2.01a Population and sample: terminology

Edna wishes to investigate the energy intake from eating out at restaurants for the households in her village. She wants a sample of 100 households. She has a list of all 2065 households in the village. Ralph suggests this selection method. "Number the households 0000 to 2064. Obtain 100 different four-digit random numbers between 0000 and 2064 and select the corresponding households for inclusion in the investigation."

1. What is the population for this investigation? Circle your answer. [1 mark]

   Edna and Ralph

   The 2065 households in the village

   The energy intake for the village from eating out

   The 100 households selected

2. What is the sampling method suggested by Ralph? Circle your answer. [1 mark]

   Opportunity

   Random number

   Continuous random variable

   Simple random

Ali conducted an investigation into the distances ridden by those members of a cycling club who rode at least 120 km in a training week. She grouped all the distances into intervals of length 10 km and then constructed a cumulative frequency diagram, which is shown below. \includegraphics{figure_8}

1. Explain whether the data Ali used is a sample or a population. [1]

The club is taking part in a competition. Eight team members and one reserve are to be selected. The club captain decides that the team members should be those cyclists who rode the furthest during the training week, and that the reserve should be the cyclist who rode the next furthest.

1. Use the graph to estimate the shortest distance cycled by a team member. [1]

The captain's best friend rode 156 km in the training week and was selected as reserve. Ali complained that this was unjustifiable.

1. Explain whether there is sufficient evidence in the diagram to support Ali's complaint. [1]

The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.

\(x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(P(X = x)\)	\(0.05\)	\(0.2\)	\(0.5\)	\(0.2\)	\(0.05\)

1. 1. Explain why \(\text{E}(X) = 2\). [1]
   2. Find \(\text{Var}(X)\). [3]
2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250X - 80\). Find

The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.

1. Explain why it would be inappropriate to test all the remote controls in this way. [1]
2. State an advantage of using random sampling in this context. [1]

An exercise gym opens at 6:00 a.m. every day. The manager decides to use a questionnaire to gather the opinions of the gym members. The first 30 members arriving at the gym on a particular morning are asked to complete the questionnaire.

1. What is the intended population in this context? [1]
2. What type of sampling is this? [1]
3. How could the sampling process be improved? [1]

Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

1. State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students. [1]
2. Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. [1]

Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.

1. Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier. [3]
2. Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the 5\% significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test. [7]