32}{15} \approx 2.13
\]

### Step 2: Find the probability corresponding to the z-score
We need to find the probability that a standard normal variable \(Z\) is greater than 2.13.

Using standard normal distribution tables or a calculator:

\[
P(Z > 2.13) \approx 0.0166
\]

### Step 3: Interpretation
The probability that a pregnancy lasts at least 300 days is approximately 0.0166.

### Answer:
The correct answer is **a. 0.0166**.

Would you like more details or have any other questions?

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Here are some related questions you might find interesting:
1. How do you calculate the z-score for any value in a normal distribution?
2. What does the z-score represent in a standard normal distribution?
3. How do you use a z-score table to find probabilities?
4. What is the probability that a pregnancy lasts between 250 and 280 days?
5. What is the probability that a pregnancy lasts less than 260 days?
6. How does changing the standard deviation affect the probability calculations?
7. How can you calculate the mean and standard deviation from raw data?
8. How do normal distributions apply to other real-world scenarios?

**Tip:** Always double-check the z-score formula and your calculations when solving problems involving the normal distribution!

Learn how to calculate the probability that a pregnancy lasts at least 300 days using the normal distribution. Understand the z-score concept, standard deviation, and mean values. Detailed step-by-step solution provided.

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