Year 12 hypothesis testing test answers to download as PDF

These Year 12 hypothesis testing test answers give the worked solutions and the marks for every question of the hypothesis testing test. The test is marked out of 20, so you can work out your score.

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Answer to Question 1 (4 marks)
The claim is tested against a lower probability, so the test is one-tailed with the lower tail \(P(X\le9)\).
\(H_0:p=0.8\), \(H_1:p\lt 0.8\) (1). \(X\sim B(15,0.8)\) (1). \(P(X\le9)=0.0611\) (1) \(\gt 0.05\), do not reject \(H_0\); insufficient evidence that the trains are less punctual than claimed (1).
Although 9 out of 15 is only 60%, a result this low happens about 6% of the time when \(p=0.8\), which is not unusual enough at the 5% level.
Answer to Question 2 (4 marks)
For an upper-tail test the critical region is \(X\ge c\); work down from the top using \(P(X\ge c)=1-P(X\le c-1)\).
We need the smallest \(c\) with \(P(X\ge c)\le0.05\) (1). \(P(X\ge12)=0.0442\le0.05\) and \(P(X\ge11)=0.0978\gt 0.05\) (2). Critical region \(X\ge12\), actual significance level 4.42% (1).
So any value of 12 or more leads to rejecting \(H_0\), and the probability of wrongly doing so is 4.42%.
Answer to Question 3 (4 marks)
The question asks whether the proportion differs, so the test is two-tailed and each tail is allowed 2.5%. Under \(H_0\), \(X\sim B(20,0.5)\) and 5 is in the lower tail.
\(H_0:p=0.5\), \(H_1:p\ne0.5\) (1). \(P(X\le5)=0.0207\) (1). \(0.0207\lt 0.025\), reject \(H_0\) (1). Sufficient evidence that the proportion preferring A is not 50% (1).
The data suggest that fewer than half of customers prefer flavour A.
Alternatively use the critical region: \(P(X\le5)=0.0207\) but \(P(X\le6)=0.0577\), so the lower critical region is \(X\le5\) and the observed value 5 lies in it. Comparing with 0.05 instead of 0.025 is a common mistake in two-tailed tests.
Answer to Question 4 (4 marks)
1. The significance level is the probability of rejecting \(H_0\) when it is true. There is a 5% (or less) probability of observing a result this extreme if \(H_0\) were true (2). So rejecting \(H_0\) means the data would be unusual if \(H_0\) were true.
2. A hypothesis test is based on probability, not certainty. The result could still happen by chance when \(H_0\) is true (a Type I error); the test only gives evidence (2). So we say there is evidence for \(H_1\), never that \(H_1\) is proved.
Answer to Question 5 (4 marks)
1. The vaccine should reduce the infection rate, so the test is one-tailed (lower tail) with \(X\sim B(20,0.3)\). \(H_0:p=0.3\), \(H_1:p\lt 0.3\) (1). \(P(X\le2)=0.0355\lt 0.05\) (1), reject \(H_0\): evidence that the vaccine reduces the infection rate (1) Only 2 out of 20 (10%) catching the virus is unlikely if the vaccine had no effect.
2. The critical region is the largest lower tail with probability at most 5%: \(P(X\le2)=0.0355\) but \(P(X\le3)=0.1071\). The critical region is \(X\le2\), so the actual level is 3.55% (1) This is the probability of wrongly concluding that the vaccine works.
The conclusion in (a) must mention the vaccine to earn the final mark.

Total : 20 marks

How to use these Year 12 hypothesis testing test answers

First, sit the test under timed conditions before you read anything here. Then mark each answer and give yourself the marks shown, including method marks. When you lose marks, read the worked solution and find the step you missed. After that, revise the hypothesis testing lesson and redo the worksheet.

Also, the Year 12 hypothesis testing test answers are available as a PDF, so a teacher can mark printed tests quickly. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday.