Sampling and estimation：練習問題

A random sample of 100 observations is taken from a population with a known standard deviation of \(\sigma = 5\). The sample mean is calculated to be \(\bar{x} = 24.2\). Calculate a \(95\%\) confidence interval for the population mean \(\mu\).

A population has a mean \(\mu\) and variance \(\sigma^2\). A random sample of size \(n\) is taken from this population. If \(\bar{X}\) is the sample mean, which of the following statements about the expectation and variance of \(\bar{X}\) is correct?

A random sample of size \(n\) observations is taken from a population. The unbiased estimate of the population mean is \(\hat{\mu} = 45\). The sum of squares of the observations is \(\sum x^2 = 53025\). If the unbiased estimate of the population variance is \(\hat{\sigma}^2 = 100\), find the sample size \(n\).

A random sample of size \(n=15\) taken from a large population yields the following sums: \(\sum x = 120\) and \(\sum x^2 = 1005\). Calculate the unbiased estimate of the population variance, \(\hat{\sigma}^2\).

In a random sample of \(n=400\) adult residents, 160 stated that they own a dog. Determine the approximate \(98\%\) confidence interval for the true proportion \(p\) of adult residents who own a dog.

A random sample of size \(n=64\) is taken from a normal population with standard deviation \(\sigma = 8\). The sample mean is \(\bar{x} = 45.2\). Calculate the \(98\%\) confidence interval for the population mean \(\mu\).

Two independent normal populations have standard deviations \(\sigma_A=5\) and \(\sigma_B=7\). Independent random samples of sizes \(n_A=25\) and \(n_B=30\) are taken. If the true means are equal, \(\mu_A = \mu_B\), calculate the probability that the sample mean \(\bar{X}_A\) exceeds the sample mean \(\bar{X}_B\) by more than 3 hours, i.e., \(P(\bar{X}_A - \bar{X}_B > 3)\).

A random sample of size \(n\) is drawn from a population with a known standard deviation \(\sigma\). If the standard error of the sample mean, \(\text{SE}(\bar{X})\), is \(E\), determine the new sample size required to reduce the standard error to \(E/2\).

In a large political survey conducted in a city, a random sample of 300 registered voters was selected. Of these 300 voters, 180 stated that they supported Candidate Z.

(i) Calculate the approximate 90\% confidence interval for \(p\), the true population proportion of voters who support Candidate Z.

(ii) Explain, in the context of this problem, what the phrase "90\% confidence interval" means.

(iii) If the total number of registered voters in the city is 1 million, estimate the minimum number of voters who support Candidate Z based on the calculated confidence interval.

A market researcher wants to estimate the mean spending \(\mu\) of customers at a large shopping centre. A preliminary random sample of 40 customers yields an unbiased estimate of the population variance, \(s^2 = 25\) (dollars\(^2\)).

(i) Using this estimate for the population variance \(\sigma^2\), calculate the minimum total sample size \(N\) required so that a 99\% confidence interval for \(\mu\) has a margin of error of at most \(\$1.00\).

(ii) How many additional customers need to be sampled beyond the initial 40 customers to meet this requirement?

(iii) Briefly explain why calculating the required sample size using a 95\% confidence level instead of 99\% would result in a smaller minimum sample size.