4.11 Hypothesis testing: chi-squared and t-testsIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   One hundred and fifty students in Year $10$ and Year $11$ were each asked how they usually travel to school. Year $10$ (75 students): walk $20$, bus $30$, car $25$. Year $11$ (75 students): walk $30$, bus $25$, car $20$. A $\chi^2$ test for independence is to be carried out between year group and method of travel.

   (a)

   Write down the number of degrees of freedom for the test.

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$5$

   (b)

   Which is the correct null hypothesis?

   [1 mark]

   • A$H_0$: year group and method of travel are independent
   • B$H_0$: year group and method of travel are associated
   • C$H_0$: the mean number of students per method is equal
   • D$H_0$: the two year groups have equal means

   (c)

   Use your GDC to find the $p$-value of the test, and state your conclusion at the $5\%$ significance level.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A café sells four desserts. The manager claims that customers choose the four desserts equally often. In one week, $200$ customers chose cheesecake $68$, brownie $52$, sorbet $44$ and tart $36$ times. A $\chi^2$ goodness of fit test is used to test the claim.

   (a)

   If the manager's claim is true, how many customers would be expected to choose each dessert?

   [1 mark]

   • A$25$
   • B$40$
   • C$68$
   • D$50$

   (b)

   The $\chi^2$ statistic is $11.2$ and the critical value at the $5\%$ significance level is $7.81$. Which conclusion is correct?

   [1 mark]

   • ADo not reject $H_0$, because $11.2>7.81$
   • BReject $H_0$, because the $p$-value is greater than $0.05$
   • CReject $H_0$: there is evidence that the desserts are not chosen equally often
   • DDo not reject $H_0$: the desserts are chosen equally often

   (c)

   Write down the null and alternative hypotheses, and the number of degrees of freedom.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A teacher compares the scores of two classes in the same test. Class A has $12$ students with mean score $68.4$ and sample standard deviation $7.2$. Class B has $10$ students with mean score $62.1$ and sample standard deviation $8.1$. The scores in both classes are normally distributed with equal variances. Use a pooled two-sample $t$-test on your GDC.

   (a)

   Test, at the $5\%$ significance level, whether the mean score of class A is greater than that of class B. State the hypotheses, the $p$-value and your conclusion.

   [3 marks]

   (b)

   A colleague asks instead whether the mean scores are different. Carry out a two-tailed test at the $5\%$ level, and explain why the conclusion differs from part (a).

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A gym has $250$ members. A survey recorded each member's plan and preferred session. Basic: $30$ prefer mornings and $50$ prefer evenings. Standard: $45$ prefer mornings and $55$ prefer evenings. Premium: $45$ prefer mornings and $25$ prefer evenings. The gym also compares weekly visits for members who joined in January ($15$ members, mean $3.8$, sample standard deviation $1.2$) and in July ($12$ members, mean $3.1$, sample standard deviation $1.0$). Weekly visits are normally distributed with equal variances in the two groups. Use your GDC.

   (a)

   Carry out a $\chi^2$ test for independence at the $5\%$ significance level between plan and preferred session.
   (i) State the null hypothesis $H_0$ and the alternative hypothesis $H_1$.
   (ii) Show how to find the expected frequency for Basic members who prefer mornings.
   (iii) Write down the degrees of freedom and, using your GDC, the $p$-value.
   (iv) State your conclusion.

   [6 marks]

   (b)

   Test, at the $10\%$ significance level, whether the mean number of weekly visits differs between members who joined in January and in July.
   (i) State the hypotheses.
   (ii) Find the $p$-value.
   (iii) State your conclusion.
   (iv) State the two conditions that must hold for this $t$-test to be valid.

   [6 marks]

   Total for question 4: 12 marks

End of questions