10.	The average human birth weight in Australia in 1991 was 3350 g. Since then, the mean age of the mothers when giving birth has increased. Has this impacted the mean birth weight? What is the P-value for a test of the hypotheses H0: μ = 3350 against Ha: μ ≠ 3350 if the calculated statistic is z = 2.56? 

To determine whether the mean birth weight has changed since 1991, we can conduct a hypothesis test. The hypotheses are:

- **Null Hypothesis (H₀):** \(\mu = 3350\) g (The mean birth weight has not changed from 3350 g.)
- **Alternative Hypothesis (Hₐ):** \(\mu \neq 3350\) g (The mean birth weight has changed from 3350 g.)

Given that the calculated test statistic \(z = 2.56\), we need to find the P-value corresponding to this z-value in a two-tailed test.

### Step 1: Find the P-value
The P-value is the probability of observing a test statistic as extreme as the one observed, assuming that the null hypothesis is true. For a two-tailed test, the P-value is calculated as:

\[
\text{P-value} = 2 \times P(Z > |z|)
\]

Where \(Z\) follows the standard normal distribution.

### Step 2: Calculate the P-value
For \(z = 2.56\):

\[
P(Z > 2.56) \approx 0.0052
\]

Thus, the P-value is:

\[
\text{P-value} = 2 \times 0.0052 = 0.0104
\]

### Conclusion
The P-value is approximately 0.0104. If we set a significance level (α) of 0.05, since the P-value (0.0104) is less than 0.05, we would reject the null hypothesis. This suggests that there is significant evidence that the mean birth weight has changed from 3350 g.

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Do you want more details on this test or have any other questions?

Here are some related questions:

1. What is the significance level, and how is it used in hypothesis testing?
2. How do you interpret a P-value in the context of a hypothesis test?
3. What does a two-tailed test imply in hypothesis testing?
4. Can a P-value be higher than 1? Why or why not?
5. What is the relationship between the z-value and the P-value?
6. How does the change in the mean age of mothers potentially affect birth weights?
7. What assumptions must be met to use the z-test in hypothesis testing?
8. What are Type I and Type II errors in hypothesis testing?

**Tip:** In hypothesis testing, the P-value helps us determine whether to reject the null hypothesis, but it does not measure the size of the effect or the importance of the results. Always consider the practical significance of your findings as well.

Explore how changes in the mean age of mothers impact birth weight through hypothesis testing. Calculate the P-value for a z-test with a test statistic of z = 2.56, testing H₀: μ = 3350 against Ha: μ ≠ 3350.

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