To conduct a hypothesis test regarding the brown trout's IQ, we can follow these steps, assuming a one-sample $t$-test is appropriate for this small sample size.

1. Set up the Hypotheses

Let:

• $\mu$ = the true mean IQ of brown trout.

Our hypotheses are:

• Null Hypothesis $H_0$: $\mu = 4$ (The mean IQ of brown trout is 4)
• Alternative Hypothesis $H_1$: $\mu > 4$ (The mean IQ of brown trout is greater than 4)

2. Collect Sample Data

The sample data for IQs of the 12 brown trout is: $5, 4, 7, 3, 6, 4, 5, 3, 6, 3, 8, 5$

3. Calculate Sample Mean and Standard Deviation

Let's calculate the sample mean $\bar{x}$ and sample standard deviation $s$.

The sample mean $\bar{x}$ is given by: $\bar{x} = \frac{\sum x}{n}$ where $x$ are the individual IQ scores, and $n = 12$.

The sample standard deviation $s$ is: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$

4. Calculate the Test Statistic

Using a one-sample $t$-test, the test statistic $t$ is given by: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$ where:

• $\mu_0 = 4$ (the hypothesized mean),
• $\bar{x}$ is the sample mean, and
• $s$ is the sample standard deviation.

5. Determine the Critical Value and Make a Decision

We will use a significance level of $\alpha = 0.05$ for a one-tailed test. For $n - 1 = 11$ degrees of freedom, we can find the critical value $t_{0.05, 11}$ from the $t$-distribution table.

If $t$ exceeds the critical value, we reject $H_0$; otherwise, we do not reject $H_0$.

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Let's proceed with these calculations.The calculations yield the following results:

• Sample Mean ($\bar{x}$) = 4.917
• Sample Standard Deviation ($s$) = 1.621
• Test Statistic ($t$) = 1.959
• Critical Value at $\alpha = 0.05$ for a one-tailed test with 11 degrees of freedom = 1.796

Conclusion:

Since the test statistic $t = 1.959$ is greater than the critical value $1.796$, we reject the null hypothesis $H_0$. This suggests that there is statistically significant evidence at the 5% significance level to support the belief that the mean IQ of brown trout is greater than 4.

Would you like further details on any specific step, or have additional questions?

Here are some related questions:

1. What would be the conclusion if we used a different significance level, like 1%?
2. How does increasing the sample size affect the reliability of the hypothesis test?
3. What are Type I and Type II errors in hypothesis testing?
4. How would this test change if the alternative hypothesis were $\mu \neq 4$?
5. Can hypothesis tests be used to make decisions in fields other than psychology or biology?

Tip: Always check assumptions of normality when using $t$-tests, especially with small sample sizes.