To construct a 98% confidence interval for the mean noise level at the given locations, we need to follow these detailed steps:
1. Sample Data: The measurements of noise levels (in decibels) at 5 volcanoes are: 127, 174, 157, 120, and 161.
2. Calculate the Sample Mean:
[tex]\[
\text{Sample Mean} (\bar{x}) = \frac{\sum x_i}{n}
\][/tex]
where [tex]\( x_i \)[/tex] are the sample values and [tex]\( n \)[/tex] is the number of observations. For this data:
[tex]\[
\bar{x} = \frac{127 + 174 + 157 + 120 + 161}{5} = 147.8
\][/tex]
3. Calculate the Sample Standard Deviation:
[tex]\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\][/tex]
Calculate the deviations from the mean, square them, sum them up, divide by [tex]\(n-1\)[/tex], and then take the square root (degrees of freedom [tex]\(df = n-1\)[/tex]):
[tex]\[
s \approx 23.5
\][/tex]
4. Determine the t-score for the 98% Confidence Level:
Since the sample size is 5, the degrees of freedom ([tex]\(df\)[/tex]) is [tex]\(n - 1 = 4\)[/tex]. Look up the t-score that corresponds to a 98% confidence level (which is 0.98/2 + 0.5 = 0.99 in one tail). For [tex]\( df = 4 \)[/tex], the t-score [tex]\( t_{0.99,4} \)[/tex] is approximately 3.747.
5. Calculate the Margin of Error (E):
[tex]\[
E = t_{\alpha/2,df} \times \frac{s}{\sqrt{n}}
\][/tex]
[tex]\[
E = 3.747 \times \frac{23.5}{\sqrt{5}} \approx 38.9
\][/tex]
6. Construct the Confidence Interval:
[tex]\[
\text{Confidence Interval} = \left( \bar{x} - E, \bar{x} + E \right)
\][/tex]
[tex]\[
\left( 147.8 - 38.9, 147.8 + 38.9 \right) = (108.9, 186.7)
\][/tex]
Thus, the 98% confidence interval for the mean noise level at the given volcano locations, rounded to one decimal place, is:
- Lower endpoint: 108.9
- Upper endpoint: 186.7
