You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the
population standard deviation is approximately o = 48.7. You would like to be 98% confident that your
estimate is within 10 of the true population mean. How large of a sample size is required?
n =

To determine the required sample size \( n \) for estimating a population mean with a specified confidence level and margin of error, we use the following formula:

\[
n = \left( \frac{Z \cdot \sigma}{E} \right)^2
\]

where:
- \( Z \) is the Z-score corresponding to the desired confidence level,
- \( \sigma \) is the population standard deviation (in this case, 48.7),
- \( E \) is the margin of error (in this case, 10).

### Step 1: Determine the Z-score for a 98% confidence level
For a 98% confidence level, we look up the Z-score that leaves 1% in each tail (because \( 100\% - 98\% = 2\% \), split between two tails). The Z-score for 98% confidence is approximately \( Z = 2.33 \).

### Step 2: Plug values into the formula
Substitute \( Z = 2.33 \), \( \sigma = 48.7 \), and \( E = 10 \):

\[
n = \left( \frac{2.33 \cdot 48.7}{10} \right)^2
\]

### Step 3: Calculate the sample size

1. Calculate the numerator: \( 2.33 \cdot 48.7 = 113.471 \).
2. Divide by the margin of error: \( \frac{113.471}{10} = 11.3471 \).
3. Square this result: \( 11.3471^2 \approx 128.74 \).

Since the sample size \( n \) must be a whole number, we round up to ensure the desired confidence level. 

**Final Answer:**
\[
n = 129
\]

This is the required sample size to be 98% confident that the estimate is within 10 units of the true population mean.

Would you like further details on this calculation?

---

1. How does the Z-score change for different confidence levels?
2. What happens if the margin of error is decreased?
3. Why do we use the Z-score and not a t-score in this case?
4. How would this calculation change if the population standard deviation were unknown?
5. Can this formula be applied to other confidence intervals, like 90% or 95%?

**Tip:** Always round up to the next whole number for sample sizes to maintain the desired confidence level and precision.

You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately σ = 48.7. You would like to be 98% confident that your estimate is within 10 of the true population mean. How large of a sample size is required?

Determine the sample size needed for a 98% confidence level, where the population standard deviation is 48.7 and the desired margin of error is 10. This calculation uses statistical concepts like confidence intervals and sample size determination.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation