Answer :
Final answer:
To estimate the mean weight of babies born in a hospital with a 99 percent confidence interval 1 pound wide, a sample of at least 27 birth records is required. If the confidence level is lowered to 95 percent, a sample of at least 16 birth records is needed, assuming a population standard deviation of 1 pound.
Explanation:
To determine the sample size needed for a 99 percent confidence interval that is 1 pound wide, with a known population standard deviation of 1 pound, we can use the formula for the margin of error (E) of a confidence interval: E = Z * (σ/√n), where Z is the Z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.
For a 99% confidence interval, the Z-score (from a Z-table or calculator) is approximately 2.576. To have a margin of error of 0.5 pounds (since the interval is 1 pound wide and the error is half of the width), we calculate the sample size (n) needed:
n = (Zσ/E)^2 = (2.576 * 1 / 0.5)^2 = (5.152)^2 = 26.54. Therefore, the required sample size is at least 27 birth records (since we always round up to the nearest whole number when dealing with sample sizes).
For a 95% confidence interval, the Z-score is approximately 1.96. Repeating the calculation gives:
n = (1.96 * 1 / 0.5)^2 = (3.92)^2 = 15.3664. Therefore, the required sample size is at least 16 birth records.
It's important to note that these calculations assume the central limit theorem applies and that the population weights of babies are normally distributed or the sample size is large enough for the distribution of the sample mean to be approximately normal.
Final answer:
The hospital administrator should take a sample size of 28 birth records for a 99 percent confidence interval and a sample size of 16 for a 95 percent confidence interval, to estimate the mean weight of babies born in the hospital with a 1-pound wide margin of error.
Explanation:
The construction of confidence intervals and the determination of the necessary sample size in statistics, which falls under the realm of Mathematics, specifically statistical mathematics. The grade level can be considered as College due to the complexity of the topic and the understanding required.
To compute the sample size needed, we typically use the formula for determining sample size which is n = (Zα/2 * σ/E)^2. Where:
- Zα/2 is the z-score for the desired confidence level
- σ is the population standard deviation
- E is the margin of error (half the width of the confidence interval).
For a 99 percent confidence level, the z-score is approximately 2.576. For a 95 percent confidence level, the z-score is approximately 1.96. Given that the estimated standard deviation is 1 pound (σ=1), and she wants the confidence interval to be 1 pound wide, that means the margin of error E is 1/2 pound.
Plugging into the formula: For 99% confidence, n = (2.576*1/.5)^2 = 27.868, therefore we would need a sample size of 28 as sample sizes should always be rounded up. For 95% confidence, n = (1.96*1/.5)^2 = 15.366, and we would need a sample size of 16.
Learn more about Confidence Intervals and Sample Size here:
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