An educator wants to estimate the proportion of school children in Boston who are living with only one parent. Since their report is to be published, they want a reasonably accurate estimate. However, since their funding is limited, they do not want to collect a larger sample than necessary. They hope to use a sample size such that, with probability 0.95, the error will not exceed 0.04. What sample size will ensure this, regardless of what sample proportion value occurs when they gather the sample?
Try this formula:
n = [(z-value)^2 * p * q]/E^2
Note: n = sample size needed; .5 for p and .5 for q are used if no value is stated in the problem. E = maximum error, which is .04 in the problem. Z-value is found using a z-table (for 95%, the value is 1.96).
When you find n, round up to the next highest whole number.
To determine the sample size that will ensure the desired level of accuracy, we need to use the formula for sample size calculation when estimating a proportion.
The formula for calculating the sample size for estimating a proportion is:
n = (Z^2 * p * (1-p)) / E^2
Where:
n is the required sample size
Z is the z-score corresponding to the desired level of confidence (0.95, in this case)
p is the estimated proportion (unknown in this case)
E is the desired margin of error (0.04, in this case)
Since the proportion is unknown, we use the worst-case scenario of p = 0.5, which maximizes the sample size, ensuring that it will cover all possible values of p.
Plugging in the known values into the formula:
n = (Z^2 * p * (1-p)) / E^2
n = (1.96^2 * 0.5 * (1-0.5)) / 0.04^2
n = 3.8416 * 0.25 / 0.0016
n = 9.604 / 0.0016
n = 6002.5
Therefore, a sample size of approximately 6002 (rounded up to the nearest whole number) would ensure that, regardless of the sample proportion value, the error in the estimate will not exceed 0.04, with a 95% probability.