Describing sampling methods

1. A lake contains three different types of carp.

There are an estimated 450 mirror carp, 300 leather carp and 850 common carp.
Tim wishes to investigate the health of the fish in the lake.
He decides to take a sample of 160 fish.

1. Give a reason why stratified random sampling cannot be used.
2. Explain how a sample of size 160 could be taken to ensure that the estimated populations of each type of carp are fairly represented. You should state the name of the sampling method used. As part of the health check, Tim weighed the fish.
   His results are given in the table below.

   Weight (wkg)	Frequency (f)	Midpoint (m kg)
   \(2 \leqslant w < 3.5\)	8	2.75
   \(3.5 \leqslant w < 4\)	32	3.75
   \(4 \leqslant w < 4.5\)	64	4.25
   \(4.5 \leqslant w < 5\)	40	4.75
   \(5 \leqslant w < 6\)	16	5.5

   $$\left( \text { You may use } \sum \mathrm { f } m = 692 \quad \text { and } \quad \sum \mathrm { f } m ^ { 2 } = 3053 \right)$$
3. Calculate an estimate for the standard deviation of the weight of the carp. Tim realised that he had transposed the figures for 2 of the weights of the fish.
   He had recorded in the table 2.3 instead of 3.2 and 4.6 instead of 6.4
4. Without calculating a new estimate for the standard deviation, state what effect
   1. using the correct figure of 3.2 instead of 2.3
   2. using the correct figure of 6.4 instead of 4.6
      would have on your estimated standard deviation.
      Give a reason for each of your answers.

3 A researcher is conducting an investigation into the number of portions of fruit adults consume each day. The researcher decides to ask 50 men and 50 women to complete a simple questionnaire.

1. State the type of sampling procedure the researcher is using.
2. Write down one disadvantage of this sampling procedure. The researcher represents the data in Fig. 3.1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Number of portions of fruit consumed by adults} \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-06_531_991_701_248}
   \end{figure} Fig. 3.1
3. Describe the shape of the distribution. The data are summarised in the frequency table in Fig. 3.2. \begin{table}[h]

   Number of portions of fruit	0	1	2	3	4	5
   Number of adults	18	34	26	11	7	4

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
4. For the data in Fig. 3.2, use your calculator to find
   Give your answers correct to 2 decimal places. A second researcher chooses a proportional stratified sample of 100 children from years 5 and 6 in a certain primary school. There are 220 children to choose from. In year 5 there are 125 children, of whom 81 are boys.
5. How many year 5 girls should be included in the sample? The second researcher found that the mean number of portions of fruit consumed per day by the children in this sample was 1.61 and the standard deviation was 0.53 .
6. Comment on the amount of fruit consumed per day by the children compared to the amount of fruit consumed per day by the adults.

1. Sara is investigating the variation in daily maximum gust, \(t \mathrm { kn }\), for Camborne in June and July 1987.

She used the large data set to select a sample of size 20 from the June and July data for 1987. Sara selected the first value using a random number from 1 to 4 and then selected every third value after that.

1. State the sampling technique Sara used.
2. From your knowledge of the large data set explain why this process may not generate a sample of size 20 . The data Sara collected are summarised as follows $$n = 20 \quad \sum t = 374 \quad \sum t ^ { 2 } = 7600$$
3. Calculate the standard deviation.

Jo is investigating the popularity of a certain band amongst students at her school. She decides to survey a sample of 100 students.

1. State an advantage of using a stratified sample rather than a simple random sample. [1]
2. Explain whether it would be reasonable for Jo to use her results to draw conclusions about all students in the UK. [1]

A researcher wants to find out how many adults in a large town use the internet at least once a week. The researcher has formulated a suitable question to ask. For each of the following methods of taking a sample of the adults in the town, give a reason why the method may be biased. Method A: Ask people walking along a particular street between 9 am and 5 pm on one Monday. Method B: Put the question through every letter box in the town and ask people to send back answers. Method C: Put the question on the local council website for people to answer online. [3]