2.05a Hypothesis testing language: null, alternative, p-value, significance

8 A company records the number of complaints, \(X\), that it receives over 60 months. The summarised results are $$\sum x = 102 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 103.25$$ 8

1. Using this data, explain why it may be appropriate to model the number of complaints received by the company per month by a Poisson distribution with mean 1.7
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2. The company also receives enquiries as well as complaints. The number of enquiries received is independent of the number of complaints received. The company models the number of complaints per month with a Poisson distribution with mean 1.7 and the number of enquiries per month with a Poisson distribution with mean 5.2 The company starts selling a new product.
   The company records a total of 3 complaints and enquiries in one randomly chosen month. Investigate if the mean total number of complaints and enquiries received per month has changed following the introduction of the new product, using the \(10 \%\) level of significance.
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3. It is later found that the mean total number of complaints and enquiries received per month is 6.1 Find the power of the test carried out in part (b), giving your answer to four decimal places. \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-15_2492_1721_217_150}
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1. Test the scientist's claim, using the 10\% level of significance.
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2. For the context of the test carried out in part (a), state the meaning of a Type I error. [1 mark]

7 Company \(A\) uses a machine to produce toys. The number of toys in a week that do not pass Company \(A\) 's quality checks is modelled by a Poisson distribution \(X\) with standard deviation 5 The machine producing the toys breaks down.
After it is repaired, 16 toys in the next week do not pass the quality checks.
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1. Investigate whether the average number of toys that do not pass the quality checks in a week has changed, using the \(5 \%\) level of significance.
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2. For the test carried out in part (a), state in context the meaning of a Type II error. 7
3. Company \(B\) uses a different machine to produce toys.
   The number of toys in a week that do not pass Company B's quality checks is modelled by a Poisson distribution \(Y\) with mean 18 The variables \(X\) and \(Y\) are independent.
   Find the distribution of the total number of toys in a week produced by companies \(A\) and \(B\) that do not pass their quality checks. 7
4. State two reasons why a Poisson distribution may not be a valid model for the number of toys that do not pass the quality checks in a week. Reason 1 \(\_\_\_\_\) Reason 2 \(\_\_\_\_\)

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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.

5. A laboratory carrying out screening for a certain blood disorder claims that the average time taken for test results to be returned is 38 hours. A reporter for a national newspaper suspects that the results take longer, on average, to be returned than claimed by the laboratory. The reporter finds the time, \(x\) hours, for 50 randomly selected results, in order to conduct a hypothesis test. The following summary statistics were obtained. $$\sum x = 2163 \quad \sum x ^ { 2 } = 98508$$

1. Calculate the \(p\)-value for the reporter's hypothesis test, and complete the test using a \(5 \%\) level of significance. Hence write a headline for the reporter to use.
2. Explain the relevance or otherwise of the Central Limit Theorem to your answer in part (a).
3. Briefly explain why a random sample is preferable to taking a batch of 50 consecutive results.
4. On another occasion, the reporter took a different random sample of 10 results.
   1. State, with a reason, what type of hypothesis test the reporter should use on this occasion.
   2. State one assumption required to carry out this test.

At a certain large school it was found that the proportion of students not wearing correct uniform was 0.15. The school sent a letter to parents asking them to ensure that their children wear the correct uniform. The school now wishes to test whether the proportion not wearing correct uniform has been reduced.

1. It is suggested that a random sample of the students in Grade 12 should be used for the test. Give a reason why this would not be an appropriate sample. [1]
2. State suitable null and alternative hypotheses. [1]
3. Use a binomial distribution to find the probability of a Type I error. You must justify your answer fully. [5]
4. In fact 4 students out of the 50 are not wearing correct uniform. State the conclusion of the test, explaining your answer. [2]
5. State, with a reason, which of the errors, Type I or Type II, may have been made. [2]

A suitable sample of 50 students is selected and the number not wearing correct uniform is noted. This figure is used to carry out a test at the 5% significance level.

The number of faults in cloth made on a certain machine has a Poisson distribution with mean 2.4 per 10 m\(^2\). An adjustment is made to the machine. It is required to test at the 5% significance level whether the mean number of faults has decreased. A randomly selected 30 m\(^2\) of cloth is checked and the number of faults is found.

1. State suitable null and alternative hypotheses for the test. [1]
2. Find the probability of a Type I error. [3]

Exactly 3 faults are found in the randomly selected 30 m\(^2\) of cloth.

1. Carry out the test at the 5% significance level. [2]

Later a similar test was carried out at the 5% significance level, using another randomly selected 30 m\(^2\) of cloth.

1. Given that the number of faults actually has a Poisson distribution with mean 0.5 per 10 m\(^2\), find the probability of a Type II error. [2]

In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution Po(0.31). Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase. In order to test whether this is the case, the total number of enquiries during a randomly chosen 5-minute period is noted. You should assume that a Poisson model is still appropriate. Given that the total number of enquiries is 5, carry out the test at the 2.5% significance level. [5]

The number of accidents per year on a certain road has the distribution \(\text{Po}(\lambda)\). In the past the value of \(\lambda\) was \(3.3\). Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5\%\) significance level.

1. Calculate the probability of a Type I error. [4]
2. Given that \(X = 2\), carry out the test. [3]
3. The council decides to carry out another similar test at the \(5\%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is \(0.6\), calculate the probability of a Type II error. [3]
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than \(10\) accidents in \(30\) years. [4]

Jeevan thinks that a six-sided die is biased in favour of six. In order to test this, Jeevan throws the die 10 times. If the die shows a six on at least 4 throws out of 10, she will conclude that she is correct.

1. State appropriate null and alternative hypotheses. [1]
2. Calculate the probability of a Type I error. [3]
3. Explain what is meant by a Type II error in this situation. [1]
4. If the die is actually biased so that the probability of throwing a six is \(\frac{1}{3}\), calculate the probability of a Type II error. [3]

The number of sightings of a golden eagle at a certain location has a Poisson distribution with mean 2.5 per week. Drilling for oil is started nearby. A naturalist wishes to test at the 5\% significance level whether there are fewer sightings since the drilling began. He notes that during the following 3 weeks there are 2 sightings.

1. Find the critical region for the test and carry out the test. [5]
2. State the probability of a Type I error. [1]
3. State why the naturalist could not have made a Type II error. [1]

The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.

1. A hypothesis test at the 10% level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test. [5]
2. Find the probability of a Type I error. [2]

Records show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km. New driving regulations are introduced and in the next 20 weeks he drives a total of 21 800 km.

1. Stating any assumption(s), test, at the 1% significance level, whether his mean weekly driving distance has decreased. [6]
2. A similar test at the 1% significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context. [2]

The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1

1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods. [2]
2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. [2]

The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.

1. Use the tables to find the value of \(w\) [3]

A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.

1. State the null hypothesis for this test. [1]
2. Determine the critical region for the test at the 5\% level of significance. [2]

A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution. The fisherman takes 5 fishing trips each lasting 1 hour.

1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. [6]

The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,

1. carry out the test at the 5\% level of significance. State your hypotheses clearly. [6]

Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the 5\% level of significance. State your hypotheses clearly. [6]

1. Explain what you understand by a critical region of a test statistic. [2]

The number of breakdowns per day in a large fleet of hire cars has a Poisson distribution with mean \(\frac{1}{7}\).

1. Find the probability that on a particular day there are fewer than 2 breakdowns. [3]
2. Find the probability that during a 14-day period there are at most 4 breakdowns. [3]

The cars are maintained at a garage. The garage introduced a weekly check to try to decrease the number of cars that break down. In a randomly selected 28-day period after the checks are introduced, only 1 hire car broke down.

1. Test, at the 5% level of significance, whether or not the mean number of breakdowns has decreased. State your hypotheses clearly. [7]

Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.

1. Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week. [4]

Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.

1. Test, at the 5\% level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly. [7]

It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.

1. Using a 5\% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to 2.5\% as possible. [6]
2. State the actual significance level of the above test. [1]

At a later date, a random sample of 20 bowls was taken and 2 of them were found to have defects.

1. Test, at the 10\% level of significance, whether or not there is evidence that the proportion of bowls with defects has decreased. State your hypotheses clearly. [7]

A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company's website at their first attempt.

1. Explain why the Poisson distribution may be a suitable model in this case. [1]

Find the probability that, in a randomly chosen 2 hour period,

1. 1. all users connect at their first attempt,
   2. at least 4 users fail to connect at their first attempt.
   [5]

The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60.

1. Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a 5\% level of significance and state your hypotheses clearly. [9]

A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.

1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample. [2]
2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac{1}{4}\). The probability of rejection in either tail should be as close as possible to 0.025 [3]
3. Find the actual significance level of this test. [2]

In the sample of 50 the actual number of faulty bolts was 8.

1. Comment on the company's claim in the light of this value. Justify your answer. [2]

The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.

1. Test at the 1\% level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly. [6]

The proportion of houses in Radville which are unable to receive digital radio is 25\%. In a survey of a random sample of 30 houses taken from Radville, the number, \(X\), of houses which are unable to receive digital radio is recorded.

1. Find P(5 \(\leq X < 11\)) [3]

A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.

1. Test, at the 10\% level of significance, the radio company's claim. State your hypotheses clearly. [5]

\emph{Liftsforall} claims that the lift they maintain in a block of flats breaks down at random at a mean rate of 4 times per month. To test this, the number of times the lift breaks down in a month is recorded.

1. Using a 5\% level of significance, find the critical region for a two-tailed test of the null hypothesis that 'the mean rate at which the lift breaks down is 4 times per month'. The probability of rejection in each of the tails should be as close to 2.5\% as possible. [3]

Over a randomly selected 1 month period the lift broke down 3 times.

1. Test, at the 5\% level of significance, whether \emph{Liftsforall}'s claim is correct. State your hypotheses clearly. [2]
2. State the actual significance level of this test. [1]

The residents in the block of flats have a maintenance contract with \emph{Liftsforall}. The residents pay \emph{Liftsforall} £500 for every quarter (3 months) in which there are at most 3 breakdowns. If there are 4 or more breakdowns in a quarter then the residents do not pay for that quarter. \emph{Liftsforall} installs a new lift in the block of flats. Given that the new lift breaks down at a mean rate of 2 times per month,

1. find the probability that the residents do not pay more than £500 to \emph{Liftsforall} in the next year. [6]

A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:

1. Show that the mean number of errors per page in this sample of pages is 2. [2]
2. Find the variance of the number of errors per page in this sample. [2]
3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution. [1]

Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2.

1. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a 5\% level of significance. [6]