1. Answer true or false for each of the following:
a) one way of reducing the sampling error is to reduce the sample size.
c) the finite correction factor = $\frac{N-n}{N-1}$ where N is the population size and n is the sample size.
d) A sample of size n is taken from a population whose variance is $\sigma^2$. $\sigma/\sqrt{n}$ is the sampling error of the distribution of the sample mean.
e) If the population is not a normal distribution, and the sample size is less than 30, n<30, then by the Central Limit Theorem, $\bar{X}$ will have a normal distribution.
f) the t distribution is used to calculate a confidence interval when
i. the population is normal,
ii. the population variance is known,
iii. the sample size is small.
2. A normal population has a mean, $\mu = 45$ and a variance, $\sigma^2 = 12$. Samples of size 12 are taken from the population. Write down mean and variance of the sample mean, $\bar{X}$.
3. The marks of an examination taken by a large number of students has a normal distribution with the mean mark of 74 and a variance of 144. The mean mark for a random sample of 80 students is denoted by $\bar{X}$
i. Write down the standard error of $\bar{X}$.
ii. Calculate $P(\bar{X} < 72)$
4. The length of steel rods used to reinforce concrete is known to have a normal distribution with mean of 600 centimetres and a standard deviation of 8 centimetres. A random sample of 40 steel rods was measured. What is the probability that the mean length of the sample is less than 598 cm?
5. A random sample of 84 medium sized oranges was found to have a mean Vitamin C content of 64.5 mg, with a standard deviation of 3.14 mg. Construct a 97% confidence interval for the mean Vitamin C content of those medium sized oranges.
6. The lengths of a sample of 12 fish were measured to the nearest centimetre as follows:
23.7 28.9 24.5 28.2 30.5 22.6 35.5 32.1 27.6 25.4 26.4 25.3
a) Calculate unbiased estimates for the population mean and the population standard deviation.
b) Making appropriate assumptions, calculate a 94% confidence interval for the mean length of the fish. State the conditions for which a T distribution will be appropriate to calculate a confidence interval.
