Questions: Hospital Noise Levels for a sample of 5 operating rooms taken in a hospital study, the mean noise level was 42 decibels and the standard deviation was 11.9. Find the 90% confidence interval of the true mean of the noise levels in the operating rooms. Assume the variable is normally distributed. Round your answers to at least two decimal places.

Hospital Noise Levels for a sample of 5 operating rooms taken in a hospital study, the mean noise level was 42 decibels and the standard deviation was 11.9. Find the 90% confidence interval of the true mean of the noise levels in the operating rooms. Assume the variable is normally distributed. Round your answers to at least two decimal places.

Transcript text: Hospital Noise Levels for a sample of 5 operating rooms taken in a hospital study, the mean noise level was 42 decibels and the standard deviation was 11.9. Find the $90\%$ confidence interval of the true mean of the noise levels in the operating rooms. Assume the variable is normally distributed. Round your answers to at least two decimal places.

Solution

Solution Steps

Step 1: Given Data

We are provided with the following data from a study of noise levels in operating rooms:

Sample mean ($\bar{x}$): $42$ dB
Sample standard deviation ($s$): $119$ dB
Sample size ($n$): $5$
Confidence level: $90\%$

Step 2: Determine the Significance Level

The significance level ($\alpha$) is calculated as: \[ \alpha = 1 - \text{confidence level} = 1 - 0.90 = 0.10 \]

Step 3: Calculate the Critical Value

Since the sample size is small ($n < 30$), we use the $t$-distribution. For a $90\%$ confidence level and $n - 1 = 4$ degrees of freedom, the critical value ($t$) is approximately $2.13$.

Step 4: Calculate the Margin of Error

The margin of error ($E$) is calculated using the formula: \[ E = t \cdot \frac{s}{\sqrt{n}} = 2.13 \cdot \frac{119}{\sqrt{5}} \approx 113.45 \]

Step 5: Calculate the Confidence Interval

The confidence interval for the true mean ($\mu$) is given by: \[ \bar{x} \pm E \] Substituting the values: \[ 42 \pm 113.45 \] This results in: \[ (-71.45, 155.45) \]