Exercises: Statistical investigation

The statistical enquiry cycle has five stages. They are listed below in the wrong order.

A. Explore and analyse the data using appropriate displays and numerical summaries

B. Organise and manage the data

C. Answer the original question, giving reasons based on the analysis

D. Formulate a question

E. Plan and collect unbiased, representative data

1. a) Write the letters in the correct order.
2. b) Imagine your investigation is "How long do 3rd Years in our school take to travel to school?" For each stage, write one sentence describing what you would actually do.
3. c) From the specification: over how long a period do students carry out CBA 2, and when must it be completed?
4. d) True or false? Explain your answer in one or two sentences: "The CBA 2 report is marked by the State Examinations Commission."

Here are five questions a student might want to investigate.

1. Is Ireland a better country to live in than Spain?

2. How many hours per week do 3rd Years in our school spend on homework?

3. Do you like maths?

4. Do 3rd Year students who travel to school by bus take longer to get there than students who walk?

5. What is the best team in the League of Ireland?

1. a) For each question, say whether it could be investigated by collecting and analysing data. Give a short reason.
2. b) Choose one question that cannot be investigated as written. Rewrite it as a statistical question. Your new question must name the group (population) and the thing you will measure or record.
3. c) For question 4, write a hypothesis (a prediction you could test). Say what data you would need to collect to test it.

1. a) Classify each variable as categorical (nominal), categorical (ordinal), numerical (discrete) or numerical (continuous).

	Variable
(i)	County of residence
(ii)	Number of siblings
(iii)	Time to run 400 m, in seconds
(iv)	Rating of the canteen food from 1 (poor) to 5 (excellent)
(v)	Height in cm
(vi)	Way of travelling to school (bus, car, walk, cycle)

1. b) Niamh investigates the question: "Do students react faster with their dominant hand than with their other hand?" Each student catches a falling ruler. Niamh records the distance, in cm, that the ruler falls before it is caught.

1. i) Which variable is being compared (the two conditions), and which variable is measured as the outcome?

ii) List three things Niamh should keep the same (controls) so the test is fair.

iii) Explain why the same person should drop the ruler every time.

Ciarán wants to answer the question: "How many hours per week do 3rd Years at Coláiste na Mara in Tralee spend on homework?" There are 180 students in 3rd Year, in six classes (3A to 3F) of 30. Each class list is numbered 1 to 30. He considers three methods of choosing a sample.

• Method 1: ask his 12 friends at lunchtime.

• Method 2: post an online poll in the year WhatsApp group and use whoever replies.

• Method 3: use a random number generator to choose 5 students from each class list (30 students in total).

1. a) Evaluate each method. Say whether it is likely to be unbiased and representative, and explain why.
2. b) Describe, step by step, how Ciarán would carry out Method 3. Say what he does if the same number comes up twice for one class.
3. c) Write a short data-collection plan with these headings:

• the exact question he will ask, with the unit

• a recording table (give the column headings)

• when and how the data will be collected

• how he will treat participants responsibly (for example anonymity, avoiding sensitive questions, permission from the teacher)

• how he will store the data safely

1. d) Give one reason why 30 students is better than 12. Then explain why even a good sample of 30 might not match the whole year exactly.

A random sample of 20 students from 3rd Year recorded their travel time to school, in minutes:

5, 8, 10, 10, 12, 15, 15, 15, 18, 20, 20, 22, 25, 25, 30, 30, 35, 40, 45, 60

1. a) Copy and complete a grouped frequency table with these class intervals: $0-<10$, $10-<20$, $20-<30$, $30-<40$, $40-<50$, $50-<60$, $60-<70$. Check that your frequencies add up to 20.
2. b) Calculate the mean, the median, the mode and the range of the 20 values.
3. c) Draw a histogram in your copy. In your written answer, state the height of each bar and the labels you put on each axis.
4. d) Describe the shape of the distribution in one sentence.
5. e) The mean is larger than the median. Explain why. Say which value you would use as the "typical" travel time.

A student investigates: "Do 3rd Years spend more time on sport per week than 1st Years?" She takes a random sample of 25 students from each year and records their weekly hours of sport. Her summary:

Measure	1st Year sample	3rd Year sample
Mean	6.2 hours	4.8 hours
Median	5.5 hours	4.5 hours
Mode	5 hours	3 hours
Range	14 hours	9 hours

1. a) State in one sentence what the data suggest about the original question.
2. b) Write a conclusion of 4 to 6 sentences. It must answer the original question, use at least two pieces of numerical evidence (comparing like with like), use careful wording about the sample, and give one reason why you cannot be completely certain.
3. c) Use this practice rubric to assess your own conclusion. Give a score and a one-line justification for each criterion. Then write one improvement target.

Rubric (practice guide only, not the official features of quality)

• Answers the original question directly: 0 = not answered; 1 = answered but unclear; 2 = clear, direct answer.

• Uses numerical evidence from the analysis: 0 = no numbers; 1 = one number or numbers not compared; 2 = at least two numbers compared.

• Compares like with like: 0 = mixes different measures; 1 = partly; 2 = mean with mean, median with median.

• Careful wording about sample and uncertainty: 0 = claims it is true for all students; 1 = hint of caution; 2 = clearly refers to "this sample" and gives a reason for caution.

Total: /8

Use the travel-time data again (in minutes, $n = 20$):

5, 8, 10, 10, 12, 15, 15, 15, 18, 20, 20, 22, 25, 25, 30, 30, 35, 40, 45, 60

1. a) Calculate the mean and the median of all 20 values.
2. b) A classmate says: "60 minutes is too big. Delete it." State two checks that should be made before deleting any value. Say what should happen if the value turns out to be genuine.
3. c) Calculate the mean and median of the 19 values without 60 (mean correct to one decimal place). Which of the two measures changed more? Explain why.
4. d) Stretch. A common rule for flagging an outlier is to find the quartiles. $Q_1$ is the median of the lower 10 values and $Q_3$ is the median of the upper 10 values. The interquartile range is $IQR = Q_3 - Q_1$. A value above $Q_3 + 1.5 \times IQR$ is flagged as an outlier. Show whether 60 is flagged. Show also that there is no low outlier.
5. e) Write 2 to 3 sentences for the analysis section. They should give a typical travel time and report the 60-minute value honestly, saying how you dealt with it.

Róisín wants to answer the question "Do Irish teenagers sleep enough?" Her plan:

• She asks 10 teammates on her camogie team after Saturday morning training.

• She asks: "You sleep a lot, don't you? How many hours did you sleep last night?"

• Their answers (hours) are: 9, 8, 9, 10, 8, 9, 9, 7, 10, 9. The mean is 8.8 hours.

• She draws a bar chart of the 10 answers in which the vertical axis starts at 6 hours.

• Her conclusion: "Irish teenagers sleep 8.8 hours, so they all get enough sleep."

1. a) Identify at least five problems with Róisín's investigation. Cover the question, the sample, the wording, the data and the conclusion.
2. b) For three of the problems, explain how each could push the result in a particular direction or limit what she can conclude.
3. c) Rewrite her question so that it is specific. Then describe an improved plan in 4 to 5 bullet points (population, sample, question wording, data to record, display).
4. d) Imagine she must still use her original 10 values. Write an "evaluation" paragraph of 4 to 5 sentences that honestly states the limitations.

A student investigates: "Do 3rd Years who walk to school have shorter travel times than 3rd Years who take the bus?" Her random sample has 8 walkers and 12 bus users.

Measure	Walk ($n = 8$)	Bus ($n = 12$)
Mean	11.5 min	29.0 min
Median	12 min	27.5 min
Range	15 min	55 min

Her draft conclusion is:

"The walkers get to school faster than the bus people. The bus is bad. 11.5 is smaller than 29, so the bus makes students late. This proves everyone should walk to school."

1. a) Score the draft with the practice rubric below (0, 1 or 2 per criterion). For each score, quote or point to the words that justify it.
2. b) Rewrite the conclusion and add a 2 to 3 sentence evaluation of the limitations. Improve the weakest criteria.
3. c) Score your rewrite with the same rubric. Write one target for your own real CBA 2 report.

Rubric (practice guide only, not the official features of quality)

1. Question and population stated: 0 = missing; 1 = vague; 2 = clearly stated.

2. Numerical evidence compared like with like: 0 = none; 1 = some; 2 = at least two measures compared properly.

3. Answer linked to the question and reasoned: 0 = no link; 1 = partly; 2 = clear answer with reasons.

4. Claims match the evidence: 0 = overclaims (says "proves", "everyone"); 1 = some overclaiming; 2 = careful ("in this sample", association not cause).

5. Limitations identified, with their effect: 0 = none; 1 = named only; 2 = named and effect explained.

6. Clear mathematical language and units: 0 = unclear; 1 = mostly clear; 2 = clear, with units.

Total: /12