To determine the required sample size for achieving a precision of the margin of error with a 95% confidence interval, we can use the formula for the margin of error in a confidence interval:

$\text{Margin of Error} = z \times \frac{\sigma}{\sqrt{n}}$

Where:

• $z$ is the z-score corresponding to the confidence level,
• $\sigma$ is the population standard deviation,
• $n$ is the sample size.

Given Values

• Current margin of error (at 90% confidence): $E = 1.060$
• Population standard deviation: $\sigma = 17.2$
• Confidence level to achieve: 95%

1. Step 1: Determine the z-score for a 95% confidence interval. The z-score for a 95% confidence interval is approximately $z = 1.96$.

2. Step 2: Set up the margin of error formula for the desired precision.

   Since we know the margin of error and want the same precision (1.060), we can set up the equation as follows:

   $1.060 = 1.96 \times \frac{17.2}{\sqrt{n}}$

3. Step 3: Solve for $n$.

   Rearranging to isolate $n$:

   $\sqrt{n} = \frac{1.96 \times 17.2}{1.060}$

   $n = \left( \frac{1.96 \times 17.2}{1.060} \right)^2$

Let's calculate the exact value for $n$:import math

Given values

z_score_95 = 1.96 population_std_dev = 17.2 desired_margin_of_error = 1.060

Calculate the required sample size n

n_required = (z_score_95 * population_std_dev / desired_margin_of_error) ** 2 n_required = math.ceil(n_required) # Round up to ensure sufficient sample size

n_required10121012successThe required sample size to achieve the same margin of error (1.060) with a 95% confidence interval is 1,012.

Would you like further details on this calculation, or have any additional questions?

Here are some related questions:

1. How does increasing the confidence level affect the margin of error?
2. What role does the population standard deviation play in determining sample size?
3. Why is it necessary to round up the sample size in statistical calculations?
4. How would the sample size change if the margin of error was decreased to 0.5?
5. What are some practical limitations of achieving very high confidence intervals with small margins of error?

Tip: When calculating sample size, rounding up ensures the desired precision and confidence, as even a slightly smaller sample might not meet the required criteria.