Sampling Distributions & Introduction to Hypothesis Testing (Edexcel International A Level (IAL) Maths: Statistics 2): Exam Questions

Scientists at an animal health organisation are looking to collect data on a particular non-contagious disease in sheep throughout New Zealand.  The chance of any individual sheep in the country having the disease is believed to be constant and independent of location.

(i) Give two reasons why the scientists would not choose to use a census for their data collection.

(ii) Suggest a suitable sampling frame for the population.

For purposes of the scientists’ study, every sheep in the total population may be classified simply as either ‘has the disease’ or ‘does not have the disease’.  A random sample of  sheep is taken from the population and each sheep is tested to see whether it has the disease.

For a given random sample of size $n$, write down an expression for the total number of possible samples that could occur as a result of the testing.

The random variable $X$ is defined as

$X=\{\begin{array}{ll}1,& \mathrm{if}\mathrm{the}\mathrm{sheep}\mathrm{has}\mathrm{the}\mathrm{disease},\\ 0,& \mathrm{if}\mathrm{the}\mathrm{sheep}\mathrm{does}\mathrm{not}\mathrm{have}\mathrm{the}\mathrm{disease}.\end{array}$

Write down the sampling distribution of the statistic

$Y=\sum _1^{n}X$

Explain what you understand by a critical region of a test statistic.

Nationally 44% of A Level mathematics students identify as female.  The headteacher of a particular school claims that the proportion of A Level mathematics students in the school who identify as female is higher than the national average.

(i) State a suitable null hypothesis to test the headteacher’s claim.

(ii) State a suitable alternative hypothesis to test the headteacher’s claim.

The headteacher takes a random sample of 60 A Level mathematics students and records the number of them who identify as female, $x$.  For a test at the 10% significance level the critical region is $X\ge 32$.

Given that $x=36$, comment on the headteacher’s claim.

The probability of a chicken laying an egg on any given day is 65%.  Two farmers, Amina and Bert, have 30 chickens each.  They believe that the probability of their chickens laying an egg on any given day is different to 65%.

(i) State a suitable null hypothesis to test the farmers’ belief.

(ii) State a suitable alternative hypothesis for a two-tailed test.

During a specific day, Amina and Bert each record the number of their 30 chickens that lay an egg.  At the 5% significance level the critical regions for this test are $X\le 13$ and $X\ge 25$.

Write down the critical values for the hypothesis test.

(i) Given that for Amina $x=12$, comment on her belief.

(ii) Given that for Bert $x=24$, comment on his belief.

A memory experiment involves having participants read a list of 20 words for two minutes and then recording how many of the words they can recall.  Peter, a psychologist, claims that more than 60% of teenagers can recall all the words.  Peter takes a random sample of 40 teenagers and records how many of them recall all the words.

(i) State a suitable null hypothesis to test the psychologist’s claim.

(ii) State a suitable alternative hypothesis to test the psychologist’s claim.

Given that the critical value for the test is $x=19$, state the outcome of the test if

(i) 18 out of the 40 teenagers recall all the words

(ii) 19 out of the 40 teenagers recall all the words

(iii) 20 out of the 40 teenagers recall all the words.

A machine produces toys for a company.  It was found that 8% of the toys it was producing were faulty.  After an engineer works on the machine, she claims that the proportion of faulty toys should now have decreased.

State suitable null and alternative hypotheses to test this claim.

After the engineer is finished, the manager of the company takes a random sample of 100 toys and finds that 2 of them are faulty.

Given that $P(X\le 2)=0.01127$ when$X~B(100,0.08)$, determine the outcome of the hypothesis test using a 1% level of significance.  Give your conclusion in context.

After it was estimated that only 72% of patients were turning up for their appointments at Pearly Teeth dental surgery, the owner began sending text message reminders to the patients on the day before their appointments.  In order to test whether the reminders have increased the proportion of patients turning up to their appointments, the owner decides to conduct a hypothesis test at the 5% level of significance using the next 160 patients scheduled for appointments as a sample.

State suitable null and alternative hypotheses to test this claim.

Given that for this hypothesis test $X~B(160,p)$ describe, in context, the random variable $X.$

Out of the 160 patients used for the sample, 127 turned up for their appointments. It is given that, assuming the null hypothesis is true, $P(X\ge 127)=0.02094$.

Determine the outcome of the hypothesis test, giving your conclusion in context.

Chase buys a board game which contains a six-sided dice. He rolls the dice 150 times and obtains the number six on 15 occasions. Chase wishes to test his belief that the dice is not fair.

(i) State a suitable null hypothesis to test Chase’s belief.

(ii) State a suitable alternative hypothesis for a two-tailed test.

Given that $P(X\le 15)=0.01452$ when  $X~B\left(150,\frac{1}{6}\right)$, test Chase’s belief that the dice is not fair, using a 2% level of significance.

Flight delays at a certain small airport are found to happen randomly and independently at an average rate of 32 delays per 7-day week. The airport manager puts a new scheme into place to reduce delays. The next day there are only 2 delayed flights.  A hypothesis test is carried out at the 5% significance level.

(i) Suggest a suitable test statistic.

(ii) Write down a suitable sampling distribution to use along with your test statistic from part (i).

(iii) Write down suitable null and alternative hypotheses the airport manager could use to see if the average rate of flight delays per week has been reduced.

(iv) Explain the condition under which the null hypothesis would be rejected.

A two-tailed test of the null hypothesis $H_0:p=0.23$ is carried out for the random variable $X~B(60,p)$. 

Write down the alternative hypothesis.

One of the critical regions is $X\ge 20$.  You are given the following probabilities:

$P(X\le 8)=0.04603$

$P(X\le 9)=0.08932$

$P(X\le 10)=0.15526$

Given that a 10% level of significance is used, determine the other critical region. Give a reason for your answer by using a relevant probability.

You are also given that $P(X\ge 20)=0.04427$.

Find the actual level of significance of this test.

Joel is a manager at a swimming pool and claims that less than half of customers wear goggles in the water.  Joel forms a sample using the next 100 swimmers and he notes that 42 of them wear goggles.

If $X~B(100,0.5)$ then:

$\begin{array}{l}P(X<42)=0.0443\\ P(X\le 42)=0.0666\\ P(X=42)=0.0223\\ P(X\ge 42)=0.9557\\ P(X>42)=0.9334\end{array}$

Stating your hypotheses clearly, test Joel’s claim using a 5% level of significance.

Joel discovers that there was a family of 12 people included in the sample, all of whom wore goggles.

Explain how this information affects the conclusion to the hypothesis test.

At Hilbert’s Hotel three quarters of customers leave feedback upon departure by writing a comment in a book on the reception desk.  Karla, the manager, decides to get rid of the feedback book and instead leaves a feedback form in each room.  To test whether this new system has made a difference to the proportion of guests who leave feedback, Karla forms a sample using the next 80 room bookings.  Once the 80 sets of guests leave Hilbert’s Hotel, Karla counts that 65 feedback forms have been completed.

When $X~B\left(80,\frac{3}{4}\right)$ the following probabilities are given:

$\begin{array}{l}P(X<65)=0.8792\\ P(X\le 65)=0.9260\\ P(X=65)=0.0468\\ P(X\ge 65)=0.1208\\ P(X>65)=0.0740\end{array}$

Test, using a 10% level of significance, whether there is evidence to suggest that the feedback forms have changed the proportion of guests who leave feedback. State your hypotheses clearly.

Karla repeats the same test, with the same hypotheses, the following week and finds that 53 out of the 80 sets of guests fill in their feedback forms. This leads to the null hypothesis being rejected. Karla claims that this shows that there is evidence that the proportion of guests leaving feedback has decreased.

Explain whether Karla’s claim is valid.

A spinner with seven equal sections labelled 0, 1, 1, 3, 3, 3 and 3 is being used in a game show where contestants spin the spinner twice and the two individual values it lands on, $S_1$ and $S_2$, are recorded.

List all the possible samples that could result when a contestant spins the spinner twice.

Assuming that the spinner is fair, find the sampling distribution for the sum of $S_1\mathrm{and}{S}_2$ $\left(\sum S\right)$.

If the sum of the two spins is 0, the contestant wins a grand prize.  After a day of playing the game, the grand prize has been won six times and the game show host thinks that someone may have tampered with the spinner so as to make a grand prize win more likely.  A hypothesis test is carried out at the 5% level of significance to test if the spinner is fair.  

Write suitable null and alternative hypotheses for the test.

Given that the game was played 100 times on the day in question,

(i) suggest a suitable approximating distribution that could be used to carry out the hypothesis test

(ii) describe the circumstances under which the null hypothesis would be rejected.

A group of high school statistics students are investigating the probability of winning a game called Chi Squares.  Their teacher claims that they have more than a 60% chance of winning the game.  To test the claim, they play 30 games of Chi Squares and win 80% of them.  They perform a hypothesis test using a 5% level of significance.  Below are shown the solutions of two students, Gertrude and Nate:

You are given that the students have correctly calculated their probabilities.

Identify and explain the three mistakes made by Gertrude.

Identify and explain the two mistakes made by Nate.

Use the information above to find the $p$-value for the test statistic $x=24$, showing your calculation clearly.

Explain what you understand by the significance level of a hypothesis test.

(i) Give an advantage of using a lower significance level for a hypothesis test.

(ii) Give a disadvantage of using a lower significance level for a hypothesis test.

For each of the following scenarios, explain whether a 1%, 5% or 10% level of significance would be most appropriate.

(i) A shopkeeper takes a sample of 10 cartons of milk to test whether the amount of milk in a carton has decreased.

(ii) A doctor takes a sample of 100 patients to test whether there is an improvement to the recovery rate of an illness when a new drug is used, compared with the current best treatment regime.

(iii) A manager takes a sample of 100 employees to test whether their level of job satisfaction has changed after new working hours have been introduced.

A chocolatier produces his trademark homemade chocolate truffle, the Deliciously Decadent, in batches of 50.  He always tastes two randomly-selected chocolates from each batch to check the quality of the batch.

Explain why the chocolatier takes a sample rather than a census to check for quality.

Suggest a suitable sampling frame and identify the sampling units.

If a chocolate is up to standard, the chocolatier assigns it the number 1.  If not then the chocolate is assigned the number 0.

Using this numbering system, list all of the possible samples that could result when the chocolatier quality tests a batch of chocolates.

After conducting many quality tests the chocolatier finds that 5% of chocolates are not up to standard, regardless of which batch they have come from.

Using your answer from part (c), and clearly defining any random variables you use, find the sampling distribution of the mean value of the sample when quality testing a single batch of chocolates.

It is known that historically 40% of all bees in a certain part of the UK belonged to pollinating species.  Farmers in the area, however, believe that that percentage has decreased in the past ten years.  They design an experiment in which they will safely catch 200 bees in the area, check which species they belong to, and then release them.  The farmers carry out a hypothesis test at the 10% significance level. 

They calculate that for $X~B(200,0.4)$

$\begin{array}{l}P(X<70)=0.063903\\ P(X=70)=0.020495\\ P(X=71)=0.025018\end{array}$

Clearly defining any parameters, state the null and alternative hypotheses for the farmers’ test.

Find the probability of incorrectly rejecting the null hypothesis.

It is discovered subsequently that in fact only 30% of bees in the area now belong to pollinating species. Given that for $X~B(200,0.3)$

$\begin{array}{l}P(X\ge 70)=0.072786\\ P(X=70)=0.018579\\ P(X<70)=0.927214\end{array}$

find the probability that the farmers’ hypothesis test results in incorrectly accepting the null hypothesis.

In the context of hypothesis testing, explain the term:

(i) critical region

(ii) critical value.

The table below shows the probabilities for different values that $X~B(40,0.8)$ can take:

A test of the null hypothesis $H_0:p=0.8$ against the alternative hypothesis $H_1:p>0.8$ is carried out for the random variable $X~B(40,p).$

Using a 5% level of significance, find the values of $X$ which would lead to the rejection of the null hypothesis.

A second test is carried out with the same null hypothesis against the alternative hypothesis $H_1:p\ne 0.8.$

Given that $x=38$ is a critical value, find the minimum level of significance for the test.

Meditest is a company manufacturing medical tests which are used to determine whether a patient has a certain illness.  Meditest claims that the tests are 95% accurate, however a particular hospital will only purchase the tests if they are more than 95% accurate.  Meditest test the accuracy of their product using a sample of 250 patients with the illness and agree on a 1% level of significance.  They discover that the tests are accurate for 245 out of the 250 patients.

If $X~B(250,0.95)$ then $P(X=245)=0.008515$ and $P(X>245)=0.004571$.

Stating your hypotheses clearly, test whether Meditest’s product is more than 95% accurate using a 1% level of significance.

Meditest notice that they would have had sufficient evidence to reject the null hypothesis using a 5% level of significance.  They change the level of significance from 1% to 5% and report to the hospital that their product is more than 95% accurate.

Comment on the validity of Meditest’s report to the hospital.

Frank is the owner of a factory which has recently opened near a school where Hilda is the headteacher.  Before the factory opened, the attendance rate at the school was good 90% of the time.  Hilda claims that the proportion of days when the attendance rate is good has decreased and she suspects this is due to the fumes from the factory making the children sick.  Frank disagrees and claims that the factory has made no difference to the attendance rate.  To test their claims a sample of 40 days is taken and on 32 days the attendance rate is good.

If $X~B(40,0.9)$ then:

$\begin{array}{l}P(X<32)=0.015495\\ P(X=32)=0.026407\\ P(X>32)=0.958098\end{array}$

Stating your hypotheses clearly, test Hilda’s claim using a 5% level of significance. Give your answer in context.

Explain whether the outcome of the test supports Hilda’s suspicion.

Stating your hypotheses clearly, test Frank’s claim using a 5% level of significance.

Suggest a reason why Frank might have chosen to use a two-tailed test.

State, with a reason, whether a one-tailed test or a two-tailed test would have been more appropriate for this scenario.

A teacher keeps bronze, silver and gold star stickers as prizes for his students.  The teacher takes a sample of five stickers at random from a full pack.

(i) Identify the population and the sampling units.

(ii) Write down the number of possible samples and find how many distinct samples the teacher could take.

The school office contains a large number of the bronze, silver and gold star stickers in the ratio 5:3:2. At the end of each term the students are awarded 50 points for a gold star, 20 points for a silver star and 10 points for a bronze star. Their final score is the product of the points their stars are worth. The teacher takes a sample of 3 stickers at random to show his students how to find the product of the points their stickers are worth.

Given that the teacher chose his sample only from bronze and silver stars, write down the sampling distribution of the product of the values the stickers are worth. Clearly define your random variable and statistic.

Given that $X~B(40,0.3)$ then:

$\begin{array}{l}P(X\le 5)=0.008618\\ P(X=6)=k\\ P(X=7)=0.031522\end{array}$

When a sample of size 40 is used to test $H_0:p=0.3$ against $H_1:p<0.3$, it is known that $x=6$ is the critical value using a 5% level of significance.  Use the probabilities above to find upper and lower bounds for the value of $k$.

When a sample of size 40 is used to test  $H_0:p=0.3$ against $H_1:p\ne 0.3$, it is known that $x=6$ is one of the two critical values using a 5% level of significance.  Use the probabilities above to find an improvement for one of the bounds for the value of $k$.

If $X~B(n,p)$ then $P(X=0)=(1-p{)}^n$ and $P(X=n)=p^n$.

A sample of size 30 is used to test the null hypothesis $H_0:p=0.9$ against the alternative hypothesis $H_1:p>0.9$ using a $k$% level of significance.

Given that there is at least one value that leads to the rejection of the null hypothesis, find the range of values for $k$.

A sample of size 100 is used to test the null hypothesis $H_0:p=q$ against the alternative hypothesis $H_1:p<q$ using a 5% level of significance.

Given that there are no critical values for this test, find the range of values for $q.$

A sample of size $m$ is used to test the null hypothesis $H_0:p=0.2$ against the alternative hypothesis $H_1:p\ne 0.2$ using a 1% level of significance.

Given that there is exactly one critical region for this test, find the range of values for $m$.