Let's go through the calculation again and round the final answer to two decimal places as requested.

Given:

• Sample mean ($\bar{x}$) = 4.19
• Population mean under the null hypothesis ($\mu$) = 4.25
• Sample standard deviation ($s$) = 0.51
• Sample size ($n$) = 9494

Formula for the $t$-statistic:

$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$

Step-by-Step Calculation:

1. Calculate the standard error (SE):

$\text{SE} = \frac{s}{\sqrt{n}} = \frac{0.51}{\sqrt{9494}}$

2. Calculate the square root of the sample size:

$\sqrt{9494} \approx 97.44$

3. Divide the standard deviation by the square root of the sample size to get the standard error:

$\text{SE} = \frac{0.51}{97.44} \approx 0.00523$

4. Compute the $t$-statistic using the sample mean, population mean, and standard error:

$t = \frac{4.19 - 4.25}{0.00523} = \frac{-0.06}{0.00523} \approx -11.47$

Rounded to Two Decimal Places:

$t \approx -11.47$

So, the test statistic, rounded to two decimal places, is $-11.47$.

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Do you want details or have any questions?

Here are 5 questions you might want to ask next:

1. How do you interpret the P-value in hypothesis testing?
2. What are Type I and Type II errors in hypothesis testing?
3. Can you explain how to find critical values in a $t$-distribution table?
4. How would the conclusion change if the significance level was 0.05?
5. What are the assumptions of a $t$-test for a single sample mean?

Tip: Always ensure you use the correct degrees of freedom when consulting the $t$-distribution table for accurate critical values and P-values.