Howdoyoucalculate the sample size needed for a study to detect a
specific effect size with 80% power at a 5% significance level?

To calculate the sample size needed for a study to detect a specific effect size with 80% power at a 5% significance level, you can follow these steps:

### 1. **Determine the Effect Size (Cohen's d):**
   - The effect size represents the magnitude of the difference you expect to detect between groups. It's often denoted by Cohen's d, which can be calculated as:
     \[
     d = \frac{\text{Mean difference between groups}}{\text{Standard deviation within groups}}
     \]
   - Common benchmarks for Cohen's d are:
     - Small effect: \( d = 0.2 \)
     - Medium effect: \( d = 0.5 \)
     - Large effect: \( d = 0.8 \)

### 2. **Set the Power (1 - β):**
   - The power of a study is the probability of correctly rejecting the null hypothesis when it is false. 
   - A typical power value is 0.80 (80%).

### 3. **Set the Significance Level (α):**
   - The significance level is the probability of rejecting the null hypothesis when it is true (Type I error).
   - A common value is 0.05 (5%).

### 4. **Use the Appropriate Statistical Formula or Software:**
   - For a two-sample t-test, the sample size for each group can be approximated by:
     \[
     n = \left( \frac{(Z_{\alpha/2} + Z_{\beta})^2 \times 2 \times \sigma^2}{\Delta^2} \right)
     \]
     where:
     - \( Z_{\alpha/2} \) is the critical value of the standard normal distribution at α/2 (for a 5% significance level, \( Z_{\alpha/2} \approx 1.96 \))
     - \( Z_{\beta} \) is the critical value for power (for 80% power, \( Z_{\beta} \approx 0.84 \))
     - \( \sigma^2 \) is the variance within each group (assumed equal)
     - \( \Delta \) is the expected difference in means (effect size)

   - The formula simplifies for known effect size \( d \) to:
     \[
     n = \left( \frac{(Z_{\alpha/2} + Z_{\beta})^2}{d^2} \right)
     \]

### 5. **Adjust for Study Design:**
   - If the study involves more complex designs (e.g., matched pairs, repeated measures), adjustments may be necessary. 

### 6. **Calculate or Use a Sample Size Calculator:**
   - You can manually calculate using the above formulas or use online calculators/software like G*Power, which automates this process.

### Example Calculation:
Let's say we want to detect a medium effect size (Cohen's d = 0.5) with 80% power and a significance level of 0.05.

\[
n = \left( \frac{(1.96 + 0.84)^2}{0.5^2} \right) = \frac{7.84}{0.25} = 31.36
\]
Thus, you would need about **32 subjects per group**.

### Do you want further details or have any questions?

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**Related Questions:**

1. How do you calculate the sample size for a one-sided test instead of a two-sided test?
2. What is the impact of increasing the power from 80% to 90% on the sample size?
3. How does the variance within groups affect the required sample size?
4. What sample size is required if you expect a small effect size (e.g., \( d = 0.2 \))?
5. How can you calculate sample size for studies involving multiple groups?
6. What role does the dropout rate play in determining the final sample size?
7. Can you calculate the sample size needed for a regression analysis?
8. What are some tools or software available for sample size calculations?

**Tip:** Always ensure your estimated effect size is based on previous studies or pilot data for more accurate sample size calculations.

Learn how to calculate the sample size needed for a study to detect a specific effect size with 80% power at a 5% significance level. This problem covers key statistical concepts, including Cohen's d, power, significance level, and the formula for sample size calculation. Suitable for advanced researchers and statisticians.

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