One-tailed hypothesis test

1 Each multiple choice question in a test has 4 suggested answers, exactly one of which is correct. Rehka knows nothing about the subject of the test, but claims that she has a special method for answering the questions that is better than just guessing. There are 60 questions in the test, and Rehka gets 22 correct.

1. State null and alternative hypotheses for a test of Rehka's claim.
2. Using a normal approximation, test at the \(5 \%\) significance level whether Rehka's claim is justified.

2 An engineering test consists of 100 multiple-choice questions. Each question has 5 suggested answers, only one of which is correct. Ashok knows nothing about engineering, but he claims that his general knowledge enables him to get more questions correct than just by guessing. Ashok actually gets 27 answers correct. Use a suitable approximating distribution to test at the \(5 \%\) significance level whether his claim is justified.

7 Past experience shows that \(35 \%\) of the senior pupils in a large school know the regulations about bringing cars to school. The head teacher addresses this subject in an assembly, and afterwards a random sample of 120 senior pupils is selected. In this sample it is found that 50 of these pupils know the regulations. Use a suitable approximation to test, at the \(10 \%\) significance level, whether there is evidence that the proportion of senior pupils who know the regulations has increased. Justify your approximation.

4 In a certain country it is known that 11\% of people are left-handed.

1. Calculate the probability that, in a random sample of 98 people from this country, 5 or fewer are found to be left-handed, giving your answer correct to 3 significant figures. An anthropologist believes that the proportion of left-handed people is lower in a particular ethnic group. The anthropologist collects a random sample of 98 people from this particular ethnic group in order to test the hypothesis that the proportion of left-handed people is less than \(11 \%\). The anthropologist carries out the test at the \(1 \%\) level.
2. Determine the critical region for this test.

1. A seed producer claims that \(96 \%\) of its bean seeds germinate.

To test the producer's claim, a random sample of 75 bean seeds was planted and 66 of these seeds germinated. Use a suitable approximation to test, at the \(1 \%\) level of significance, whether or not the producer is overstating the probability of its bean seeds germinating. State your hypotheses clearly.

12 The table shows information for England and Wales, taken from the UK 2011 census. A random sample of 10000 people in another country was chosen in 2011 , and the number, \(m\), of children aged 5-17 was noted.
It was found that there was evidence at the \(2.5 \%\) level that the proportion of children aged \(5 - 17\) in the same year was higher than in the UK.
Unfortunately, when the results were recorded the value of \(m\) was omitted. Use an appropriate normal distribution to find an estimate of the smallest possible value of \(m\).