One-tail z-test (lower tail)

It is known that the lifetime of a certain species of animal in the wild has mean 13.3 years. A zoologist reads a study of 50 randomly chosen animals of this species that have been kept in zoos. According to the study, for these 50 animals the sample mean lifetime is 12.48 years and the population variance is 12.25 years\(^2\).

1. Test at the 5% significance level whether these results provide evidence that animals of this species that have been kept in zoos have a shorter expected lifetime than those in the wild. [7]
2. Subsequently the zoologist discovered that there had been a mistake in the study. The quoted variance of 12.25 years\(^2\) was in fact the sample variance. Determine whether this makes a difference to the conclusion of the test. [5]
3. Explain whether the Central Limit Theorem is needed in these tests. [1]

The mass, in kilograms, of a species of fish in the UK has population mean 4.2 and standard deviation 0.25. An environmentalist believes that the fish in a particular river are smaller, on average, than those in other rivers in the UK. A random sample of 100 fish of this species, taken from the river, has sample mean 4.16 kg. Stating a necessary assumption, test at the 5% significance level whether the environmentalist is correct. [8]

In a region of England, the government decides to use an advertising campaign to encourage people to eat more healthily. Before the campaign, the mean consumption of chocolate per person per week was known to be 66.5g, with a standard deviation of 21.2g

1. After the campaign, the first 750 available people from this region were surveyed to find out their average consumption of chocolate.
   1. State the sampling method used to collect the survey. [1 mark]
   2. Explain why this sample should not be used to conduct a hypothesis test. [1 mark]
2. A second sample of 750 people revealed that the mean consumption of chocolate per person per week was 65.4g Investigate, at the 10% level of significance, whether the advertising campaign has decreased the mean consumption of chocolate per person per week. Assume that an appropriate sampling method was used and that the consumption of chocolate is normally distributed with an unchanged standard deviation. [6 marks]

The greatest weight \(W\) N that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80. A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N.

1. Test at the 1% significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support. [7]
2. State an assumption needed in carrying out the test in part (a). [1]
3. Explain whether it is necessary to use the central limit theorem in carrying out the test. [1]