1. Population parameter and sample statistic
Difficulty: ★☆☆A factory produces a batch of \(5000\) lamps. A random sample of \(80\) lamps has mean lifetime \(1240\) hours. The aim is to estimate the mean lifetime of the entire batch. Which quantity is the population parameter?
Hint
2. Standard error from a known standard deviation
Difficulty: ★☆☆Independent random observations come from a population with standard deviation \(15\) minutes. A sample contains \(36\) observations. What is the standard error of its mean?
Hint
3. Estimating a population mean
Difficulty: ★☆☆An independent random sample of \(18\) parcels has total mass \(378\) kg. What is the usual unbiased estimate of the population mean parcel mass?
Hint
4. Exact distribution of a normal sample mean
Difficulty: ★★☆Individual scores are modelled by \(N(72,81)\), with the second parameter denoting variance. For an independent random sample of \(9\) scores, which distribution does the sample mean follow?
Hint
5. Unbiased variance from summaries
Difficulty: ★★☆An independent random sample has \(n=8\), \(\sum x_i=40\) and \(\sum x_i^2=228\). Find the unbiased estimate \(s^2\) of the population variance.
Hint
6. An interval probability for a sample mean
Difficulty: ★★☆Independent random measurements come from \(N(100,144)\), with variance as the second parameter. For a sample of \(36\) measurements, find \(\operatorname{P}(98<\bar X<103)\), correct to 4 decimal places.
Hint
7. A central-limit approximation
Difficulty: ★★☆Journey times have a non-normal distribution with mean \(40\) minutes and standard deviation \(18\) minutes. For \(81\) independent random journeys, a normal approximation for the sample mean is suitable. Estimate \(\operatorname{P}(\bar X<37)\), correct to 4 decimal places.
Hint
8. Estimated standard error from raw data
Difficulty: ★★☆An independent random sample consists of \(3,7,9,10,11\). The population variance is unknown. Using the unbiased variance estimate \(s^2\), find the estimated standard error \(s/\sqrt n\) of the sample mean.
Hint
9. Planning accuracy of a sample mean
Difficulty: ★★★A normal population has mean \(50\) and known standard deviation \(12\). Independent random observations will be taken. Using the standard normal value \(1.96\) for central probability \(0.95\), what is the smallest sample size for which \(\operatorname{P}(|\bar X-50|<3)\) is at least \(0.95\)?
Hint
10. Comparing independent sample means
Difficulty: ★★★Two independent random samples of sizes \(9\) and \(16\) come from the same normal population \(N(\mu,144)\). Let their sample means be \(\bar X_9\) and \(\bar X_{16}\). Find \(\operatorname{P}(\bar X_9-\bar X_{16}>5)\), correct to 4 decimal places.