Thank you for visiting 1 Assume that a sample is used to estimate a population mean mu Find the margin of error M E that corresponds to a sample. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
The 80% confidence interval for estimating the mean temperature is approximately (66.43, 89.77), which is an open-interval represented by parentheses.
To find the margin of error (M.E.) for estimating a population mean at a given confidence level, we can use the formula:
M.E. = Critical value * Standard deviation / √(sample size)
Sample size (n) = 18
Sample mean (x(bar)) = 30.8
Standard deviation (σ) = 9.7
Confidence level = 90%
First, we need to find the critical value corresponding to a 90% confidence level. This critical value is associated with the z-score for the desired confidence level.
Using a standard normal distribution table or a calculator, we find that the critical value for a 90% confidence level is approximately 1.645 (rounded to 3 decimal places).
Now, we can calculate the margin of error:
M.E. = 1.645 * 9.7 / √(18)
M.E. ≈ 3.058 (rounded to 3 decimal places)
Therefore, the margin of error (M.E.) for estimating the population mean at a 90% confidence level is approximately 3.058 (rounded to 3 decimal places).
To find the 80% confidence interval for estimating the mean temperature, we can use the formula:
Confidence Interval = Sample mean ± Margin of Error
Given the sample temperatures: 81.4, 64.2, 73.8, 78.6, 66.5, 97.7, 84.5.
First, we need to calculate the sample mean and standard deviation.
Sample mean (x(bar)) = (81.4 + 64.2 + 73.8 + 78.6 + 66.5 + 97.7 + 84.5) / 7 ≈ 78.1
Next, we need to calculate the margin of error (M.E.) using the formula:
M.E. = Critical value * Standard deviation / √(sample size)
The critical value for an 80% confidence level can be found using a t-distribution table or calculator. For a sample size of 7, the critical value is approximately 1.894 (rounded to 3 decimal places).
Standard deviation (s) can be calculated as the square root of the sample variance.
Sample variance = [(81.4 - 78.1)² + (64.2 - 78.1)² + (73.8 - 78.1)² + (78.6 - 78.1)² + (66.5 - 78.1)² + (97.7 - 78.1)² + (84.5 - 78.1)²] / (7-1) ≈ 152.757
Standard deviation (s) = √(152.757) ≈ 12.358
Now, we can calculate the margin of error:
M.E. = 1.894 * 12.358 / √(7)
M.E. ≈ 11.672 (rounded to 3 decimal places)
Finally, we can construct the confidence interval:
Confidence Interval = Sample mean ± Margin of Error
Confidence Interval = 78.1 ± 11.672
Confidence Interval ≈ (66.428, 89.772) (rounded to two decimal places)
Therefore, the 80% confidence interval for estimating the mean temperature is approximately (66.43, 89.77) (rounded to two decimal places), which is an open-interval represented by parentheses.
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