Constructing a 95% Confidence Interval for a Population Mean
A school wants to estimate the average test score of its students. A random sample of 36 students yielded a sample mean of 75 with a sample standard deviation of 8. Construct a 95% confidence interval for the true mean test score.
Application of the Central Limit Theorem in Manufacturing Quality Control
A manufacturing process produces items whose weights are assumed to be normally distributed with a known standard deviation of 2 grams, but with an unknown mean. To ensure quality control, samples of size 30 are taken regularly. Using the Central Limit Theorem, answer the following:
Understanding the Central Limit Theorem
Explain the Central Limit Theorem (CLT) and describe its significance in the context of sampling distributions. Include in your answer the conditions under which the CLT holds.
Variance Estimation: Known vs Unknown Mean
Consider a sample \(X_1, X_2, \ldots, X_n\) drawn from a normally distributed population with unknown variance.