總體平均值的置信區間 ：練習題

A researcher calculates a \(90\%\) confidence interval for a population mean. If the sample size is increased from \(n\) to \(9n\) while the sample standard deviation remains the same, how will the width of the new confidence interval compare to the original one?

A random sample of size \( n = 100 \) is taken from a normal population with a known standard deviation of \( \sigma = 20 \). If the sample mean is found to be \( \bar{x} = 150 \), calculate the \( 98\% \) confidence interval for the population mean \( \mu \).
(Given: \( z_{0.01} = 2.33 \))

A \(95\%\) confidence interval for a population mean is constructed using a random sample of size \(n_1\). Another researcher wants to construct a \(99\%\) confidence interval for the same population mean with the exact same width as the first interval. If the population standard deviation is known and remains constant, find the ratio of the new sample size \(n_2\) to the original sample size \(n_1\).
(Use \(z_{0.025} = 1.96\) and \(z_{0.005} = 2.575\))

A random sample of size \( n \) is used to construct a \( 95\% \) confidence interval for the population mean, and the width of the interval is \( W \). If the researcher wants to reduce the width of the interval to \( \frac{1}{4}W \) while maintaining the same confidence level and assuming the population standard deviation remains constant, what should be the new sample size?

A researcher is studying the battery life of a new smartphone model. A random sample of \(64\) smartphones is tested, and their battery lives (in hours) are recorded. The sum of the battery lives is \(\sum x = 1152\) hours, and the sum of the squares of the battery lives is \(\sum x^2 = 21016\).

(a) Calculate the unbiased estimates of the population mean and the population variance of the battery life. (2 points)

(b) Construct a \(98\%\) confidence interval for the population mean battery life. (2 points)

(c) If the researcher wants the margin of error to be within \(0.25\) hours with the same confidence level, what should be the minimum sample size required? (2 points)

(Relevant z-value: \(z_{0.01} = 2.326\))

A quality control manager is investigating the weights of a specific brand of organic coffee beans. A random sample of 100 bags is selected, and the weights, \(x\) grams, are recorded. The following summary statistics are obtained:
\(\sum x = 5020\) and \(\sum x^2 = 252416\)

(a) Find the unbiased estimates of the population mean and population variance of the weights of the coffee bags. Give your answers to 4 decimal places where necessary.

(b) Construct a 95% confidence interval for the population mean weight of the coffee bags. Give the limits correct to 2 decimal places.

(c) The manager wants to estimate the population mean weight to within a margin of error of 0.25 grams with 99% confidence. Based on the sample standard deviation obtained in part (a), find the minimum sample size required.

(d) If the manager decided to use the original sample of 100 bags to construct a 90% confidence interval instead of a 95% confidence interval, explain whether the width of the interval would increase, decrease, or remain the same.