Home / IB Mathematics SL 4.11 Formulation of null and alternative hypotheses AI HL Paper 1- Exam Style Questions

IB Mathematics SL 4.11 Formulation of null and alternative hypotheses AI HL Paper 1- Exam Style Questions- New Syllabus

IB DP Maths AI HL Paper 1 - All Topics

Question

Matt, a customer service manager, is investigating whether a customer’s level of satisfaction depends on the specific service channel they used.
He gathers data from a random sample of 250 customers, recording their satisfaction across three interaction types: in-person, online chat bots, and website contact forms. The results are classified into three categories—satisfied, neutral, and dissatisfied—and summarized in the contingency table below.
Type of service interactionSatisfaction level
NeutralSatisfiedDissatisfied
In-person303523
Online chat bots313923
Website contact forms281922
Matt conducts a \(\chi^2\) test for independence at a 5% level of significance. The critical value for this test is 9.488. The null hypothesis, \(H_0\), states that customer satisfaction level and the type of service interaction are independent.
(a) State the alternative hypothesis, \(H_1\), for this test.
(b) Calculate the degrees of freedom for this investigation.
(c) Determine \(\chi_{calc}^2\), the chi-squared test statistic.
Following his calculations, Matt determines that there is sufficient evidence to reject the null hypothesis.
(d) (i) Determine if Matt’s conclusion is valid. Provide a justification.
(ii) State the final conclusion of the test in the context of the study.

Most-appropriate topic codes:

• SL 4.11: The \(\chi^2\) test for independence — parts (a), (b), (c), and (d)
▶️ Answer/Explanation
Detailed solution

(a)
\(H_1\): Satisfaction level and the type of service interaction are not independent.

(b)
Degrees of freedom formula: \(df = (r – 1)(c – 1)\).
There are 3 rows (service types) and 3 columns (satisfaction levels).
\(df = (3 – 1)(3 – 1) = 2 \times 2 = 4\).

(c)
Using a Graphic Display Calculator (GDC) to perform the \(\chi^2\) test on the matrix:
\(\chi_{calc}^2 = 3.32\) (accept \(3.31906\dots\)).

(d)
(i) Compare the calculated statistic to the critical value:
\(\chi_{calc}^2 = 3.32\) and \(\chi_{crit}^2 = 9.488\).
Since \(3.32 < 9.488\) (or since p-value \(0.506 > 0.05\)), there is insufficient evidence to reject \(H_0\).
Therefore, No, he isn’t correct.

(ii) The conclusion in context is that there is insufficient evidence to say that “satisfaction level and the type of service interaction are not independent” (or insufficient evidence to suggest they are dependent).

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