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Calculate the confidence interval for the population mean.
Published on February 11, 2025
Statistics
The provided text details step-by-step calculations for confidence intervals of means and variances, along with sample size determination, under different scenarios (known and unknown population standard deviation). The solutions involve using z-scores, t-scores, and chi-squared values to construct intervals for population parameters.
Question
Answer
Here are the step-by-step solutions to the problems:
Problem 1: Confidence interval for the mean
Given:
Sample size
Sample mean
Known standard deviation
Confidence level
Sample size
Sample mean
Known standard deviation
Confidence level
For a known
Here,
We need to find
So,
We need to find
So,
Confidence interval =
Lower bound =
Upper bound =
Upper bound =
- untere Grenze des Konfidenzintervalls: 17.42
- obere Grenze des Konfidenzintervalls: 18.30
Problem 2: Confidence interval for the mean
Given:
Sample size
Sample mean
Sample standard deviation
Confidence level
Sample size
Sample mean
Sample standard deviation
Confidence level
For an unknown
Here,
We need to find
So,
We need to find
So,
Confidence interval =
Lower bound =
Upper bound =
Upper bound =
- untere Grenze des Konfidenzintervalls: 16.78
- obere Grenze des Konfidenzintervalls: 18.94
Problem 3: Confidence interval for the variance
Given:
Sample size
Sample variance
Confidence level
Sample size
Sample variance
Confidence level
The confidence interval for
Here,
From the tables, for
From the tables, for
Confidence interval for
Lower bound =
Upper bound =
Upper bound =
- untere Grenze des Konfidenzintervalls: 0.34
- obere Grenze des Konfidenzintervalls: 5.55
Problem 4: Sample size
Given:
Known standard deviation
Confidence level
Maximum length of confidence interval
Known standard deviation
Confidence level
Maximum length of confidence interval
The length of the confidence interval for
We want this length to be at most
We want this length to be at most
For
Since
Final Answer: The final answer is