Question
What is the minimum sample size required for a 95% confidence interval with a 3% margin of error to estimate the proportion of college students who have taken a course in data science?
Answers
GPT-4o mini
To determine the minimum sample size required for a 95% confidence interval with a 3% margin of error when estimating a proportion, we can use the formula for sample size estimation for proportions. The formula is:
\[
n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right)
\]
Where:
- \( n \) = required sample size
- \( Z \) = Z-score corresponding to the desired confidence level
- \( p \) = estimated proportion of the population (if unknown, use 0.5 for maximum variability)
- \( E \) = margin of error
### Step 1: Determine the Z-score
For a 95% confidence level, the Z-score is approximately 1.96.
### Step 2: Estimate the proportion (p)
If you don't have prior data, it is common practice to use \( p = 0.5 \) to get the maximum required sample size.
### Step 3: State the margin of error (E)
The margin of error is given as 3%, which can be expressed as a decimal:
\[
E = 0.03
\]
### Step 4: Substitute the values into the formula
Using these values:
- \( Z = 1.96 \)
- \( p = 0.5 \)
- \( E = 0.03 \)
Now, substitute these values into the sample size formula:
\[
n = \left( \frac{(1.96)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.03)^2} \right)
\]
Calculating this step-by-step:
1. Calculate \( Z^2 \):
\[
(1.96)^2 = 3.8416
\]
2. Calculate \( p(1 - p) \):
\[
0.5 \cdot (1 - 0.5) = 0.5 \cdot 0.5 = 0.25
\]
3. Calculate \( E^2 \):
\[
(0.03)^2 = 0.0009
\]
4. Plug in these values into the sample size equation:
\[
n = \frac{3.8416 \cdot 0.25}{0.0009}
\]
\[
n = \frac{0.9604}{0.0009} \approx 1067.11
\]
### Step 5: Round up to the nearest whole number
Since sample size cannot be a fraction, round up to the nearest whole number:
\[
n \approx 1068
\]
### Conclusion
The minimum sample size required for a 95% confidence interval with a 3% margin of error is **1068**.
\[
n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right)
\]
Where:
- \( n \) = required sample size
- \( Z \) = Z-score corresponding to the desired confidence level
- \( p \) = estimated proportion of the population (if unknown, use 0.5 for maximum variability)
- \( E \) = margin of error
### Step 1: Determine the Z-score
For a 95% confidence level, the Z-score is approximately 1.96.
### Step 2: Estimate the proportion (p)
If you don't have prior data, it is common practice to use \( p = 0.5 \) to get the maximum required sample size.
### Step 3: State the margin of error (E)
The margin of error is given as 3%, which can be expressed as a decimal:
\[
E = 0.03
\]
### Step 4: Substitute the values into the formula
Using these values:
- \( Z = 1.96 \)
- \( p = 0.5 \)
- \( E = 0.03 \)
Now, substitute these values into the sample size formula:
\[
n = \left( \frac{(1.96)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.03)^2} \right)
\]
Calculating this step-by-step:
1. Calculate \( Z^2 \):
\[
(1.96)^2 = 3.8416
\]
2. Calculate \( p(1 - p) \):
\[
0.5 \cdot (1 - 0.5) = 0.5 \cdot 0.5 = 0.25
\]
3. Calculate \( E^2 \):
\[
(0.03)^2 = 0.0009
\]
4. Plug in these values into the sample size equation:
\[
n = \frac{3.8416 \cdot 0.25}{0.0009}
\]
\[
n = \frac{0.9604}{0.0009} \approx 1067.11
\]
### Step 5: Round up to the nearest whole number
Since sample size cannot be a fraction, round up to the nearest whole number:
\[
n \approx 1068
\]
### Conclusion
The minimum sample size required for a 95% confidence interval with a 3% margin of error is **1068**.