Construct a 95% confidence interval for the population mean based on a sample of measurements2.6, 8.4, 3.0, 5.5, 6.4, 8.1if measurement errors have Normal distribution, and the measurement device guarantees a standard deviation of σ = 2.3.
Question
Construct a 95% confidence interval for the population mean based on a sample of measurements2.6, 8.4, 3.0, 5.5, 6.4, 8.1if measurement errors have Normal distribution, and the measurement device guarantees a standard deviation of σ = 2.3.
Solution
To construct a 95% confidence interval for the population mean, we need to follow these steps:
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First, calculate the sample mean (x̄). This is done by adding all the sample measurements together and dividing by the number of measurements. In this case, the sample mean is (2.6 + 8.4 + 3.0 + 5.5 + 6.4 + 8.1) / 6 = 5.67.
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Next, we need to find the standard error of the mean (SE). This is calculated by dividing the standard deviation (σ) by the square root of the sample size (n). Here, the standard deviation is given as 2.3, and the sample size is 6, so the standard error is 2.3 / √6 = 0.94.
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Now, we need to find the z-score for a 95% confidence interval. The z-score is a measure of how many standard deviations an element is from the mean. For a 95% confidence interval, the z-score is approximately 1.96.
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Finally, we can calculate the confidence interval. The lower limit of the confidence interval is calculated by subtracting the product of the z-score and the standard error from the sample mean, and the upper limit is calculated by adding the product of the z-score and the standard error to the sample mean.
So, the lower limit is 5.67 - (1.96 * 0.94) = 3.85, and the upper limit is 5.67 + (1.96 * 0.94) = 7.49.
Therefore, the 95% confidence interval for the population mean is (3.85, 7.49).
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