4.12 Data collection, reliability and validityIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A school council wants to find out how students feel about the new canteen menu. One member suggests asking students as they leave the canteen at lunchtime: "Don't you agree that the new menu is far healthier and tastier than the old boring one?"

   (a)

   What is the main weakness of the proposed question?

   [1 mark]

   • AIt is unrelated to the topic being investigated
   • BIt asks for personal information
   • CIt contains too many answer choices
   • DIt is leading, because the wording encourages agreement

   (b)

   Which replacement question is unbiased and structured with consistent answer choices?

   [1 mark]

   • A"Tell us what you think of the new menu." (free response)
   • B"How much healthier is the new menu? A bit healthier, healthier, much healthier, far healthier."
   • C"How satisfied are you with the new menu? Very satisfied, satisfied, neither satisfied nor dissatisfied, dissatisfied, very dissatisfied."
   • D"Do you agree that the new menu is an improvement? Yes or no."

   (c)

   Explain why asking only students who are leaving the canteen at lunchtime may give biased results.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A psychologist designs a 20-question questionnaire to measure exam anxiety. She gives it to 40 students and then gives it to the same students again two weeks later. The Pearson's product-moment correlation coefficient between the two sets of scores is $r=0.91$. She also compares the scores with a diagnosis of exam anxiety made independently by a school counsellor.

   (a)

   What type of reliability test is the repeated questionnaire?

   [1 mark]

   • ATest-retest
   • BParallel forms
   • CContent validity
   • DCriterion-related validity

   (b)

   Which conclusion is justified by $r=0.91$?

   [1 mark]

   • AThe questionnaire is valid because it measures exam anxiety
   • BThe questionnaire is reliable because the results are consistent
   • CThe questionnaire is biased because the same students were used twice
   • DThe questionnaire is unreliable because the correlation is not 1

   (c)

   Explain what is meant by validity, and state the type of validity test carried out when the scores are compared with the counsellor's diagnosis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A researcher records, for 155 students, the number of hours slept the previous night and whether the student felt alert in the first lesson. The numbers of students who felt alert and who did not feel alert were: under 5 hours: 1 and 5; 5 to under 6 hours: 9 and 15; 6 to under 7 hours: 25 and 25; 7 to under 8 hours: 32 and 15; 8 hours or more: 20 and 8. In total 87 students felt alert. A $\chi^2$ test for independence is to be carried out.

   (a)

   Show that the expected number of students who slept under 5 hours and felt alert is $3.37$ to 3 significant figures, and explain which categories should be combined, giving a reason.

   [3 marks]

   (b)

   Using the combined categories, carry out the test at the 5% significance level, stating the hypotheses, the number of degrees of freedom, the $\chi^2$ statistic and the $p$-value, and your conclusion.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A factory packs items in cartons of 5. An inspector checks 120 randomly chosen cartons and records the number of defective items in each: 0 defective in 55 cartons, 1 in 40 cartons, 2 in 18 cartons, 3 in 6 cartons, 4 in 1 carton and 5 in no cartons. She wants to test, at the 5% significance level, whether the number of defective items in a carton follows a binomial distribution $B(5,p)$, where $p$ is estimated from the data. Use your GDC where needed.

   (a)

   (i) Find an estimate of $p$.
   (ii) Find the expected number of cartons with 0 defective items and with 1 defective item.
   (iii) Explain why the categories 2, 3, 4 and 5 defective items must be combined into "2 or more", and find the expected frequency for "2 or more".

   [6 marks]

   (b)

   Carry out the test at the 5% significance level: state the hypotheses, the observed frequencies you use, the number of degrees of freedom, the $\chi^2$ statistic and the $p$-value, and your conclusion.

   [6 marks]

   Total for question 4: 12 marks

End of questions