Data, sampling and averages Problem set A One question at a time

A sports club has 480 members: 200 juniors, 160 seniors and 120 veterans. A survey of 24 members is needed, with the same proportion of each group. Choosing randomly within each group.

1. (a) Name this sampling method.

   • The population is split into groups and a random sample is taken from each in proportion to its size.
   • This is stratified sampling.

2. (b) How many seniors are in the sample?

   • \(\frac{24}{480}\times 160=8\)

Name the sampling method in each case.

1. (a) The names of all 900 students are put into a computer, which picks 60 names at random.

   • Every member of the population has an equal chance of being chosen, with no pattern: simple random.

2. (b) At a stadium, every 10th person through the turnstile is asked a question.

   • Items are taken at a fixed interval from a list or a stream: systematic.

3. (c) An interviewer must ask 20 men and 20 women. She stops people in the street until both quotas are full.

   • The sample is filled to fixed numbers in each group, and the interviewer chooses who: quota.

The 100 m times of runners at a school sports day are recorded to the nearest second. The table shows the results.

1. (a) How many runners were there in total?

   • \(3 + 6 + 10 + 14 + 9 + 5 + 4 + 2 + 1 = 54\)

2. (b) Complete a grouped table for the classes 11.5 ≤ t < 14.5, 14.5 ≤ t < 17.5 and 17.5 ≤ t < 20.5.

   • 11.5 ≤ t < 14.5: 3 + 6 + 10 = 19
   • 14.5 ≤ t < 17.5: 14 + 9 + 5 = 28
   • 17.5 ≤ t < 20.5: 4 + 2 + 1 = 7
   • Check: the total is still 54

3. (c) A recorded value is 16 seconds. Between which two values does the actual value lie?

   • The data is rounded to the nearest second, so the actual value is within half a second of 16.
   • 15.5 ≤ t < 16.5

Find the mean, the median and the mode of each list.

1. (a) The number of goals scored in 7 matches: 6, 9, 4, 9, 12, 7, 15.

   • Mean: 62 ÷ 7 = 8.86
   • Ordered: 4, 6, 7, 9, 9, 12, 15
   • Median = 9, mode = 9

2. (b) The ages of 10 members of a choir: 18, 22, 19, 25, 22, 31, 20, 22, 28, 24.

   • Mean: 231 ÷ 10 = 23.1
   • Ordered: 18, 19, 20, 22, 22, 22, 24, 25, 28, 31
   • Median = 22 (halfway between the middle two), mode = 22

Find the value of x.

1. (a) The mean of 3, 7, x, 2x, 11 is 9.

   • \(\frac{3+7+x+2 x+11}{5}=9\)
   • \(21+3x=45\)
   • \(3x=24,\ x=8\)

2. (b) The mean of x − 2, 2x, 3x + 1, 5 is 10.

   • \(\frac{x - 2+2 x+3 x + 1+5}{4}=10\)
   • \(4+6x=40\)
   • \(6x=36,\ x=6\)

The mean time taken by 9 students to run 400 m is 82.4 seconds. A tenth student runs, and the mean of the 10 times is 84 seconds.

1. (a) Find the tenth student's time.

   • Total of the 10 times: \(10\times 84=840\)
   • Total of the first 9 times: \(9\times 82.4=741.6\)
   • Tenth time: \(840-741.6=98.4\) seconds

2. (b) A student whose time was 96 seconds is then removed from the 10. Find the mean time of the other 9 students, to 3 significant figures.

   • Total of the other 9: \(840-96=744\)
   • Mean: \(\frac{744}{9}=82.7\) seconds

The table shows the number of books read last month by each of the students in a class.

1. (a) Find how many students there were in total.

   • \(2 + 5 + 9 + 7 + 4 + 2 + 1 = 30\)

2. (b) Find how many students had more than 3 books.

   • 4 + 2 + 1 = 7

3. (c) Use your calculator to find the mean.

   • Enter the values in one list and the frequencies in another.
   • From the calculator: x̄ = 2.53

4. (d) Use your calculator to find the standard deviation.

   • From the calculator: σx = 1.43 (use σx, not sx)

A lorry waits at a depot. The table shows the waiting time, in minutes, over 20 days. The mean waiting time is 6.75 minutes.

1. (a) Write down the total frequency.

   • \(4 + 7 + 5 + 4 = 20\)

2. (b) Find the value of y.

   • \(\frac{3\times4+5\times7+8\times5+4y}{20}=6.75\)
   • \(87+4y=135\)
   • \(4y=48,\ y=12\)

The masses of 100 parcels in a depot are recorded in the table.

1. (a) How many parcels have a mass less than 2.0 kg?

   • 11 + 24 + 31 = 66

2. (b) State the modal group.

   • The highest frequency is 31.
   • Modal group: 1.5 ≤ m < 2.0

3. (c) Estimate the mean mass.

   • Midpoints: 0.75, 1.25, 1.75, 2.25, 2.75
   • \(\frac{0.75\times11+1.25\times24+1.75\times31+2.25\times22+2.75\times12}{100}=1.75\)

The waiting times at a clinic, in minutes, are grouped in the table. Estimate the mean waiting time.

• Midpoints: 3, 4.5, 8
• \(\Sigma f x = 23\times3+31\times4.5+27\times8=424.5\)
• \(\bar{x}\approx\frac{424.5}{81}=5.24\)

The mean score of 20 boys in a fitness test is 61. The mean score of all 50 students in the year is 66.4.

1. (a) Find the total score of all 50 students.

   • \(50\times 66.4=3320\)

2. (b) Find the mean score of the 30 girls.

   • Total score of the boys: \(20\times 61=1220\)
   • Total score of the girls: \(3320-1220=2100\)
   • Mean score of the girls: \(\frac{2100}{30}=70\)

A phone company advertises: "Nine out of ten customers say our phones last for years." The claim comes from an email survey sent to 5000 customers. 120 customers replied, and 108 of them said yes.

1. (a) Find the percentage of the replies that said yes.

   • \(\frac{108}{120}\times 100=90\%\)

2. (b) Give two reasons why the advertisement may not be reliable.

   • \(\frac{120}{5000}\times 100=2.4\%\): almost everyone asked did not reply, so the replies are self-selected.
   • Customers who like the phone are more likely to answer, so the sample is biased towards yes.
   • The company wants a good result, so the source is not neutral.

3. (c) Suggest one way to get a more reliable result.

   • Choose customers at random from the customer list.
   • Ask a neutral question and follow up the people who do not reply.

The masses, in kg, of 15 bags of potatoes are: 2.1, 2.6, 3.4, 2.9, 3.8, 2.4, 3.1, 4.2, 2.8, 3.5, 3.0, 2.2, 3.9, 3.3, 2.7.

1. (a) Use your calculator to find the exact mean.

   • From the calculator: x̄ = 3.06

2. (b) Complete a grouped frequency table for the classes 2.0 ≤ m < 2.5, 2.5 ≤ m < 3.0, 3.0 ≤ m < 3.5, 3.5 ≤ m < 4.0, 4.0 ≤ m < 4.5.

   • 2.0 ≤ m < 2.5: 3
   • 2.5 ≤ m < 3.0: 4
   • 3.0 ≤ m < 3.5: 4
   • 3.5 ≤ m < 4.0: 3
   • 4.0 ≤ m < 4.5: 1
   • Check: total = 15

3. (c) Use your table to estimate the mean.

   • Midpoints: 2.25, 2.75, 3.25, 3.75, 4.25
   • \(\frac{2.25\times3+2.75\times4+3.25\times4+3.75\times3+4.25\times1}{15}=3.08\)

4. (d) Explain why the answers to (a) and (c) are different.

   • The estimate treats every value in a class as if it were at the midpoint.
   • The actual values are spread across each class, so the two means differ.

The marks of 12 students in a quiz are: 56, 61, 74, 68, 85, 61, 79, 92, 61, 70, 83, 77.

1. (a) Use your calculator to find the mean.

   • One-variable statistics: x̄ = 72.3

2. (b) Find the median.

   • Ordered list: 56, 61, 61, 61, 68, 70, 74, 77, 79, 83, 85, 92
   • Median = 72

3. (c) Find the mode.

   • 61 appears 3 times, more than any other value.

4. (d) Find the standard deviation σx.

   • From the calculator: σx = 10.8

A town council wants to know whether residents support a new bus route. A councillor asks the first 40 people she meets at the bus station.

1. (a) Name the sampling method used.

   • The people were chosen because they were easy to reach: convenience sampling.

2. (b) Give one reason why the sample may be biased.

   • People at the bus station use buses, so they are more likely to support a bus route; the sample is not representative.

3. (c) Suggest a better way to choose the sample.

   • Use a random sample of residents, for example chosen from the electoral register, or a stratified sample by area.