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Confidence Interval for Variance of Bolt Diameters
Published on February 11, 2025
Statistics
The solution calculates confidence intervals for the variance and mean of bolt diameters, demonstrating how reducing the confidence level results in a narrower confidence interval for the mean.
Question
zu Aufgabenteil a) es ist für das Konfidenzniveau 0,90 das Konfidenzintervall für die Varianz von x anzugeben zu Aufgabenteil b) die Varianz² wird auf 0,04 geändert, es soll wieder das Konfidenzintervall zum Konfidenzniveau 0,90 angegeben werden
es sind 9 Bolzen
in der Angabe beträgt die korrigierte empirische Varianz 0,09mm²
bei ist da Konfidenzniveau 0,90
zu Aufgabe a) das Konfidenzniveau ist 0,90 nicht 0,8 bitte ändern
bei c) wird das Konfidenzniveau auf 0,89 reduziert
Answer
Here’s the step-by-step solution to the problem:
Problem 8
A machine produces bolts. In a quality control check, the diameter of 9 bolts was measured. The sample mean is 20.14 mm and the corrected sample variance is 0.09 mm². The diameter of the bolts is modeled by a normally distributed random variable X.
a) Confidence interval for the variance of X at a confidence level of 0.90.
For a normally distributed population, the confidence interval for the variance
Given:
- Sample size
- Sample variance
- Confidence level
, so , , - Degrees of freedom
We need to find the critical chi-squared values for
Now, we can calculate the confidence interval:
Lower bound:
Lower bound:
Upper bound:
The 90% confidence interval for the variance of X is approximately
b) Confidence interval for the expectation (mean) of X at a confidence level of 0.90, given that the population variance
For a normally distributed population with known variance
Given:
- Sample mean
- Population variance
, so population standard deviation - Sample size
- Confidence level
, so ,
We need to find the z-score
Now, we can calculate the confidence interval:
Margin of error:
Margin of error:
Lower bound:
Upper bound:
The 90% confidence interval for the expectation (mean) of X is approximately
c) What happens when the confidence level is reduced to 0.89?
When the confidence level is reduced from 0.90 to 0.89, the value of
For the confidence interval of the mean (part b), the width of the interval is determined by
Since
Let’s calculate the 89% confidence interval for comparison:
For confidence level 0.89,
New margin of error:
For confidence level 0.89,
New margin of error:
New lower bound:
New upper bound:
The 89% confidence interval for the expectation (mean) of X is approximately
Comparing the 90% confidence interval
Final Answer:
a) The 90% confidence interval for the variance of X is approximately
b) The 90% confidence interval for the expectation (mean) of X is approximately
c) When the confidence level is reduced to 0.89, the confidence interval for the expectation (mean) becomes narrower. For a confidence level of 0.89, the confidence interval is approximately