Practice section 1 of 5

Suppose the mass of a man in Singapore is assumed to be normally distributed with mean 75 kg and variance $80\ \mathrm{kg}^2$. 100 men are randomly selected from the population and their sample mean mass is 77.2 kg. Alice claims that the population mean mass is actually more than 75 kg. Conduct a hypothesis test at the 5% level of significance.

Define the hypotheses $H_0$ and $H_1$ to test (a) a claim that a supermarket manager has overstated the mean time spent shopping by customers in his store as 25 minutes; (b) a claim that the mean cholesterol level of all Singaporeans as 190 has been understated; and (c) if the average height of a population is different from 168 cm.

A random variable $X$ is known to have a normal distribution with variance 36. The mean of the distribution of $X$ is denoted by $\mu$. A random sample of 50 observations of $X$ has mean 20.2. Test, at the 1% level of significance, the null hypothesis $\mu=22$ against $\mu<22$.

The lengths of metal bars produced by a particular machine are normally distributed with mean length 420 cm and standard deviation 12 cm.

The machine is serviced, after which a sample of 100 bars gives a mean length of 423 cm.

Is there evidence, at the 5% level of significance, of a change in the mean length of the bars produced by the machine, assuming that the standard deviation remains the same?

A large number of students in a college have completed a geography project. The time, $x$ hours, taken by a student to complete the project is noted for a random sample of 150 students. The results are summarised by $\sum x=4626$, $\sum x^2=147691$.

Find unbiased estimates for the population mean and variance.

Test, at the 5% significance level, whether the population mean time for a student to complete the project exceeds 30 hours.

State, giving a reason, whether any assumption about the population is needed for the test to be valid.

At an early stage in analysing the marks scored by a large number of candidates in an examination paper, the Examining Board takes a random sample of 250 candidates and finds that their marks satisfy $\sum x=11872$ and $\sum x^2=646193$. Using these figures, the null hypothesis $\mu=49.5$ is tested against the alternative $\mu<49.5$ at the $\alpha\%$ level of significance. Determine $\alpha$ for which the null hypothesis is rejected in favour of the alternative hypothesis.