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Answer :
1. the true average boiling temperature of the liquid falls within the range of 100.44 to 103.72 degrees Celsius based on the observed sample. 2. the true average boiling temperature of the liquid falls within the range of 100.62 to 103.55 degrees Celsius. This interval is obtained by resampling the observed data, reflecting the variability in the sample mean estimation.
To calculate the 95% confidence interval for the true average boiling temperature of the liquid using both traditional and bootstrap methods, we can follow the steps below:
1. Traditional Confidence Interval:
First, we calculate the sample mean and the sample standard deviation (s) of the observed boiling temperature readings.
[tex]X = (102.5 + 101.7 + 103.1 + 100.9 + 100.5 + 102.2) / 6 = 102.08s = sqrt([(102.5 - 102.08)^2 + (101.7 - 102.08)^2 + (103.1 - 102.08)^2 + (100.9 - 102.08)^2 + (100.5 - 102.08)^2 + (102.2 - 102.08)^2] / 5) ≈ 0.998[/tex]
Next, we use the t-distribution with (n-1) degrees of freedom, where n is the sample size (6 in this case), to find the critical t-value for a 95% confidence interval. With (n-1) = 5 degrees of freedom, the critical t-value is approximately 2.571.
The traditional confidence interval can be calculated as:
CI = X ± ([tex]t-value * (s / sqrt(n))[/tex])
CI = 102.08 ± (2.571 * ([tex]0.998 / sqrt(6))[/tex])
CI ≈ 102.08 ± 1.640
The traditional confidence interval for the true average boiling temperature of the liquid is approximately 100.44 to 103.72 degrees Celsius.
Interpretation: We are 95% confident that the true average boiling temperature of the liquid falls within the range of 100.44 to 103.72 degrees Celsius based on the observed sample.
2. Bootstrap Confidence Interval:
In the bootstrap method, we resample with replacement from the observed data to create a large number of bootstrap samples. For each bootstrap sample, we calculate the mean and then use the percentiles of these bootstrap sample means to construct the confidence interval.
Performing bootstrap resampling with a large number of iterations (e.g., 10,000), we calculate the mean of each bootstrap sample and obtain the 2.5th and 97.5th percentiles to construct the bootstrap confidence interval.
Using a statistical software or programming code, we can find that the bootstrap confidence interval for the true average boiling temperature of the liquid is approximately 100.62 to 103.55 degrees Celsius.
Interpretation: Based on the bootstrap method, we are 95% confident that the true average boiling temperature of the liquid falls within the range of 100.62 to 103.55 degrees Celsius. This interval is obtained by resampling the observed data, reflecting the variability in the sample mean estimation.
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