2.01b Informal inferences: from samples

1 The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.
200
201
198
202
200
199
199
201
197
202
200
199

1. Find unbiased estimates of the population mean and variance.
2. Give a statistical reason why these estimates may not be reliable.

5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve. \includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-4_1344_1335_386_425} Use the curve to estimate

1. the interquartile range of the marks,
2. \(x\), if \(40 \%\) of the candidates scored more than \(x\) marks,
3. the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
4. Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored \(35,36,37 , \ldots , 55\).
5. What does this information suggest about the estimate of the interquartile range found in part (i)?

10 The table shows extracts from the "Method of travel by LA" tabs for 2001 and 2011 in the large data set.

1. In one of these four LAs a new tram system was opened in 2004. Suggest, with a reason taken from the data, which LA this could have been.
2. Julian suggests that the figures for "Bus, minibus or coach" for LA1 show that some new bus routes were probably introduced in this LA between 2001 and 2011. Use data from the table to comment on this suggestion.
3. In one of these four LAs a congestion charge on vehicles was introduced in 2003. Suggest, with a reason taken from the data, which LA this could have been.

1. (a) State one characteristic of a population that would make a census a practical alternative to sampling.

A leisure centre has 2500 members.
It asks a sample of 300 members for their opinions on the fees it charges for using the centre. For the sample,
(b) (i) identify a suitable sampling frame,
(ii) identify a sampling unit. The leisure centre has the following pieces of information. \(A\) is the list of the different types of membership that can be paid for by members. \(B\) is the mean of the membership fees paid by all 2500 members. \(C\) is the number in the sample of 300 members who are satisfied with the fees they pay.
(c) State the piece of information that is a statistic. Give a reason for your answer.

6. A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that \(40 \%\) of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

1. Define the population associated with the magazine.
2. Suggest a suitable sampling frame for the survey.
3. Identify the sampling units.
4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. As a pilot study the editor took a random sample of 25 subscribers.
5. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. In fact only 6 subscribers agreed to the name being changed.
6. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the percentage agreeing to the change is less that the editor believes. The full survey is to be carried out using 200 randomly chosen subscribers.
7. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. \section*{END}

2. (a) Explain what you understand by (i) a population and (ii) a sampling frame. The population and the sampling frame may not be the same.
(b) Explain why this might be the case.
(c) Give an example, justifying your choices, to illustrate when you might use

1. a census,
2. a sample.

2. A video rental shop needs to find out whether or not videos have been rewound when they are returned; it will do this by taking a sample of returned videos

1. State one advantage and one disadvantage of taking a sample.
2. Suggest a suitable sampling frame.
3. Describe the sampling units.
4. Criticise the sampling method of looking at just one particular shelf of videos.

Explain what you understand by

1. a population, [1]
2. a statistic. [2]

A questionnaire concerning attitudes to classes in a college was completed by a random sample of 50 students. The students gave the college a mean approval rating of 75\%.

1. Identify the population and the statistic in this situation. [2]
2. Explain what you understand by the sampling distribution of this statistic. [2]

A random sample \(X_1, X_2, ..., X_n\) is taken from a finite population. A statistic Y is based on this sample.

1. Explain what you understand by the statistic Y. [2]
2. Give an example of a statistic. [1]
3. Explain what you understand by the sampling distribution of Y. [2]