Hypothesis test for a binomial proportion

A manufacturer makes components for the car industry. On average 1.5% of the components made are defective.

Using a suitable model, find the probability that in a random sample of 36 components there are

exactly 2 defective components,

more than 3 defective components. Once every six months, the manufacturer tests whether the proportion of defective components has changed using hypotheses H₀:p = 0.015 and H₁:p ≠ 0.015 A random sample of 260 components is taken and 8 are found to be defective.

Using a 5% level of significance, complete the test.

Amar is studying the flight of a bird from its nest. He measures the bird’s height above the ground, h metres, at time t seconds for 10 values of t Amar finds the equation of the regression line for the data to be h = 38.6 – 1.28t

Interpret the gradient of this line. The product moment correlation coefficient between h and t is – 0.510

Test whether or not there is evidence of a negative correlation between the height above the ground and the time during the flight. You should

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

Jane draws the following scatter diagram for Amar’s data.

With reference to the scatter diagram, state, giving a reason, whether or not the regression line h = 38.6 – 1.28 t is an appropriate model for these data. Jane suggests an improved model using the variable u = t k² where k is a constant. She obtains the equation h = 38.1 – 0.78u

Choose a suitable value for k to write Jane’s improved model for h in terms of t only.

The proportion of left-handed adults in a country is 10% Freya believes that the proportion of left-handed adults under the age of 25 in this country is different from 10% She takes a random sample of 40 adults under the age of 25 from this country to investigate her belief.

Find the critical region for a suitable test to assess Freya’s belief. You should

• state your hypotheses clearly

• use a 5% level of significance

• state the probability of rejection in each tail

Write down the actual significance level of your test in part (a) In Freya’s sample 7 adults were left-handed.

With reference to your answer in part (a) comment on Freya’s belief.

A machine fills packets with sweets and of the packets also contain a prize. The packets of sweets are placed in boxes before being delivered to shops. There are 40 packets of sweets in each box. The random variable T represents the number of packets of sweets that contain a prize in each box.

State a condition needed for T to be modelled by B(40, A box is selected at random.

the probability that the box has exactly 6 packets containing a prize,

the probability that the box has fewer than 3 packets containing a prize. Kamil’s sweet shop buys 5 boxes of these sweets.

Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize. Kamil claims that the proportion of packets containing a prize is less than A random sample of 110 packets is taken and 9 packets contain a prize.

Use a suitable test to assess Kamil’s claim. You should

• state your hypotheses clearly

• use a 5% level of significance

A manufacturer uses a machine to make metal rods. The length of a metal rod, Lcm, is normally distributed with

• a mean of 8 cm

• a standard deviation of x cm

Given that the proportion of metal rods less than 7.902 cm in length is 2.5%

show that x = 0.05 to 2 decimal places.

Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length. The cost of producing a single metal rod is 20p A metal rod

• where L 7.94 is sold for scrap for 5p

• where L 8.09 is shortened for an extra cost of 10p and then sold for 50p

Calculate the expected profit per 500 of the metal rods. Give your answer to the nearest pound. The same manufacturer makes metal hinges in large batches. The hinges each have a probability of 0.015 of having a fault. A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty. The manufacturer's aim is for 95% of batches to be accepted.

Explain whether the manufacturer is likely to achieve its aim.

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.

Anna is investigating the relationship between exercise and resting heart rate. She takes a random sample of 19 people in her year at school and records for each person

• their resting heart rate, h beats per minute

• the number of minutes, m, spent exercising each week

Her results are shown on the scatter diagram.

Interpret the nature of the relationship between h and m Anna codes the data using the formulae

The product moment correlation coefficient between x and y is – 0.897

Test whether or not there is significant evidence of a negative correlation between x and y You should

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

The equation of the line of best fit of y on x is y = – 0.05 x + 1.92

Use the equation of the line of best fit of y on x to find a model for h on m in the form

where a and k are constants to be found.

TOTAL FOR STATISTICS IS 50 MARKS

Marc took a random sample of 16 students from a school and for each student recorded

• the number of letters, x, in their last name

• the number of letters, y, in their first name

His results are shown in the scatter diagram on the next page.

Describe the correlation between x and y. Marc suggests that parents with long last names tend to give their children shorter first names.

Using the scatter diagram comment on Marc’s suggestion, giving a reason for your answer. The results from Marc’s random sample of 16 observations are given in the table below.

Use your calculator to find the product moment correlation coefficient between x and y for these data.

Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name. You should

• state your hypotheses clearly

• use a 5% level of significance

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.

In a region of the UK, 5% of people have red hair. In a random sample of size n, taken from this region, the expected number of people with red hair is 3

Calculate the value of n.

A random sample of 20 people is taken from this region. Find the probability that

exactly 4 of these people have red hair,

at least 4 of these people have red hair.

Patrick claims that Reddman people have a probability greater than 5% of having red hair. In a random sample of 50 Reddman people, 4 of them have red hair.

Stating your hypotheses clearly, test Patrick’s claim. Use a 1% level of significance.

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.

Find P(5 ≤ X 11)

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.

Test, at the 10% level of significance, the radio company’s claim. State your hypotheses clearly.

Before Roger will use a tennis ball he checks it using a “bounce” test. The probability that a ball from Roger’s usual supplier fails the bounce test is 0.2. A new supplier claims that the probability of one of their balls failing the bounce test is less than 0.2. Roger checks a random sample of 40 balls from the new supplier and finds that 3 balls fail the bounce test. Stating your hypotheses clearly, use a 5% level of significance to test the new supplier’s claim.

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.

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 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 student takes a multiple choice test. The test is made up of 10 questions each with 5 possible answers. The student gets 4 questions correct. Her teacher claims she was guessing the answers. Using a one tailed test, at the 5% level of significance, test whether or not there is evidence to reject the teacher’s claim. 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.