The language of hypothesis testing

Kay is studying the variables Daily Total Sunshine (x) and Daily Total Rainfall (y) from the large data set for Leeming in 2015 Kay starts with 5th May and then selects every 10th day thereafter.

State the name of the sampling technique Kay uses. Kay wants to find the regression line of y on x for these data.

Using your knowledge of the large data set, explain how Kay might need to clean these data before finding the equation of the regression line. The equation of the regression line Kay finds is y = 0.741 + 0.199x

Using your knowledge of the large data set,

state the units of the gradient of the regression line,

give an interpretation of the y-intercept of the regression line. Kay’s teacher claimed that the greater the amount of sunshine in a day the lower the amount of rain there should be.

State, giving a reason, whether the teacher’s claim is true for Kay’s data. The teacher used all the data for these variables from the large data set for Leeming in 2015, as a sample. The teacher calculated the product moment correlation coefficient for x and y to be – 0.160 In a suitable test to determine whether there is evidence to support the teacher’s claim, the p-value was 0.015

Using a 5% level of significance, state the hypotheses and conclusion for this test.

For the test in part (e) describe

the sample,

a possible population.

A dentist knows from past records that 10% of customers arrive late for their appointment. A new manager believes that there has been a change in the proportion of customers who arrive late for their appointment. A random sample of 50 of the dentist's customers is taken.

Write down

• a null hypothesis corresponding to no change in the proportion of customers who arrive late

• an alternative hypothesis corresponding to the manager's belief

Using a 5% level of significance, find the critical region for a two-tailed test of the null hypothesis in (a) You should state the probability of rejection in each tail, which should be less than 0.025

Find the actual level of significance of the test based on your critical region from part (b) The manager observes that 15 of the 50 customers arrived late for their appointment.

With reference to part (b), comment on the manager’s belief.

Customers arrive at a bank at a rate of 10 customers per 5 minutes. John suggests that a Poisson distribution would be a suitable model to describe the number of customers arriving at the bank in a 1-minute interval.

Write down two assumptions that John has made.

Given that the Poisson distribution is a suitable model,

find the probability that in any given 1-minute interval

exactly 2 customers arrive,

more than 4 customers arrive.

When there is sufficient evidence that the rate of customers arriving at the bank is higher than 10 customers per 5 minutes, the manager must open another service till to avoid long queues. In a 4-minute interval, the manager observes 9 customers arriving.

Test, at the 5% significance level, whether or not the manager has sufficient evidence to open another service till. State your hypotheses clearly.

Using a 1% significance level, determine the smallest number of customers the manager must observe in a 1-minute interval in order to have sufficient evidence to open another service till.

A company sells a variety of bars of chocolate including Pearchoc bars. In an attempt to increase the sales of Pearchoc bars, the company runs a promotion where any Pearchoc wrapper that has the word WINNER printed on the inside wins a prize. The company prints the word WINNER on 5% of Pearchoc bar wrappers. Pearchoc bars are sold in packets of 12

Find the probability that in a randomly selected packet of Pearchoc bars, the number of wrappers with the word WINNER printed on them will be

exactly one,

more than two.

A customer bought 4 packets of Pearchoc bars.

Find the probability that only 1 of the packets contained exactly 1 wrapper with the word WINNER printed on it.

Before the promotion, 4% of all bars of chocolate sold in a supermarket were Pearchoc bars. After the promotion, the company selects a random sample of 150 sales of bars of chocolate from the supermarket database and finds that 10 of these sales were Pearchoc bars.

Using a suitable approximation and stating your hypotheses clearly, test whether or not the company’s promotion has been successful. Use a 5% level of significance.

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.

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.

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.

Using your answer to part (b), state whether or not this sample supports the manager’s claim. Use a 5% level of significance.

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.

Find the value of n.

A potter believes that 20% of pots break whilst being fired in a kiln. Pots are fired in batches of 25.

Let X denote the number of broken pots in a batch. A batch is selected at random. Using a 10% significance level, find the critical region for a two tailed test of the potter’s belief. You should state the probability in each tail of your critical region.

The potter aims to reduce the proportion of pots which break in the kiln by increasing the size of the batch fired. He now fires pots in batches of 50. He then chooses a batch at random and discovers there are 6 pots which broke whilst being fired in the kiln.

Test, at the 5% level of significance, whether or not there is evidence that increasing the number of pots in a batch has reduced the percentage of pots that break whilst being fired in the kiln. State your hypotheses clearly.

A company receives telephone calls at random at a mean rate of 2.5 per hour.

Find the probability that the company receives

at least 4 telephone calls in the next hour,

exactly 3 telephone calls in the next 15 minutes.

Find, to the nearest minute, the maximum length of time the telephone can be left unattended so that the probability of missing a telephone call is less than 0.2

The company puts an advert in the local newspaper. The number of telephone calls received in a randomly selected 2 hour period after the paper is published is 10

Test at the 5% level of significance whether or not the mean rate of telephone calls has increased. State your hypotheses clearly.

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.

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.

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

Test, at the 5% level of significance, whether Liftsforall ’s claim is correct. State your hypotheses clearly.

State the actual significance level of this test.

The residents in the block of flats have a maintenance contract with Liftsforall. The residents pay 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. 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,

find the probability that the residents do not pay more than £500 to Liftsforall in the next year.

Sammy manufactures wallpaper. She knows that defects occur randomly in the manufacturing process at a rate of 1 every 8 metres. Once a week the machinery is cleaned and reset. Sammy then takes a random sample of 40 metres of wallpaper from the next batch produced to test if there has been any change in the rate of defects.

Stating your hypotheses clearly and using a 10% level of significance, find the critical region for this test. You should choose your critical region so that the probability of rejection is less than 0.05 in each tail.

State the actual significance level of this test.

Thomas claims that his new machine would reduce the rate of defects and invites Sammy to test it. Sammy takes a random sample of 200 metres of wallpaper produced on Thomas’ machine and finds 19 defects.

Using a suitable approximation, test Thomas’ claim. You should use a 5% level of significance and state your hypotheses clearly.

An online shop sells a computer game at an average rate of 1 per day.

Find the probability that the shop sells more than 10 games in a 7 day period.

Once every 7 days the shop has games delivered before it opens.

Find the least number of games the shop should have in stock immediately after a delivery so that the probability of running out of the game before the next delivery is less than 0.05

In an attempt to increase sales of the computer game, the price is reduced for six months. A random sample of 28 days is taken from these six months. In the sample of 28 days, 36 computer games are sold.

Using a suitable approximation and a 5% level of significance, test whether or not the average rate of sales per day has increased during these six months. State your hypotheses clearly.

In a manufacturing process 25% of articles are thought to be defective. Articles are produced in batches of 20

A batch is selected at random. Using a 5% significance level, find the critical region for a two tailed test that the probability of an article chosen at random being defective is 0.25 You should state the probability in each tail which should be as close as possible to 0.025

The manufacturer changes the production process to try to reduce the number of defective articles. She then chooses a batch at random and discovers there are 3 defective articles.

Test at the 5% level of significance whether or not there is evidence that the changes to the process have reduced the percentage of defective articles. State your hypotheses clearly.

Frugal bakery claims that their packs of 10 muffins contain on average 80 raisins per pack. A Poisson distribution is used to describe the number of raisins per muffin. A muffin is selected at random to test whether or not the mean number of raisins per muffin has changed.

Find the critical region for a two-tailed test using a 10% level of significance. The probability of rejection in each tail should be less than 0.05

Find the actual significance level of this test.

The bakery has a special promotion claiming that their muffins now contain even more raisins. A random sample of 10 muffins is selected and is found to contain a total of 95 raisins.

Use a suitable approximation to test the bakery’s claim. You should state your hypotheses clearly and use a 5% level of significance.

Explain what you understand by a hypothesis.

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.

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.

Using a 1% level of significance, find the smallest value of n for which the hypothesis H₀: p 0 45. will be rejected in favour of H₁: p 0 45.

A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.

Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval.

The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.

Stating your hypotheses clearly, and using a 5% level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.

Using a 5% level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.

A shopkeeper knows, from past records, that 15% of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.

Stating your hypotheses clearly, and using a 5% level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till.

During the refurbishment a new sandwich display was installed. Before the refurbishment 20% of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.

Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper’s claim. Use a 10% level of significance.

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.

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

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

all users connect at their first attempt,

at least 4 users fail to connect at their first attempt.

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.

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.

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.

Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample.

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 . The probability of rejection in either tail should be as close as possible to 0.025

Find the actual significance level of this test.

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

Comment on the company’s claim in the light of this value. Justify your answer.

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

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

Define the critical region of a test statistic.

A discrete random variable X has a Binomial distribution B(30, p). A single observation is used to test H₀: p = 0.3 against H₁: p ≠ 0.3

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

Write down the actual significance level of the test.

The value of the observation was found to be 15.

Comment on this finding in light of your critical region.

Past records suggest that 30% of customers who buy baked beans from a large supermarket buy them in single tins. A new manager questions whether or not there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.

Using a 10% level of significance, find the critical region for a two-tailed test to answer the manager’s question. You should state the probability of rejection in each tail which should be less than 0.05.

Write down the actual significance level of a test based on your critical region from part (a).

The manager found that 11 customers from the sample of 20 had bought baked beans in single tins.

Comment on this finding in the light of your critical region found in part (a).

A single observation x is to be taken from a Binomial distribution B(20, p). This observation is used to test H₀: p = 0.3 against H₁: p ≠ 0.3

Using a 5% level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to 2.5%.

State the actual significance level of this test.

The actual value of x obtained is 3.

State a conclusion that can be drawn based on this value giving a reason for your answer.

A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.

Test, at the 10% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.

State the minimum number of visits required to obtain a significant result.

State an assumption that has been made about the visits to the server.

In a random two minute period on a Saturday the web server is visited 20 times.

Using a suitable approximation, test at the 10% level of significance, whether or not the rate of visits is greater on a Saturday.