Using random number tables/generators

2 Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 5 & 6 & 7 & 1 & 0 & 9 & 8 & 4 & 3 & 1 & 0 & 9 & 6 & 6 & 5 & 0 & 2 & 1 & 7 & 6 \end{array}$$ Henri's first two student numbers are 567 and 109.

1. Use Henri's digits to find the numbers of the next two students in the sample.
   There were 30 students in Henri's sample. He asked each of them how much time, \(X\) hours, they spent on social media each week, on average. He summarised the results as follows. $$n = 30 \quad \Sigma x = 610 \quad \Sigma x ^ { 2 } = 12405$$
2. Use this information to calculate an unbiased estimate of the mean of \(X\) and show that an unbiased estimate of the variance of \(X\) is less than 0.1 .
3. Henri's friend claims that Henri has probably made a mistake in his calculation of \(\Sigma x\) or \(\Sigma x ^ { 2 }\). Use your answer to part (b) to comment on this claim.

2 Amy has to choose a random sample from the 265 students in her year at college. She numbers the students from 1 to 265 and then uses random numbers generated by her calculator. The first two random numbers produced by her calculator are 0.213165448 and 0.073165196 .

1. Use these figures to find the numbers of the first four students in her sample.
   There were 25 students in Amy's sample. She asked each of them how much money, \(
   ) x$, they earned in a week, on average. Her results are summarised below. $$n = 25 \quad \Sigma x = 510 \quad \Sigma x ^ { 2 } = 13225$$
2. Find unbiased estimates of the population mean and variance.
3. Explain briefly what is meant by 'population' in this question.

3 Jack has to choose a random sample of 8 people from the 750 members of a sports club.

1. Explain fully how he can use random numbers to choose the sample. Jack asks each person in the sample how much they spent last week in the club café. The results, in dollars, were as follows. $$\begin{array} { l l l l l l l l } 15 & 25 & 30 & 8 & 12 & 18 & 27 & 25 \end{array}$$
2. Find unbiased estimates of the population mean and variance.
3. Explain briefly what is meant by 'population' in this question.

8 A club secretary wishes to survey a sample of members of his club. He uses all members present at a particular meeting as his sample.

1. Explain why this sample is likely to be biased. Later the secretary decides to choose a random sample of members.
   The club has 253 members and the secretary numbers the members from 1 to 253 . He then generates random 3-digit numbers on his calculator. The first six random numbers generated are 156, 965, 248, 156, 073 and 181. The secretary uses each number, where possible, as the number of a member in the sample.
2. Find possible numbers for the first four members in the sample.