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Let's work through this problem step-by-step.

### Given:
- Population mean (\(\mu_0\)) = 2 kg
- Sample mean (\(\bar{x}\)) = 1.8 kg
- Sample standard deviation (\(s\)) = 0.5 kg
- Sample size (\(n\)) = 20
- Significance level (\(\alpha\)) = 0.01 for a two-tailed test

### (a) State the null and alternative hypotheses.

- **Null hypothesis** (\(H_0\)): The average weight of the fish is 2 kg.
  \[
  H_0: \mu = 2 \text{ kg}
  \]
- **Alternative hypothesis** (\(H_1\)): The average weight of the fish is not 2 kg.
  \[
  H_1: \mu \neq 2 \text{ kg}
  \]

### (b) Calculate the t-statistic.

The formula for the t-statistic is:
\[
t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}
\]
Substituting the given values:
\[
t = \frac{1.8 - 2}{0.5 / \sqrt{20}} = \frac{-0.2}{0.5 / 4.4721} \approx \frac{-0.2}{0.1118} \approx -1.79
\]
Thus, the t-statistic is approximately **-1.79**.

### (c) Determine the critical t-value at a significance level of 0.01 (two-tailed).

For a two-tailed test with \(\alpha = 0.01\) and \(n - 1 = 19\) degrees of freedom, we can look up the critical t-value in a t-distribution table.

- At \(df = 19\) and \(\alpha = 0.01\) for a two-tailed test, the critical t-value is approximately **±2.861**.

### (d) Based on the t-statistic and the critical value, determine whether the null hypothesis should be rejected.

The calculated t-statistic is \(-1.79\), and the critical values are \(\pm 2.861\).

Since \(-1.79\) lies between \(-2.861\) and \(2.861\), we **fail to reject the null hypothesis**. This means that there is not enough evidence at the 1% significance level to conclude that the average weight of the fish is different from 2 kg.

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Would you like further details or explanations on any of these steps?

Here are five related questions to expand your understanding:

1. What would change in the t-statistic if the sample size was larger?
2. How would the t-test differ if it were a one-tailed test?
3. What assumptions are required for the t-test to be valid?
4. How does the choice of significance level (e.g., 0.05 vs. 0.01) affect hypothesis testing?
5. What is the interpretation of a p-value in the context of hypothesis testing?

**Tip:** When conducting hypothesis tests, always ensure you check the conditions (e.g., normality or sample size) to justify using the t-distribution.

A researcher wants to test whether the average weight of a certain species of fish is 2 kg. A sample of 20 fish is taken, yielding a sample mean weight of 1.8 kg with a standard deviation of 0.5 kg.

(a) State the null and alternative hypotheses.
(b) Calculate the t-statistic.
(c) Determine the critical t-value at a significance level of 0.01 (two-tailed).
(d) Based on the t-statistic and the critical value, determine whether the null hypothesis should be rejected.

Step-by-step solution to a hypothesis test problem about fish weight, covering null and alternative hypotheses, calculating t-statistics, and critical t-value at a 0.01 significance level with a two-tailed test. Ideal for undergraduate students.

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