A nurse at a local hospital is interested in estimating the birth weight of infants. How large a sample must she select if she desires to be 99% confident that the true mean is within 2 ounces of the sample mean? The standard deviation of the birth weights is known to be 7 ounces.
To determine the sample size required to estimate the birth weight of infants with 99% confidence and within 2 ounces of the sample mean, we can use the formula:
n = (Z * σ / E)²
Where:
- n is the sample size required
- Z is the Z-score corresponding to the desired level of confidence (in this case, 99% confidence level, which corresponds to a Z-score of approximately 2.58)
- σ is the standard deviation of the birth weights (known to be 7 ounces)
- E is the margin of error (in this case, 2 ounces)
Plugging the given values into the formula:
n = (2.58 * 7 / 2)²
n = (18.06 / 2)²
n = 9.03²
n ≈ 81.48
Since we can't have a fractional number of samples, we will round up to the nearest whole number.
Therefore, the nurse must select a sample size of at least 82 infants in order to be 99% confident that the true mean is within 2 ounces of the sample mean.
To find the sample size needed, we can use the formula:
n = (Z * σ / E)^2
where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level
- σ is the standard deviation
- E is the maximum error or margin of error
In this case, the nurse wants to be 99% confident, so the confidence level is 0.99. The nurse also wants the true mean to be within 2 ounces of the sample mean, so the maximum error (E) is 2 ounces. The standard deviation (σ) is given as 7 ounces.
First, we need to find the Z-score for a 99% confidence level. The Z-score can be found using a Z-table or a statistical calculator. For a 99% confidence level, the Z-score is approximately 2.576.
Now we can substitute the values into the formula:
n = (2.576 * 7 / 2)^2
n = (18.032 / 2)^2
n = 9.016^2
n ≈ 81.33
Therefore, the nurse must select a sample size of at least 82 infants to be 99% confident that the true mean is within 2 ounces of the sample mean.
99% = mean ± 2.575 SEm
SEm = SD/√n