Two-tailed hypothesis test

1 In a game, a ball is thrown and lands in one of 4 slots, labelled \(A , B , C\) and \(D\). Raju wishes to test whether the probability that the ball will land in slot \(A\) is \(\frac { 1 } { 4 }\).

1. State suitable null and alternative hypotheses for Raju's test.
   The ball is thrown 100 times and it lands in slot \(A 15\) times.
2. Use a suitable approximating distribution to carry out the test at the \(2 \%\) significance level.

2 Harry has a five-sided spinner with sectors coloured blue, green, red, yellow and black. Harry thinks the spinner may be biased. He plans to carry out a hypothesis test with the following hypotheses. $$\begin{aligned} & \mathrm { H } _ { 0 } : \mathrm { P } ( \text { the spinner lands on blue } ) = \frac { 1 } { 5 }
& \mathrm { H } _ { 1 } : \mathrm { P } ( \text { the spinner lands on blue } ) \neq \frac { 1 } { 5 } \end{aligned}$$ Harry spins the spinner 300 times. It lands on blue on 45 spins.
Use a suitable approximation to carry out Harry's test at the \(5 \%\) significance level.

12 The variable \(X\) has the distribution \(\mathrm { B } \left( 50 , \frac { 1 } { 6 } \right)\). The probabilities \(\mathrm { P } ( X = r )\) for \(r = 0\) to 50 are given by the terms of the expansion of \(( a + b ) ^ { n }\) for specific values of \(a , b\) and \(n\).

1. State the values of \(a\), \(b\) and \(n\). A student has an ordinary 6 -sided dice. They suspect that it is biased so that it shows a 2 on fewer throws than it would if it were fair. In order to test the suspicion the dice is thrown 50 times and the number of 2 s is noted. The student then carries out a hypothesis test at the \(5 \%\) significance level.
2. Write down suitable hypotheses for the test.
3. Determine the rejection region for the test, showing the values of any relevant probabilities.

4. In an experiment, there are 250 trials and each trial results in a success or a failure.

1. Write down two other conditions needed to make this into a binomial experiment. It is claimed that \(10 \%\) of students can tell the difference between two brands of baked beans. In a random sample of 250 students, 40 of them were able to distinguish the difference between the two brands.
2. Using a normal approximation, test at the \(1 \%\) level of significance whether or not the claim is justified. Use a one-tailed test.
3. Comment on the acceptability of the assumptions you needed to carry out the test.

5. (a) State the conditions under which the normal distribution may be used as an approximation to the binomial distribution. A company sells seeds and claims that 55\% of its pea seeds germinate.
(b) Write down a reason why the company should not justify their claim by testing all the pea seeds they produce. To test the company's claim, a random sample of 220 pea seeds was planted.
(c) State the hypotheses for a two-tailed test of the company's claim. Given that 135 of the 220 pea seeds germinated,
(d) use a normal approximation to test, at the \(5 \%\) level of significance, whether or not the company’s claim is justified.

3. A die is rolled 60 times, and results in 16 sixes.

1. Use a suitable approximation to test, at the \(5 \%\) significance level, whether the probability of scoring a six is \(\frac { 1 } { 6 }\) or not. State your hypotheses clearly.
2. Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than \(\frac { 1 } { 6 }\). Carry out this modified test.