2.05b Hypothesis test for binomial proportion

2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers' opinions.

1. Suggest a suitable sampling frame for the sample survey.
2. Identify the sampling units.
3. Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only \(30 \%\) of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
4. Test, at the \(5 \%\) significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.

6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find

1. the probability that \(X = 5\)
2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.

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.

5. A delivery company loses packages randomly at a mean rate of 10 per month. The probability that the delivery company loses more than 12 packages in a randomly selected month is \(p\)

1. Find the value of \(p\) The probability that the delivery company loses more than \(k\) packages in a randomly selected month is at least \(2 p\)
2. Find the largest possible value of \(k\) In a randomly selected month,
3. find the probability that exactly 4 packages were lost in each half of the month. In a randomly selected two-month period, 21 packages were lost.
4. Find the probability that at least 10 packages were lost in each of these two months.
5. Using a suitable approximation, find the probability that more than 27 packages are lost during a randomly selected 4-month period.

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$f ( x ) = \begin{cases} \frac { 1 } { 16 } x ^ { 2 } & 1 \leqslant x < 3 \\ k ( 4 - x ) & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 11 } { 12 }\)
2. Sketch \(\mathrm { f } ( x )\) for \(1 \leqslant x \leqslant 4\)
3. Write down the mode of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 25 } { 9 }\)
4. use algebraic integration to find \(\operatorname { Var } ( X )\), giving your answer to 3 significant figures. The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 48 } \left( x ^ { 3 } + c \right) & 1 \leqslant x < 3 \\ \frac { 11 } { 12 } \left( 4 x - \frac { 1 } { 2 } x ^ { 2 } + d \right) & 3 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
5. 1. Find the exact value of \(C\)
   2. Find the exact value of \(d\)
6. Calculate, to 3 significant figures, the upper quartile of \(X\)
   \includegraphics[max width=\textwidth, alt={}]{a814156d-6945-4601-9cae-d28d8ae0db1e-28_2632_1826_121_121}

1. At a shop, past figures show that \(35 \%\) of customers pay by credit card. Following the shop's decision to no longer charge a fee for using a credit card, a random sample of 20 customers is taken and 11 are found to have paid by credit card.

Hadi believes that the proportion of customers paying by credit card is now greater than 35\%

1. Test Hadi's belief at the \(5 \%\) level of significance. State your hypotheses clearly. For a random sample of 20 customers,
2. show that 11 lies less than 2 standard deviations above the mean number of customers paying by credit card.
   You may assume that \(35 \%\) is the true proportion of customers who pay by credit card.
   

1. Jim farms oysters in a particular lake. He knows from past experience that \(5 \%\) of young oysters do not survive to be harvested.

In a random sample of 30 young oysters, the random variable \(X\) represents the number that do not survive to be harvested.

1. Write down a suitable model for the distribution of \(X\).
2. State an assumption that has been made for the model in part (a).
3. Find the probability that
   1. exactly 24 young oysters do survive to be harvested,
   2. at least 3 young oysters do not survive to be harvested. A second random sample, of 200 young oysters, is taken. The probability that at least \(n\) of these young oysters do not survive to be harvested is more than 0.8
4. Using a suitable approximation, find the maximum value of \(n\). Jim believes that the level of salt in the lake water has changed and it has altered the survival rate of his oysters. He takes a random sample of 25 young oysters and places them in the lake.
   When Jim harvests the oysters, he finds that 21 do survive to be harvested.
5. Use a suitable test, at the \(5 \%\) level of significance, to assess whether or not there is evidence that the proportion of oysters not surviving to be harvested is more than \(5 \%\). State your hypotheses clearly.

3. The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7

1. Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas. Aja collects 5 separate samples, each of 100 ml , of the pond water.
2. Find the probability that exactly 1 of these samples contains exactly 4 water fleas. Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of \(n \mathrm { ml }\) of the pond water is 0.9394 correct to 4 significant figures.
3. 1. Show that \(n - 1.55 \sqrt { \frac { n } { 0.07 } } - 50 = 0\)
   2. Hence find the value of \(n\) After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
4. Using a suitable test, at the \(1 \%\) level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-11_2255_50_314_34}
   

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3 A photocopier in a school is known to break down at random at a mean rate of 8 times per week.

1. Give a reason why a Poisson distribution could be used to model the number of breakdowns. The headteacher of the school replaces the photocopier with a refurbished one and wants to find out if the rate of breakdowns has increased or decreased.
2. Write down suitable null and alternative hypotheses that the headteacher should use. The refurbished photocopier was monitored for the first week after it was installed.
3. Using a \(5 \%\) level of significance, find the critical region to test whether the rate of breakdowns has now changed.
4. Find the actual significance level of a test based on the critical region from part (c). During the first week after it was installed there were 4 breakdowns.
5. Comment on this finding in the light of the critical region found in part (c).

5 Applicants for a pilot training programme with a passenger airline are screened for colour blindness. Past records show that the proportion of applicants identified as colour blind is 0.045

1. Write down a suitable model for the distribution of the number of applicants identified as colour blind from a total of \(n\) applicants.
2. State one assumption necessary for this distribution to be a suitable model of this situation.
3. Using a suitable approximation, find the probability that exactly 5 out of 120 applicants are identified as colour blind.
4. Explain why the approximation that you used in part (c) is appropriate. Jaymini claims that 75\% of all applicants for this training programme go on to become pilots. From a random sample of 96 applicants for this training programme 67 go on to become pilots.
5. Using a suitable approximation, test Jaymini's claim at the \(5 \%\) level of significance. State your hypotheses clearly.

A shop sells shoes at a mean rate of 4 pairs of shoes per hour on a weekday.

1. Suggest a suitable distribution for modelling the number of sales of pairs of shoes made per hour on a weekday.
2. State one assumption necessary for this distribution to be a suitable model of this situation.
3. Find the probability that on a weekday the shop sells
   1. more than 4 pairs of shoes in a one-hour period,
   2. more than 4 pairs of shoes in each of 3 consecutive one-hour periods. The area manager visits the shop on a weekday, the day after an advert for the shop appears in a local paper. In a one-hour period during the manager's visit, the shop sells 7 pairs of shoes. This leads the manager to believe that the advert has increased the shop's sales of pairs of shoes.
4. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of an increase in sales of pairs of shoes following the appearance of the advert.

1. Superbounce is a manufacturer of tennis balls.

It knows from past records that 10\% of its tennis balls fail a bounce test.

1. Find the probability that from a random sample of 10 of these tennis balls
   1. at least 4 fail the bounce test
   2. more than 1 but fewer than 5 fail the bounce test. The managing director makes changes to the production process and claims that these changes will reduce the probability of its tennis balls failing the bounce test. After the changes were made a random sample of 50 of the tennis balls were tested and it was found that 2 failed the bounce test.
2. Test, at the \(5 \%\) significance level, whether or not this result supports the managing director's claim. In a second random sample of \(n\) tennis balls it was found that none failed the bounce test. As a result of this sample, the managing director's claim is supported at the 1\% significance level.
3. Find the smallest possible value of \(n\)

1. Rowan believes that \(35 \%\) of type \(A\) vacuum tubes shatter when exposed to alternating high and low temperatures.

Rowan takes a random sample of 15 of these type \(A\) vacuum tubes and uses a two-tailed test, at the \(5 \%\) level of significance, to test his belief.

1. Give two assumptions, in context, that Rowan needs to make for a binomial distribution to be a suitable model for the number of these type \(A\) vacuum tubes that shatter when exposed to alternating high and low temperatures.
2. Using a binomial distribution, find the critical region for the test. You should state the probability of rejection in each tail, which should be as close as possible to 0.025
3. Find the actual level of significance of the test based on your critical region from part (b) Rowan records that in the latest batch of 15 type \(A\) vacuum tubes exposed to alternating high and low temperatures, 4 of them shattered.
4. With reference to part (b), comment on Rowan's belief. Give a reason for your answer. Rowan changes to type \(B\) vacuum tubes. He takes a random sample of 40 type \(B\) vacuum tubes and finds that 8 of them shatter when exposed to alternating high and low temperatures.
5. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of type \(B\) vacuum tubes that shatter when exposed to alternating high and low temperatures is lower than \(35 \%\) You should state your hypotheses clearly.

5. A company claims that \(35 \%\) of its peas germinate. In order to test this claim Ann decides to plant 15 of these peas and record the number which germinate.

1. 1. State suitable hypotheses for a two-tailed test of this claim.
   2. Using a \(5 \%\) level of significance, find an appropriate critical region for this test. The probability in each of the tails should be as close to \(2.5 \%\) as possible.
2. Ann found that 8 of the 15 peas germinated. State whether or not the company's claim is supported. Give a reason for your answer.
3. State the actual significance level of this test.

1. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\) This observation is to be used to test, at a \(5 \%\) level of significance,

$$\mathrm { H } _ { 0 } : \lambda = k \quad \mathrm { H } _ { 1 } : \lambda \neq k$$ where \(k\) is a positive integer.
Given that the critical region for this test is \(( X = 0 ) \cup ( X \geqslant 9 )\)

1. find the value of \(k\), justifying your answer.
2. Find the actual significance level of this test.
   

3. A single observation \(x\) is to be taken from \(X \sim \mathrm {~B} ( 12 , p )\) This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.45\) against \(\mathrm { H } _ { 1 } : p > 0.45\)

1. Using a \(5 \%\) level of significance, find the critical region for this test.
2. State the actual significance level of this test. The value of the observation is found to be 9
3. State the conclusion that can be made based on this observation.
4. State whether or not this conclusion would change if the same test was carried out at the
   1. 10\% level of significance,
   2. \(1 \%\) level of significance.

A mobile phone company claims that each year \(5 \%\) of its customers have their mobile phone stolen. An insurance company claims this percentage is higher. A random sample of 30 of the mobile phone company's customers is taken and 4 of them have had their mobile phone stolen during the last year.

1. Test the insurance company's claim at the \(10 \%\) level of significance. State your hypotheses clearly. A new random sample of 90 customers is taken. A test is carried out using these 90 customers, to see if the percentage of customers who have had a mobile phone stolen in the last year is more than 5\%
2. Using a suitable approximation and a \(10 \%\) level of significance, find the critical region for this test.
   

4. A bag contains a large number of marbles, each of which is blue or red. A random sample of 3 marbles is taken from the bag. The random variable \(D\) represents the number of blue marbles taken minus the number of red marbles taken. Given that 20\% of the marbles in the bag are blue,

1. show that \(\mathrm { P } ( D = - 1 ) = 0.384\)
2. find the sampling distribution of \(D\)
3. write down the mode of \(D\) Takashi claims that the true proportion of blue marbles is greater than 20\% and tests his claim by selecting a random sample of 12 marbles from the bag.
4. Find the critical region for this test at the 10\% level of significance.
5. State the actual significance level of this test. \includegraphics[max width=\textwidth, alt={}, center]{d2f40cdb-917a-4377-88f4-396766a299e2-15_2255_47_314_37}

3. A manufacturer produces plates. The proportion of plates that are flawed is \(45 \%\), with flawed plates occurring independently. A random sample of 10 of these plates is selected.

1. Find the probability that the sample contains
   1. fewer than 2 flawed plates,
   2. at least 6 flawed plates.
      (4) George believes that the proportion of flawed plates is not \(45 \%\). To assess his belief George takes a random sample of 120 plates. The random variable \(F\) represents the number of flawed plates found in the sample.
2. Using a normal approximation, find the maximum number of plates, \(c\), and the minimum number of plates, \(d\), such that $$\mathrm { P } ( F \leqslant c ) \leqslant 0.05 \text { and } \mathrm { P } ( F \geqslant d ) \leqslant 0.05$$ where \(F \sim \mathrm {~B} ( 120,0.45 )\) The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.
3. Use a suitable hypothesis test, at the \(5 \%\) level of significance, to assess the manufacturer's claim. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-11_2255_50_314_34}

4. In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)

1. Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per \(\mathrm { m } ^ { 2 }\) of the peat bog to follow a Poisson distribution.
   (1) Given that the number of Common Spotted-orchids in \(1 \mathrm {~m} ^ { 2 }\) of the peat bog can be modelled by a Poisson distribution,
2. find the probability that in a randomly selected \(1 \mathrm {~m} ^ { 2 }\) of the peat bog
   1. there are exactly 6 Common Spotted-orchids,
   2. there are fewer than 10 but more than 4 Common Spotted-orchids.
      (4) Juan believes that by introducing a new management scheme the number of Common Spotted-orchids in the peat bog will increase. After three years under the new management scheme, a randomly selected \(2 \mathrm {~m} ^ { 2 }\) of the peat bog contains 11 Common Spotted-orchids.
3. Using a \(5 \%\) significance level assess Juan's belief. State your hypotheses clearly. Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)
4. use a normal approximation to find the probability that in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog there are fewer than 70 Common Spotted-orchids. Following a period of dry weather, the probability that there are fewer than 70 Common Spotted-orchids in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog is 0.012 A random sample of 200 non-overlapping \(20 \mathrm {~m} ^ { 2 }\) areas of the peat bog is taken.
5. Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-15_2255_50_314_34}

1. A research project into food purchases found that \(35 \%\) of people who buy eggs do not buy free range eggs.

A random sample of 30 people who bought eggs is taken. The random variable \(F\) denotes the number of people who do not buy free range eggs.

1. Find \(\mathrm { P } ( F \geqslant 12 )\)
2. Find \(\mathrm { P } ( 8 \leqslant F < 15 )\) A farm shop gives 3 loyalty points with every purchase of free range eggs. With every purchase of eggs that are not free range the farm shop gives 1 loyalty point. A random sample of 30 customers who buy eggs from the farm shop is taken.
3. Find the probability that the total number of points given to these customers is less than 70 The manager of the farm shop believes that the proportion of customers who buy eggs but do not buy free range eggs is more than \(35 \%\) In a survey of 200 customers who buy eggs, 86 do not buy free range eggs. Using a suitable test and a normal approximation,
4. determine, at the \(5 \%\) level of significance, whether there is evidence to support the manager's belief. State your hypotheses clearly.

The number of cars entering a safari park per 10 -minute period can be modelled by a Poisson distribution with mean 6

1. Find the probability that in a given 10 -minute period exactly 8 cars will enter the safari park.
2. Find the smallest value of \(n\) such that the probability that at least \(n\) cars enter the safari park in 10 minutes is less than 0.05 The probability that no cars enter the safari park in \(m\) minutes, where \(m\) is an integer, is less than 0.05
3. Find the smallest value of \(m\) A car enters the safari park.
4. Find the probability that there is less than 5 minutes before the next car enters the safari park. Given that exactly 15 cars entered the safari park in a 30-minute period,
5. find the probability that exactly 1 car entered the safari park in the first 5 minutes of the 30-minute period. Aston claims that the mean number of cars entering the safari park per 10-minute period is more than 6 He selects a 15-minute period at random in order to test whether there is evidence to support his claim.
6. Determine the critical region for the test at the \(5 \%\) level of significance.

1. A company produces packets of sunflower seeds. Each packet contains 40 seeds. The company claims that, on average, only 35\% of its sunflower seeds do not germinate.

A packet is selected at random.

1. Using a \(5 \%\) level of significance, find an appropriate critical region for a two-tailed test that the proportion of sunflower seeds that do not germinate is 0.35 You should state your hypotheses clearly and state the probability, which should be as close as possible to \(2.5 \%\), for each tail of your critical region.
2. Write down the actual significance level of this test. Past records suggest that \(2.8 \%\) of the company's sunflower seeds grow to a height of more than 3 metres.
   A random sample of 250 of the company's sunflower seeds is taken and 11 of them grow to a height of more than 3 metres.
3. Using a suitable approximation test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than \(2.8 \%\) State your hypotheses clearly.

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.