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The question in the image involves selecting the appropriate test statistic and calculating it for two scenarios. Here's the solution:

### Part (a):
#### Given:
- Sample size \(n = 11\),
- Population is normally distributed,
- Population standard deviation (\(\sigma\)) is known: \(2.3\).

Since the population standard deviation is known, **use the \(z\)-statistic**.

The formula for \(z\) is:
\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]
Here:
- \(\bar{x} = 29.6\) (sample mean),
- \(\mu = 30.3\) (population mean),
- \(\sigma = 2.3\),
- \(n = 11\).

Now calculate:
\[
z = \frac{29.6 - 30.3}{\frac{2.3}{\sqrt{11}}} = \frac{-0.7}{\frac{2.3}{3.3166}} = \frac{-0.7}{0.6934} \approx -1.01
\]

Answer for (a): \(z = -1.01\).

---

### Part (b):
#### Given:
- Sample size \(n = 20\),
- Population distribution is unknown.

Since the population distribution is unknown and the sample size is relatively small (\(<30\)), **use the \(t\)-statistic**.

The formula for \(t\) is:
\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\]
Here:
- \(\bar{x} = 29.6\),
- \(\mu = 30.3\),
- \(s = 2.3\) (sample standard deviation),
- \(n = 20\).

Now calculate:
\[
t = \frac{29.6 - 30.3}{\frac{2.3}{\sqrt{20}}} = \frac{-0.7}{\frac{2.3}{4.4721}} = \frac{-0.7}{0.5145} \approx -1.36
\]

Answer for (b): \(t = -1.36\).

---

### Summary of Answers:
- (a) \(z = -1.01\),
- (b) \(t = -1.36\).

---

Do you need further clarifications or detailed steps?  
Here are 5 questions to deepen your understanding:

1. Why do we use \(z\)-statistics when the population standard deviation is known?
2. What are the critical differences between \(z\)-statistics and \(t\)-statistics?
3. How does the sample size influence the decision to use \(z\) or \(t\)?
4. What assumptions are required for using the \(t\)-statistic in small samples?
5. How would the test statistics change if the sample sizes were larger?

**Tip**: Always verify if the population standard deviation or sample standard deviation is provided—it directly affects the choice of test statistic!

We want to conduct a hypothesis test to determine if the population mean reading speed of second graders is different from 30.3 words per minute, using sample data provided in the scenarios.

Understand hypothesis testing for population means using sample data. Learn how to determine the correct test statistic (Z or T) based on sample size and standard deviation. Includes step-by-step calculations for Z = -1.01 and T = -1.36.

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