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.