Interpret test conclusion or error

6 It is known that \(8 \%\) of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.

1. He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5\% significance level.
2. Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).
   Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5\% significance level.
3. Find the probability of a Type I error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Stephan is an athlete who competes in the high jump. In the past, Stephan has succeeded in \(90 \%\) of jumps at a certain height. He suspects that his standard has recently fallen and he decides to carry out a hypothesis test to find out whether he is right. If he succeeds in fewer than 17 of his next 20 jumps at this height, he will conclude that his standard has fallen.

1. Find the probability of a Type I error.
2. In fact Stephan succeeds in 18 of his next 20 jumps. Which of the errors, Type I or Type II, is possible? Explain your answer.

1. Past records show that \(15 \%\) of customers at a shop buy chocolate. The shopkeeper believes that moving the chocolate closer to the till will increase the proportion of customers buying chocolate.

After moving the chocolate closer to the till, a random sample of 30 customers is taken and 8 of them are found to have bought chocolate. Julie carries out a hypothesis test, at the 5\% level of significance, to test the shopkeeper's belief.
Julie's hypothesis test is shown below. \(\mathrm { H } _ { 0 } : p = 0.15\) \(\mathrm { H } _ { 1 } : p \geqslant 0.15\) Let \(X =\) the number of customers who buy chocolate. \(X \sim \mathrm {~B} ( 30,0.15 )\) \(\mathrm { P } ( X = 8 ) = 0.0420\) \(0.0420 < 0.05\) so reject \(\mathrm { H } _ { 0 }\) There is sufficient evidence to suggest that the proportion of customers buying chocolate has increased.

1. Identify the first two errors that Julie has made in her hypothesis test.
2. Explain whether or not these errors will affect the conclusion of her hypothesis test. Give a reason for your answer.
3. Find, using a 5\% level of significance, the critical region for a one-tailed test of the shopkeeper's belief. The probability in the tail should be less than 0.05
4. Find the actual level of significance of this test.

3 Kutz and Styler are two unisex hair salons. An analysis of a random sample of 150 customers at Kutz shows that 28 per cent are male. An analysis of an independent random sample of 250 customers at Styler shows that 34 per cent are male.

1. Test, at the \(5 \%\) level of significance, the hypothesis that there is no difference between the proportion of male customers at Kutz and that at Styler.
2. State, with a reason, the probability of making a Type I error in the test in part (a) if, in fact, the actual difference between the two proportions is 0.05 .

It is believed that 25% of the customers at a bakery buy a loaf of bread. In an attempt to increase this proportion, the manager of the bakery provided free samples for the customers to taste. To decide whether providing free samples had been effective, a large random sample of customers leaving the bakery were asked whether they had purchased a loaf of bread. A hypothesis test at the 5% significance level was carried out on the data collected. The test statistic calculated was found to be in the critical region.

1. State the Null and Alternative hypotheses for this test. [1 mark]
2. State, in context, the conclusion to this test. [2 marks]