2.05b Hypothesis test for binomial proportion

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts. A supermarket receives complaints at a mean rate of 6 per week.

1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
2. Find the probability that, in a given week, there are
   1. fewer than 3 complaints received by the supermarket,
   2. at least 6 complaints received by the supermarket. In a randomly selected week, the supermarket received 12 complaints.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
4. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
   1. The continuous random variable \(Y\) has cumulative distribution function given by
   $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$

1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
2. Find the value of \(k\)
3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)
   1. The discrete random variable \(X\) is given by
   $$X \sim \mathrm {~B} ( n , p )$$ The value of \(n\) and the value of \(p\) are such that \(X\) can be approximated by a normal random variable \(Y\) where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$

1. show that \(\sigma = 6\)
2. Hence find the value of \(n\) and the value of \(p\)

6. A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that \(40 \%\) of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

1. Define the population associated with the magazine.
2. Suggest a suitable sampling frame for the survey.
3. Identify the sampling units.
4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. As a pilot study the editor took a random sample of 25 subscribers.
5. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. In fact only 6 subscribers agreed to the name being changed.
6. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the percentage agreeing to the change is less that the editor believes. The full survey is to be carried out using 200 randomly chosen subscribers.
7. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. \section*{END}

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.

6. Over a long period of time, accidents happened on a stretch of road at random at a rate of 3 per month. Find the probability that

1. in a randomly chosen month, more than 4 accidents occurred,
2. in a three-month period, more than 4 accidents occurred. At a later date, a speed restriction was introduced on this stretch of road. During a randomly chosen month only one accident occurred.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the claim that this speed restriction reduced the mean number of road accidents occurring per month. The speed restriction was kept on this road. Over a two-year period, 55 accidents occurred.
4. Test, at the \(5 \%\) level of significance, whether or not there is now evidence that this speed restriction reduced the mean number of road accidents occurring per month.

7. A teacher thinks that \(20 \%\) of the pupils in a school read the Deano comic regularly. He chooses 20 pupils at random and finds 9 of them read the Deano.

1. 1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20\%. State your hypotheses clearly.
   2. State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the \(5 \%\) level.
      (9) The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
2. By combining all 5 samples and using a suitable approximation test, at the \(5 \%\) level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20\%.
3. Comment on your results for the tests in part (a) and part (b).

6. Past records from a large supermarket show that \(20 \%\) of people who buy chocolate bars buy the family size bar. On one particular day a random sample of 30 people was taken from those that had bought chocolate bars and 2 of them were found to have bought a family size bar.

1. Test at the \(5 \%\) significance level, whether or not the proportion \(p\), of people who bought a family size bar of chocolate that day had decreased. State your hypotheses clearly. The manager of the supermarket thinks that the probability of a person buying a gigantic chocolate bar is only 0.02 . To test whether this hypothesis is true the manager decides to take a random sample of 200 people who bought chocolate bars.
2. Find the critical region that would enable the manager to test whether or not there is evidence that the probability is different from 0.02 . The probability of each tail should be as close to \(2.5 \%\) as possible.
3. Write down the significance level of this test.

1. Dhriti grows tomatoes. Over a period of time, she has found that there is a probability 0.3 of a ripe tomato having a diameter greater than 4 cm . She decides to try a new fertiliser. In a random sample of 40 ripe tomatoes, 18 have a diameter greater than 4 cm . Dhriti claims that the new fertiliser has increased the probability of a ripe tomato being greater than 4 cm in diameter.

Test Dhriti's claim at the 5\% level of significance. State your hypotheses clearly.

6.

1. Define the critical region of a test statistic. A discrete random variable \(X\) has a Binomial distribution \(\mathrm { B } ( 30 , p )\). A single observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)
2. Using a \(1 \%\) level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005
3. Write down the actual significance level of the test. The value of the observation was found to be 15 .
4. Comment on this finding in light of your critical region.
   

2. David claims that the weather forecasts produced by local radio are no better than those achieved by tossing a fair coin and predicting rain if a head is obtained or no rain if a tail is obtained. He records the weather for 30 randomly selected days. The local radio forecast is correct on 21 of these days. Test David's claim at the \(5 \%\) level of significance. State your hypotheses clearly.

6.

1. Explain what you understand by a hypothesis.
2. Explain what you understand by a critical region. Mrs George claims that 45\% of voters would vote for her.
   In an opinion poll of 20 randomly selected voters it was found that 5 would vote for her.
3. Test at the \(5 \%\) level of significance whether or not the opinion poll provides evidence to support Mrs George's claim. In a second opinion poll of \(n\) randomly selected people it was found that no one would vote for Mrs George.
4. Using a \(1 \%\) level of significance, find the smallest value of \(n\) for which the hypothesis \(\mathrm { H } _ { 0 } : p = 0.45\) will be rejected in favour of \(\mathrm { H } _ { 1 } : p < 0.45\)

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.

5. The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week. Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is

1. exactly 4,
2. more than 5 . Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.
3. Find the probability that the department can meet all requests for replacement light bulbs before the end of term. The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.
4. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased.

7. A drugs company claims that \(75 \%\) of patients suffering from depression recover when treated with a new drug. A random sample of 10 patients with depression is taken from a doctor's records.

1. Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug. Given that the claim is correct,
2. find the probability that the treatment will be successful for exactly 6 patients. The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.
3. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, the doctor's belief.
4. From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the \(1 \%\) level.

5. Past records show that the proportion of customers buying organic vegetables from Tesson supermarket is 0.35 During a particular day, a random sample of 40 customers from Tesson supermarket was taken and 18 of them bought organic vegetables.

1. Test, at the \(5 \%\) level of significance, whether or not this provides evidence that the proportion of customers who bought organic vegetables has increased. State your hypotheses clearly. The manager of Tesson supermarket claims that the proportion of customers buying organic eggs is different from the proportion of those buying organic vegetables. To test this claim the manager decides to take a random sample of 50 customers.
2. Using a \(5 \%\) level of significance, find the critical region to enable the Tesson supermarket manager to test her claim. The probability for each tail of the region should be as close as possible to \(2.5 \%\) During a particular day, a random sample of 50 customers from Tesson supermarket is taken and 8 of them bought organic eggs.
3. Using your answer to part (b), state whether or not this sample supports the manager's claim. Use a \(5 \%\) level of significance.
4. State the actual significance level of this test. The proportion of customers who buy organic fruit from Tesson supermarket is 0.2 During a particular day, a random sample of 200 customers from Tesson supermarket is taken. Using a suitable approximation, the probability that fewer than \(n\) of these customers bought organic fruit is 0.0465 correct to 4 decimal places.
5. Find the value of \(n\).

4. From past records a manufacturer of glass vases knows that \(15 \%\) of the production have slight defects. To monitor the production, a random sample of 20 vases is checked each day and the number of vases with slight defects is recorded.

1. Using a 5\% significance level, find the critical regions for a two-tailed test of the hypothesis that the probability of a vase with slight defects is 0.15 . The probability of rejecting, in either tail, should be as close as possible to \(2.5 \%\).
2. State the actual significance level of the test described in part (a). A shop sells these vases at a rate of 2.5 per week. In the 4 weeks of December the shop sold 15 vases.
3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales per week had increased in December.
   (6 marks)

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.

3. A coin is tossed 20 times, giving 16 heads.

1. Test at the \(1 \%\) significance level whether the coin is fair, stating your hypotheses clearly.
2. Find the critical region for the same test at the \(0.1 \%\) significance level.

3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.

1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
3. Determine the expected number of answers that Sarah will get right.
4. Find the probability that Sarah gets more than 25 correct answers out of 30.

3. The random variable \(X\) is modelled by a binomial distribution \(\mathrm { B } ( n , p )\), with \(n = 20\) and \(p\) unknown. It is suspected that \(p = 0 \cdot 4\).

1. Find the critical region for the test of \(\mathrm { H } _ { 0 } : p = 0.4\) against \(\mathrm { H } _ { 1 } : p \neq 0.4\), at the \(5 \%\) significance level.
2. Find the critical region if, instead, the alternative hypothesis is \(\mathrm { H } _ { 1 } : p < 0.4\).

5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results: Q. 5 continued on next page ... \section*{STATISTICS 2 (A) TEST PAPER 9 Page 2} continued ...

1. Show that the mean of \(X\) is 3 , and find the variance of \(X\).
2. Give two reasons for thinking that \(X\) can be modelled by a Poisson distribution. (2 marks) After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.
3. Test whether the group's claim is valid. Work at the \(5 \%\) significance level, and state your hypotheses clearly.
   

3. A primary school teacher finds that exactly half of his year 6 class have mobile phones. He decides to investigate whether the proportion of pupils with mobile phones is different from 0.5 in the year 5 class at his school. There are 25 pupils in the year 5 class.

1. State the hypotheses that he should use.
2. Find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
3. Determine the significance level of this test. He finds that eight of the year 5 pupils have mobile phones and concludes that there is not sufficient evidence of the proportion being different from 0.5
4. Stating the new hypotheses clearly, find if the number of year 5 pupils with mobile phones would have been significant if he had tested whether or not the proportion was less than 0.5 and used the largest critical region with a probability of less than \(5 \%\).
   (3 marks)

6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.

1. Find the significance level of this test.
2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
   (6 marks)

5. Four coins are flipped together and the random variable \(H\) represents the number of heads obtained. Assuming that the coins are fair,

1. suggest with reasons a suitable distribution for modelling \(H\) and give the value of any parameters needed,
2. show that the probability of obtaining more heads than tails is \(\frac { 5 } { 16 }\). The four coins are flipped 5 times and more heads are obtained than tails 4 times.
3. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the probability of getting more heads than tails being more than \(\frac { 5 } { 16 }\). Given that the four coins are all biased such that the chance of each one showing a head is 50\% more than the chance of it showing a tail,
4. find the probability of obtaining more heads than tails when the four coins are flipped together.

2. (a) Define

1. a Type I error,
2. a Type II error. A manufacturer sells socks in boxes of 50 .
   The mean number of faulty socks per box is 7.5 . In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.
   (b) (i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a \(5 \%\) significance level.
3. Stating your hypotheses clearly, carry out the test in part (i).
   (c) Find the probability of the Type I error for this test.
   (d) Given that the true mean number of faulty socks per box on the new machine is 5 , calculate the probability of a Type II error for this test.
   (e) Explain what would have been the effect of changing the significance level for the test in part (b) to \(2 \frac { 1 } { 2 } \%\).

4. The number of accidents that occur at a crossroads has a mean of 3 per month. In order to improve the flow of traffic the priority given to traffic is changed. Colin believes that since the change in priority the number of accidents has increased. He tests his belief by recording the number of accidents \(x\) in the month following the change. Colin sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 3\) and \(\mathrm { H } _ { 1 } : \lambda > 3\), where \(\lambda\) is the mean number of accidents per month, and rejects the null hypothesis if \(x > 4\).

1. Find the size of the test. The table gives the values of the power function of the test to two decimal places.

   \(\lambda\)	4	5	6	7
   Power	\(r\)	0.56	\(s\)	0.83

2. Calculate the value of \(r\) and the value of \(s\).
3. Comment on the suitability of the test when \(\lambda = 4\).