The t-test is a statistical test used to compare the means of two groups and determine if they are significantly different from each other. Here's a simplified step-by-step calculation for a one-sample t-test:

### Step-by-Step Calculation of a One-Sample t-Test

1. **Formulate Hypotheses**:
   - Null Hypothesis (\( H_0 \)): The sample mean is equal to the population mean (\( \mu_0 \)).
   - Alternative Hypothesis (\( H_a \)): The sample mean is not equal to the population mean (\( \mu_0 \)).

2. **Collect Data**: 
   - Sample Mean (\( \bar{x} \))
   - Sample Size (\( n \))
   - Sample Standard Deviation (\( s \))
   - Population Mean (\( \mu_0 \)).

3. **Calculate the t-Statistic**:
   - Formula: 

     \[
     t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
     \]

   - Where:
     - \( \bar{x} \): Sample mean
     - \( \mu_0 \): Population mean
     - \( s \): Sample standard deviation
     - \( n \): Sample size

4. **Determine Degrees of Freedom**:
   - \( df = n - 1 \)

5. **Compare t-Statistic with Critical Value**:
   - Use a t-distribution table to find the critical value based on your degrees of freedom (\( df \)) and the desired significance level (e.g., 0.05).

6. **Make a Decision**:
   - If \( |t| \) is greater than the critical value, reject the null hypothesis.

### Example Calculation:

- Sample Mean (\( \bar{x} \)) = 50
- Population Mean (\( \mu_0 \)) = 45
- Sample Standard Deviation (\( s \)) = 10
- Sample Size (\( n \)) = 16

**Step 3: Calculate the t-statistic**:

\[
t = \frac{50 - 45}{\frac{10}{\sqrt{16}}} = \frac{5}{\frac{10}{4}} = \frac{5}{2.5} = 2
\]

**Step 4: Degrees of Freedom**:

\[
df = 16 - 1 = 15
\]

**Step 5 & 6: Decision**:

- If the critical value at \( df = 15 \) and a 0.05 significance level is, say, 2.13, since 2 is less than 2.13, we do not reject the null hypothesis.

Would you like more details or have any questions? Here are some related questions to expand on this:

1. How do you perform a two-sample t-test?
2. What are the assumptions of a t-test?
3. How do you interpret p-values in a t-test?
4. How do you conduct a paired t-test?
5. What are the differences between a t-test and a z-test?

**Tip:** Always check the assumptions of normality and equal variances when using a t-test to ensure accurate results.

Learn how to perform a one-sample t-test using a simplified step-by-step calculation. This example includes hypothesis testing, calculating the t-statistic, and interpreting results for statistical significance. Perfect for college-level statistics students.

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