High School

Noise levels at 77 concerts were measured in decibels, yielding the following data:

197, 141, 141, 152, 145, 187, 166, 197, 141, 141, 152, 145, 187, 166.

Construct the 80% confidence interval for the mean noise level at such locations. Assume the population is approximately normal.

Step 2 of 4: Calculate the sample standard deviation for the given sample data. Round your answer to one decimal place.

Answer :

To calculate the sample standard deviation for the given noise level data, follow these steps:

  1. **List the Data: **

    • The noise levels are: 197, 141, 141, 152, 145, 187, 166, 197, 141, 141, 152, 145, 187, 166.

  2. Calculate the Mean (Average):

    • First, add up all the noise levels:
      [tex]197 + 141 + 141 + 152 + 145 + 187 + 166 + 197 + 141 + 141 + 152 + 145 + 187+ 166 = 2058[/tex]

    • Then, divide the sum by the number of data points (14):
      [tex]\text{Mean} = \frac{2058}{14} \approx 147.0[/tex]

  3. Calculate Each Deviation from the Mean and Square It:

    • For each data point, subtract the mean and then square the result. For example, for the first noise level 197:
      [tex](197 - 147)^2 = 2500[/tex]

  4. Find the Sum of Squared Deviations:

    • Repeat step 3 for each data point and sum all the squared deviations.
    • Total sum of squared deviations is:
      [tex](2500 + (141-147)^2 + (141-147)^2 + \ldots + (166-147)^2 ) = 7342.0[/tex]

  5. Calculate the Variance:

    • Divide the sum of squared deviations by the number of data points minus one (n-1, which is 13 in this case, because it’s a sample):
      [tex]\text{Variance} = \frac{7342.0}{13} \approx 564.8[/tex]

  6. Calculate the Sample Standard Deviation:

    • Take the square root of the variance:
      [tex]\text{Sample Standard Deviation} = \sqrt{564.8} \approx 23.8[/tex]

So, the sample standard deviation for the given noise level data is approximately 23.8 decibels.

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