The ideal (daytime) noise-level for hospitals is 45 decibels with a standard deviation of 10 dB. A simple random sample of 81 hospitals at a moment during the day gives a mean noise level of 47 dB. Assume that the standard deviation of noise level is really 10 dB. All answers to two places after the decimal.
Understanding the noise levels in hospitals is critical for ensuring patient comfort and safety. The National Institute for Occupational Safety and Health (NIOSH) recommends that the daytime noise level in hospitals should not exceed 45 decibels to promote a healing environment. This study explores various statistical analyses to determine the confidence intervals and margins of error associated with measuring the mean noise level across hospitals, based on a sample of 81 hospitals with an observed mean of 47 decibels, assuming a known population standard deviation of 10 decibels.
Firstly, the sample provides a basis for constructing confidence intervals to estimate the true average noise level in all hospitals. Given the data, a 99% confidence interval for the population mean can be computed using the z-distribution since the population standard deviation is known. The formula involves the sample mean, the z-value corresponding to the confidence level, and the standard error (standard deviation divided by the square root of the sample size). For a 99% confidence level, the z-value is approximately 2.576.
Calculating the margin of error (ME) for the 99% confidence interval involves multiplying the z-value by the standard error: ME = 2.576 * (10 / √81) = 2.576 * (10 / 9) ≈ 2.864. The confidence interval is then: (47 - 2.864, 47 + 2.864), which results in approximately (44.14 dB, 49.86 dB).
Next, a 90% confidence level uses a z-value of approximately 1.645, leading to a margin of error of 1.645 * (10/9) ≈ 1.828. This yields an interval of (47 - 1.828, 47 + 1.828), or approximately (45.17 dB, 48.83 dB). We can interpret this as being 90% confident that the true mean noise level falls within this range.
The concept of most unusual samples relates to the probability of observing a sample mean as extreme as the one obtained, assuming the true mean is at the ideal level of 45 dB. If the sample is not among the most unusual 2% of samples, it suggests the population mean is within the bounds defined by the corresponding confidence interval—here, approximately between 44.14 dB and 49.86 dB for a 99% confidence level.
A more stringent 99.9% confidence interval, with a z-value of approximately 3.291, results in a margin of error of 3.291 * (10/9) ≈ 3.657. The interval becomes (47 - 3.657, 47 + 3.657), or roughly (43.34 dB, 50.66
dB). This provides a higher level of certainty about the population mean but results in a wider interval.
Assuming the sample is among the most typical half of all possible samples, the best estimate for the true mean is around the sample mean of 47 dB, with a margin of error corresponding to a 50% confidence level, which is approximately 0.674 * (10/√81) ≈ 0.674 * 1.111 ≈ 0.75. Therefore, the interval is approximately (47 - 0.75, 47 + 0.75), or (46.25 dB, 47.75 dB).
For a 95% confidence level, the z-value is approximately 1.96, giving a margin of error of 1.96 * (10/9) ≈ 2.178. The confidence interval would then be (47 - 2.178, 47 + 2.178), or approximately (44.82 dB, 49.18 dB).
To determine the sample size required to achieve a margin of error of 1 dB at 95% confidence, we rearrange the margin of error formula to solve for n: n = (z * σ / E)². Substituting the known values (z = 1.96, σ = 10, E = 1), we get n = (1.96 * 10 / 1)² ≈ (19.6)² ≈ 384.16. Therefore, about 385 hospitals need to be examined.
Similarly, for 99.9% confidence (z ≈ 3.291), the required sample size is n = (3.291 * 10 / 1)² ≈ (32.91)² ≈ 1083.17, so approximately 1084 hospitals must be examined to achieve the desired margin of error.
In conclusion, analyzing and interpreting the noise levels through various confidence intervals and sample size calculations provides vital insights for hospital planning and policy. Ensuring the noise environment meets recommended standards can enhance patient outcomes and staff well-being.
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