A Nissan Motor Corporation advertisement read, “The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief.

Answer the above questions, show all work.

To conduct a hypothesis test on the claim that the mean I.Q. of brown trout is greater than 4, we need to set up our hypotheses, calculate the test statistic, and compare it to the critical value for a one-tailed test. Let’s go through each step:

### Step 1: State the Hypotheses
1. **Null Hypothesis** \((H_0)\): The mean I.Q. of brown trout is 4, \( \mu = 4 \).
2. **Alternative Hypothesis** \((H_a)\): The mean I.Q. of brown trout is greater than 4, \( \mu > 4 \).

### Step 2: Collect Data
The I.Q. scores for the 12 brown trout are:
\[
5, 4, 7, 3, 6, 4, 5, 3, 6, 3, 8, 5
\]

### Step 3: Calculate the Sample Mean and Sample Standard Deviation
To perform the test, we first calculate the sample mean \(\bar{x}\) and the sample standard deviation \(s\).

1. **Sample Mean** \(\bar{x}\):
   \[
   \bar{x} = \frac{5 + 4 + 7 + 3 + 6 + 4 + 5 + 3 + 6 + 3 + 8 + 5}{12} = \frac{59}{12} \approx 4.92
   \]

2. **Sample Standard Deviation** \(s\):
   \[
   s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
   \]
   where \(x_i\) are the individual I.Q. scores and \(n = 12\). Calculating each squared deviation and summing:

   \[
   \sum (x_i - \bar{x})^2 \approx 24.67
   \]
   \[
   s \approx \sqrt{\frac{24.67}{11}} \approx 1.50
   \]

### Step 4: Calculate the Test Statistic
The test statistic for a one-sample t-test is given by:
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\]
Substitute the values:
\[
t = \frac{4.92 - 4}{1.50 / \sqrt{12}} \approx \frac{0.92}{0.433} \approx 2.13
\]

### Step 5: Determine the Critical Value and Compare
For a one-tailed test with \(n - 1 = 11\) degrees of freedom and a typical significance level of \(\alpha = 0.05\), we can look up the critical t-value in a t-table:
\[
t_{\text{critical}} \approx 1.796
\]

### Step 6: Make a Decision
Since our calculated \(t\)-value (2.13) is greater than the critical \(t\)-value (1.796), we reject the null hypothesis.

### Conclusion
There is sufficient evidence to support the belief that the mean I.Q. of brown trout is greater than 4.

---

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

### Further Questions
1. What would happen if the sample size increased?
2. How does the significance level affect the outcome of this test?
3. Can a hypothesis test conclude absolute truth about the mean I.Q.?
4. What are the implications of using a one-tailed versus a two-tailed test here?
5. How does variability in the sample affect the reliability of our conclusion?

### Tip
When conducting hypothesis tests, always check if your sample meets the assumptions for the test (e.g., normality in the case of small sample sizes for t-tests).

A Nissan Motor Corporation advertisement read, 'The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?' Suppose you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief.

Conduct a hypothesis test to determine if the mean I.Q. of brown trout is greater than 4. This problem involves key statistical concepts, including t-tests, sample mean and standard deviation calculations, and hypothesis testing. Suitable for advanced high school students and above.

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