Perform one-tailed hypothesis test

6. A television company believes that the proportion of households that can receive Channel C is 0.35 .

1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35 .
2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
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6. A television company believes that the proportion of households that can receive Channel C is 0.35 .

1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35.
2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
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6. A television company believes that the proportion of households that can receive Channel C is 0.35 .

1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the \(2.5 \%\) significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35 .
2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working.
   [0pt] [BLANK PAGE]

9 Research has shown that drug A is effective in \(32 \%\) of patients with a certain disease. In a trial, drug B is given to a random sample of 1000 patients with the disease, and it is found that the drug is effective in 290 of these patients. Test at the \(2.5 \%\) significance level whether there is evidence that drug B is effective in a lower proportion of patients than drug A .

12 It is known that under the standard treatment for a certain disease, \(9.7 \%\) of patients with the disease experience side effects within one year. In a trial of a new treatment, 450 patients with this disease were selected and the number, \(X\), that experienced side effects within one year was noted. It was found that 51 of the 450 patients experienced side effects within one year.

1. Test, at the \(10 \%\) significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.
2. It was later discovered that all 450 patients selected for the trial were treated in the same hospital. Comment on the validity of the model used in part (a).

3. In a sack containing a large number of beads \(\frac { 1 } { 4 }\) are coloured gold and the remainder are of different colours. A group of children use some of the beads in a craft lesson and do not replace them. Afterwards the teacher wishes to know whether or not the proportion of gold beads left in the sack has changed. He selects a random sample of 20 beads and finds that 2 of them are coloured gold. Stating your hypotheses clearly test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of gold beads has changed.
nd the probability of
(c) no accidents in exactly 2 of the next 4 months.
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3 The manufacturers of a brand of chocolates claim that, on average, \(30 \%\) of their chocolates have hard centres. In a random sample of 8 chocolates from this manufacturer, 5 had hard centres. Test, at the \(5 \%\) significance level, whether there is evidence that the population proportion of chocolates with hard centres is not \(30 \%\), stating your hypotheses clearly. Show the values of any relevant probabilities.

10 Pierre is a chef. He claims that \(90 \%\) of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5\% significance level.

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1. 
   \end{tabular} &

   Andrea is the manager of a company which makes mobile phone chargers.

   In the past, she had found that \(12 \%\) of all chargers are faulty.

   Andrea decides to move the manufacture of chargers to a different factory.

   Andrea tests 60 of the new chargers and finds that 4 chargers are faulty.

   Investigate, at the \(10 \%\) level of significance, whether the proportion of faulty chargers has reduced.

   [7 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

   
   \hline \end{tabular} \end{center} 16
2. State, in context, two assumptions that are necessary for the distribution that you have used in part (a) to be valid.

19 It is known from historical data that 15\% of the residents of a town buy the local weekly newspaper, 'Local News'. A new free weekly paper is introduced into the town.
The owners of 'Local News' are interested to know whether the introduction of the free newspaper has changed the proportion of residents who buy their paper. In a random sample of 50 residents of the town taken after the free newspaper was introduced, it was found that 3 of them purchased 'Local News' regularly. Investigate, at the \(5 \%\) significance level, whether this sample provides evidence that the proportion of local residents who buy 'Local News' has changed.
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18 It is known from previous data that 14\% of the visitors to a particular cookery website are under 30 years of age. To encourage more visitors under 30 years of age a large advertising campaign took place to target this age group. To test whether the campaign was effective, a sample of 60 visitors to the website was selected. It was found that 15 of the visitors were under 30 years of age. 18

1. Explain why a one-tailed hypothesis test should be used to decide whether the sample provides evidence that the campaign was effective. 18
2. Carry out the hypothesis test at the \(5 \%\) level of significance to investigate whether the sample provides evidence that the proportion of visitors under 30 years of age has increased.
   18
3. State one necessary assumption about the sample for the distribution used in part (b) to be valid.
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16 It is believed that a coin is biased so that the probability of obtaining a head when the coin is tossed is 0.7 16

1. Assume that the probability of obtaining a head when the coin is tossed is indeed 0.7
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2. 1. Find the probability of obtaining exactly 6 heads from 7 tosses of the coin.
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3. (ii) Find the mean number of heads obtained from 7 tosses of the coin.
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4. Harry believes that the probability of obtaining a head for this coin is actually greater than 0.7 To test this belief he tosses the coin 35 times and obtains 28 heads. Carry out a hypothesis test at the \(10 \%\) significance level to investigate Harry's belief.
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17 The proportion of vegans in a city is thought to be 8\% The owner of an organic food café in this city believes that the proportion of their customers who are vegan is greater than \(8 \%\) To test this belief, a random sample of 50 customers at the café were interviewed and it was found that 7 of them were vegan. Investigate, at the \(5 \%\) level, whether this sample supports the owner's belief.
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17 Suzanne is a member of a sports club. For each sport she competes in, she wins half of the matches.
17

1. After buying a new tennis racket Suzanne plays 10 matches and wins 7 of them.
   Investigate, at the \(10 \%\) level of significance, whether Suzanne's new racket has made a difference to the probability of her winning a match. 17
2. After buying a new squash racket, Suzanne plays 20 matches. Find the minimum number of matches she must win for her to conclude, at the \(10 \%\) level of significance, that the new racket has improved her performance.

17 A council found that \(70 \%\) of its new local businesses made a profit in their first year. The council introduced an incentive scheme for its residents to encourage the use of new local businesses. At the end of the scheme, a random sample of 25 new local businesses was selected and it was found that 21 of them had made a profit in their first year. Using a binomial distribution, investigate, at the \(2.5 \%\) level of significance, whether there is evidence of an increase in the proportion of new local businesses making a profit in their first year.
\includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-33_2488_1719_219_150} Question number Additional page, if required.
Write the question numbers in the left-hand margin.

2. The discrete random variable \(X \sim \mathrm {~B} ( 30,0.28 )\)

1. Find \(\mathrm { P } ( 5 \leq X < 12 )\). Past records from a large supermarket show that \(25 \%\) of people who buy eggs, buy organic eggs. On one particular day a random sample of 40 people is taken from those that had bought eggs and 16 people are found to have bought organic eggs.
2. Test, at the \(1 \%\) significance level, whether or not the proportion \(p\) of people who bought organic eggs that day had increased. State your hypotheses clearly.
3. State the conclusion you would have reached if a \(5 \%\) significance level had been used for this test. \section*{(Total for Question 2 is 8 marks)}

3. Some packets of a certain kind of biscuit contain a free gift. The manufacturer claims that the proportion of packets containing a free gift is 1 in 4 . Marisa suspects that this claim is not true, and that the true proportion is less than 1 in 4 . She chooses 20 packets at random and finds that exactly 1 contains the free gift.

1. Use a binomial model to test the manufacturer's claim, at the \(2.5 \%\) significance level. The packets are packed in boxes, with each box containing 40 packets. Marisa chooses three boxes at random and finds that one box contains 19 packets with the free gift and the other two boxes contain no packets with the free gift.
2. Give a reason why this suggests that the binomial model used in part (a) may not be appropriate.