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Confidence Interval for the Weight of Cement Bags
Published on February 11, 2025
Statistics
The solution calculates 95% confidence intervals for a population mean using both the t-distribution (when population variance is unknown) and the z-distribution (when population variance is known). It also discusses how increasing the confidence level results in a wider confidence interval.
Question
Answer
Here’s the step-by-step solution to the problem:
Part a)
We are given the following information:
- Sample size
- Sample mean
kg - Corrected empirical variance
kg - Confidence level
Since the population variance is unknown, we use the t-distribution to calculate the confidence interval for the population mean
First, calculate the sample standard deviation
For a 95% confidence level,
The degrees of freedom are
We need to find the t-critical value
For a 95% confidence level,
The degrees of freedom are
We need to find the t-critical value
Now, calculate the margin of error:
Finally, calculate the confidence interval:
The 95% confidence interval for the expected value of X is approximately
Part b)
We are given:
- Population variance
kg , so population standard deviation kg - Confidence level
- Sample mean
kg - Sample size
Since the population variance is known, we use the z-distribution. The formula for the confidence interval is:
For a 95% confidence level,
Calculate the margin of error:
Finally, calculate the confidence interval:
The 95% confidence interval for the expected value of X is
Part c)
If the confidence level is increased to 0.97, the value of
For a higher confidence level, we need a larger critical value (
For a higher confidence level, we need a larger critical value (
Therefore, if the confidence level is increased to 0.97, the confidence interval in part a) will become longer.
Final Answer:
Part a) The 95% confidence interval for the expected value of X is approximately
Part b) The 95% confidence interval for the expected value of X is
Part c) The confidence interval will be
Part a) The 95% confidence interval for the expected value of X is approximately
Part b) The 95% confidence interval for the expected value of X is
Part c) The confidence interval will be
We can write the answer in LaTeX format as follows:
Answer:
Part a)
The 95% confidence interval for the expected value of X is calculated as follows:
The 95% confidence interval for the expected value of X is calculated as follows:
Given
The 95% confidence interval is approximately
Part b)
The 95% confidence interval for the expected value of X is calculated as follows:
The 95% confidence interval for the expected value of X is calculated as follows:
Given
The 95% confidence interval is
Part c)
Das Konfidenzintervall wird
Das Konfidenzintervall wird
Final answer: The final answer is