Determine the test statistic.

enter your response here 

​(Round to two decimal places as​ needed.)

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.

Learn how to calculate the T-statistic for hypothesis testing using a sample mean, population mean, sample standard deviation, and sample size. Step-by-step guide with detailed formula application and rounding to two decimal places.

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