Critique given sampling method

2 A certain neighbourhood contains many small houses (with small gardens) and a few large houses (with large gardens). A sample survey of all houses is to be carried out in this neighbourhood. A student suggests that the sample could be selected by sticking a pin into a map of the neighbourhood the requisite number of times, while blindfolded.

1. Give two reasons why this method does not produce a random sample.
2. Describe a better method.

8 A garden centre stocks coniferous hedging plants. These are displayed in 10 rows, each of 120 plants. An employee collects a sample of the heights of these plants by recording the height of each plant on the front row of the display.

1. Explain whether the data collected by the employee is a simple random sample. The data are shown in the cumulative frequency curve below.
   \includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-06_1376_1344_680_233} The owner states that at least \(75 \%\) of the plants are between 40 cm and 80 cm tall.
2. Show that the data collected by the employee supports this statement.
3. Explain whether all samples of 120 plants would necessarily support the owner's statement.

1. A researcher is hired by a cleaning company to survey the opinions of employees on a proposed pension scheme. The company employs 55 managers and 495 cleaners.

To collect data the researcher decides to give a questionnaire to the first 50 cleaners to leave at the end of the day.

1. Give 2 reasons why this method is likely to produce biased results.
2. Explain briefly how the researcher could select a sample of 50 employees using
   1. a systematic sample,
   2. a stratified sample. Using the random number tables in the formulae book, and starting with the top left hand corner (8) and working across, 50 random numbers between 1 and 550 inclusive were selected. The first two suitable numbers are 384 and 100 .
3. Find the next two suitable numbers.

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.

5. Gareth has a keen interest in pop music. He recently read the following claim in a music magazine. \section*{In the pop industry most songs on the radio are not longer than three minutes.}

1. He decided to investigate this claim by recording the lengths of the top 50 singles in the UK Official Singles Chart for the week beginning 17 June 2016. (A 'single' in this context is one digital audio track.) Comment on the suitability of this sample to investigate the magazine's claim.
2. Gareth recorded the data in the table below.

   Length of singles for top 50 UK Official Chart singles, 17 June 2016
   2.5-(3.0)	3.0-(3.5)	3.5-(4.0)	4.0-(4.5)	4.5-(5.0)	5.0-(5.5)	5.5-(6.0)	6.0-(6.5)	6.5-(7.0)	7.0-(7.5)
   3	17	22	7	0	0	0	0	0	1

   He used these data to produce a graph of the distributions of the lengths of singles
   \includegraphics[max width=\textwidth, alt={}, center]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-4_860_1435_1343_379} State two corrections that Gareth needs to make to the histogram so that it accurately represents the data in the table.
3. Gareth also produced a box plot of the lengths of singles. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Length of single for top 50 UK Official Singles Chart 17 June 2016} \includegraphics[alt={},max width=\textwidth]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-5_504_812_406_644}
   \end{figure} He sees that there is one obvious outlier.
   1. What will happen to the mean if the outlier is removed?
   2. What will happen to the standard deviation if the outlier is removed?
4. Gareth decided to remove the outlier. He then produced a table of summary statistics.
   1. Use the appropriate statistics from the table to show, by calculation, that the maximum value for the length of a single is not an outlier.

      		Summary statistics Length of single for top 50 UK Official Singles Chart (minutes)
      \multirow{2}{*}{Length of single}	N	Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
      	49	3.57	0.393	2.77	3.26	3.60	3.89	4.38

   2. State, with a reason, whether these statistics support the magazine's claim.
5. Gareth also calculated summary statistics for the lengths of 30 singles selected at random from his personal collection.

   		Summary statistics Length of single for Gareth's random sample of 30 singles (minutes)
   \multirow{2}{*}{Length of single}	N	Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
   	30	3.13	0.364	2.58	2.73	2.92	3.22	3.95

   Compare and contrast the distribution of lengths of singles in Gareth's personal collection with the distribution in the top 50 UK Official Singles Chart.