To construct a 95% confidence interval for the population mean, we will use the t-distribution because the sample size is small (n < 30) and the population standard deviation is unknown. First, calculate the sample mean and sample standard deviation. Then, determine the t-score for a 95% confidence level with n-1 degrees of freedom. Finally, use these values to calculate the confidence interval.
The sample size \( n \) is the number of observations in the data set. Here, the data set is \([3.0, 2.3, 2.3, 2.2, 2.8, 2.1, 1.9, 2.4]\), so the sample size is \( n = 8 \).
The sample mean \(\bar{x}\) is calculated as the sum of all observations divided by the number of observations:
\[
\bar{x} = \frac{3.0 + 2.3 + 2.3 + 2.2 + 2.8 + 2.1 + 1.9 + 2.4}{8} = 2.375
\]
The sample standard deviation \( s \) is calculated using the formula:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]
For this data, the sample standard deviation is \( s = 0.3615 \).
For a 95% confidence interval and \( n-1 = 7 \) degrees of freedom, the t-score is approximately \( t = 2.3646 \).
The margin of error \( E \) is calculated as:
\[
E = t \times \frac{s}{\sqrt{n}} = 2.3646 \times \frac{0.3615}{\sqrt{8}} = 0.3023
\]
The 95% confidence interval for the population mean is given by:
\[
(\bar{x} - E, \bar{x} + E) = (2.375 - 0.3023, 2.375 + 0.3023) = (2.0727, 2.6773)
\]
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