One-sample z-test, variance known

10 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.

18 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 66.5 g , with a standard deviation of 21.2 g 18

1. After the campaign, the first 750 available people from this region were surveyed to find out their average consumption of chocolate. 18
2. 1. State the sampling method used to collect the survey. 18
3. (ii) Explain why this sample should not be used to conduct a hypothesis test.
   18
4. A second sample of 750 people revealed that the mean consumption of chocolate per person per week was 65.4 g 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. \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-26_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-27_2492_1721_217_150} \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-28_2496_1719_214_150}

14 It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A\&E) patients. After some new initiatives were introduced, a random sample of 12 patients from the hospital's A\&E Department had the following waiting times, in hours. Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether the mean waiting time at this hospital's A\&E department has changed. You may assume that the waiting times are normally distributed with standard deviation 0.8 hours. \includegraphics[max width=\textwidth, alt={}, center]{076ea8e9-9295-46d2-b5f9-b27fa969129e-21_2488_1728_219_141}

15 A team game involves solving puzzles to escape from a room. Using data from the past, the mean time to solve the puzzles and escape from one of these rooms is 65 minutes with a standard deviation of 11.3 minutes. After recent changes to the puzzles in the room, it is claimed that the mean time to solve the puzzles and escape has changed. To test this claim, a random sample of 100 teams is selected.
The total time to solve the puzzles and escape for the 100 teams is 6780 minutes.
Assuming that the times are normally distributed, test at the \(2 \%\) level the claim that the mean time has changed.

17 The number of working hours per week of employees in a company is modelled by a normal distribution with mean of 34 hours and a standard deviation of 4.5 hours. The manager claims that the mean working hours per week of the company's employees has increased. A random sample of 30 employees in the company was found to have mean working hours per week of 36.2 hours. Carry out a hypothesis test at the \(2.5 \%\) significance level to investigate the manager's claim. \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-27_2490_1730_217_141}

5. A laboratory carrying out screening for a certain blood disorder claims that the average time taken for test results to be returned is 38 hours. A reporter for a national newspaper suspects that the results take longer, on average, to be returned than claimed by the laboratory. The reporter finds the time, \(x\) hours, for 50 randomly selected results, in order to conduct a hypothesis test. The following summary statistics were obtained. $$\sum x = 2163 \quad \sum x ^ { 2 } = 98508$$

1. Calculate the \(p\)-value for the reporter's hypothesis test, and complete the test using a \(5 \%\) level of significance. Hence write a headline for the reporter to use.
2. Explain the relevance or otherwise of the Central Limit Theorem to your answer in part (a).
3. Briefly explain why a random sample is preferable to taking a batch of 50 consecutive results.
4. On another occasion, the reporter took a different random sample of 10 results.
   1. State, with a reason, what type of hypothesis test the reporter should use on this occasion.
   2. State one assumption required to carry out this test.