Practice section 1 of 5

A school administrative manager wishes to interview a sample of 30 students from the entire population of 750 students on the quality of canteen food. Explain how simple random sampling can be carried out to choose the 30 students.

Given the data $\{52,50,65,59,70,63,67\}$, find unbiased estimates of the mean and variance of the population from which the data are drawn.

The table records the number of cars sold per day over 50 days.

Find unbiased estimates of the mean and variance of the number of cars sold.

Find unbiased estimates of the population mean and variance for each sample:

(a) $\sum x=120$, $\sum x^2=2102$, $n=8$.

(b) $\sum(t-300)=2012$, $\sum(t-300)^2=525262$, $n=200$.

The heights of a species of plants follow a normal distribution with mean 21 cm and standard deviation $\sqrt{90}$ cm. A random sample of 10 plants is taken and the mean height calculated. Find the probability that this sample mean lies between 18 cm and 27 cm.

For an independent random sample $X_1,\ldots,X_n$ from $X\sim N(\mu,1)$:

(i) State the distribution of $\bar X$.

(ii) Find the sample size required to ensure that the probability that $\bar X$ is within 0.1 of $\mu$ is greater than 0.95.