Based on an old survey 46% of blue collar workers do not contribute money to the 401k plan at their job. for this year, you would like to obtain a new sample of blue collar workers to estimate the proportion who do not contribute to their 401k accounts. You would like to be 98% confident that your estimate is within 4.6% of the true population proportion. How large of a sample size is required?
The correct Answer and Explanation is:
To determine the required sample size for estimating the proportion of blue-collar workers who do not contribute to their 401(k) accounts with a specific level of confidence and margin of error, we can use the formula for sample size in proportion estimates:
[
n = \left( \frac{Z^2 \cdot p \cdot (1 – p)}{E^2} \right)
]
Where:
- ( n ) is the required sample size.
- ( Z ) is the Z-score corresponding to the desired confidence level (98% confidence level).
- ( p ) is the estimated population proportion.
- ( E ) is the margin of error.
Step 1: Identify the values
- Confidence level = 98%. The Z-score for a 98% confidence interval can be found using statistical tables or a calculator. It is approximately 2.33.
- Estimated population proportion, ( p ) = 0.46 (from the old survey).
- Margin of error, ( E ) = 0.046 (or 4.6%).
Step 2: Substitute into the formula
Now we can plug these values into the formula:
[
n = \left( \frac{2.33^2 \cdot 0.46 \cdot (1 – 0.46)}{0.046^2} \right)
]
Step 3: Calculate the required sample size
First, calculate the components:
- ( 2.33^2 = 5.4289 )
- ( 0.46 \cdot (1 – 0.46) = 0.46 \cdot 0.54 = 0.2484 )
- ( 0.046^2 = 0.002116 )
Now, calculate ( n ):
[
n = \left( \frac{5.4289 \cdot 0.2484}{0.002116} \right)
]
[
n = \left( \frac{1.34887756}{0.002116} \right) \approx 637.38
]
Since the sample size must be a whole number, round up to the nearest whole number:
[
n \approx 638
]
Explanation:
To estimate the proportion of blue-collar workers who do not contribute to their 401(k) accounts within a 4.6% margin of error and with 98% confidence, we need a sample size of approximately 638 workers. The formula for calculating sample size in proportion estimation involves the Z-score, which accounts for the level of confidence, the estimated proportion based on previous data, and the desired margin of error. A higher confidence level (98%) requires a larger sample size than a lower confidence level, as it ensures that the estimate will likely be within the specified margin of error. In this case, using a Z-score of 2.33 and the estimated population proportion of 46%, the calculation shows that surveying 638 workers will give a precise estimate of the true proportion within the desired margin of 4.6%.