Confidence intervals for population mean

6 The heights, \(h\) centimetres, of a random sample of 100 fully grown animals of a certain species were measured. The results are summarised below. $$n = 100 \quad \Sigma h = 7570 \quad \Sigma h ^ { 2 } = 588050$$

1. Find unbiased estimates of the population mean and variance.
2. Calculate a \(99 \%\) confidence interval for the mean height of animals of this species.
   Four random samples were taken and a \(99 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
3. Find the probability that all four of these confidence intervals contain the true value of \(\mu\).

6 The daily takings, \(
) x\(, for a shop were noted on 30 randomly chosen days. The takings are summarised by \)\Sigma x = 31500 , \Sigma x ^ { 2 } = 33141816$.

1. Calculate unbiased estimates of the population mean and variance of the shop's daily takings.
2. Calculate a \(98 \%\) confidence interval for the mean daily takings. The mean daily takings for a random sample of \(n\) days is found.
3. Estimate the value of \(n\) for which it is approximately \(95 \%\) certain that the sample mean does not differ from the population mean by more than \(
   ) 6$.

3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below.

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.

3 A random sample of 150 students attending a college is taken, and their travel times, \(t\) minutes, are measured. The data are summarised by \(\Sigma t = 4080\) and \(\Sigma t ^ { 2 } = 159252\).

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(94 \%\) confidence interval for the population mean travel time.

1 A random sample of 80 values of a variable \(X\) is taken and these values are summarised below. $$n = 80 \quad \Sigma x = 150.2 \quad \Sigma x ^ { 2 } = 820.24$$ Calculate unbiased estimates of the population mean and variance of \(X\) and hence find a \(95 \%\) confidence interval for the population mean of \(X\).

3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below. $$n = 60 \quad \Sigma x = 3420 \quad \Sigma x ^ { 2 } = 195200$$

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.