Hypothesis Testing (Normal Distribution) (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Exam Questions

The brightness, $B$ lumens, of the light from a Starlighter flashlight is normally distributed with standard deviation 15. Annie claims that the mean brightness of a Starlighter is greater than 110 lumens.

To test Annie's claim, a random sample of 40 Starlighters is taken and a hypothesis test is carried out at the 5% significance level.

State suitable null and alternative hypotheses for the test.

Find the critical region for the test.

The mean brightness of the sample is found to be 114.5 lumens.

State, with a reason, the conclusion of the test.

The wingspan, $W$ cm, of a small white butterfly is normally distributed with standard deviation 0.8. A report states that the mean wingspan of a small white butterfly is 4.1 cm.

Kenzie, a butterfly enthusiast, wishes to test, at the 5% significance level, whether the mean wingspan is different from 4.1 cm.

State suitable null and alternative hypotheses for Kenzie's test.

Kenzie takes a random sample of 6 small white butterflies and finds that their mean wingspan is 2.65 cm. Kenzie's working begins as follows.

If ${\text{H}}_0$ is true, then $W~\text{N}\left(4.1,0.8^2\right)$

$\text{P}(W<2.65)=\text{P}(Z<-1.8125)=1-0.9651=0.0349<0.05$

Explain two errors in Kenzie's working.

Carry out Kenzie's test correctly at the 5% significance level.

The masses of Burmese cats are normally distributed with mean 4.2 kg and standard deviation 1.3 kg.

Kamala, a cat breeder, claims that Burmese cats that live in households with young children weigh more, on average. She takes a random sample of 25 Burmese cats that live in households with young children, and carries out a test at the 5% significance level. You should assume that the standard deviation is unchanged.

State suitable null and alternative hypotheses for the test.

Find the rejection region for the test.

The mean mass of the 25 cats in Kamala's sample is 4.65 kg.

State, with a reason, the conclusion of the test.

The time, $X$ seconds, that Pierre takes to run 400 m is normally distributed with mean 87 and variance 16.

Pierre changes his diet and claims that his mean time to run 400 m has decreased. You should assume that the variance is unchanged.

State suitable null and alternative hypotheses for a test of Pierre's claim.

After changing his diet, Pierre runs 36 separate 400 m races. His mean time for these races is 86.1 seconds.

Treating these 36 races as a random sample, test Pierre's claim at the 5% significance level.

Give a reason why the 36 races might not form a suitable sample for this test.

The lengths, $L$ cm, of unicorns' horns have mean 91 and variance 5.

Luna believes that unicorns born beneath a rainbow have longer horns, on average. To test her belief, she plans to use the mean length, $\overline{L}$ cm, of the horns of a random sample of 12 unicorns born beneath a rainbow. You should assume that the variance is unchanged.

(i) State suitable null and alternative hypotheses for Luna's test.

(ii) State a necessary assumption about the lengths of the horns for the test to be valid.

Luna decides to reject the null hypothesis if the sample mean length is greater than 92.1 cm.

Find the significance level of the test.

Given that the population mean length of the horns of unicorns born beneath a rainbow is actually 91.5 cm, find the probability of a Type II error.

The IQ scores of students at a certain school are normally distributed with mean 126 and variance 50.

The headteacher decides to play classical music at lunchtimes and suspects that this has changed the mean IQ score of the students. You should assume that the variance is unchanged.

State suitable null and alternative hypotheses for a test of the headteacher's suspicion.

The headteacher selects 10 students and asks them to complete an IQ test. Their scores are as follows.

127, 127, 129, 130, 130, 132, 132, 132, 133, 138

Test, at the 5% significance level, whether there is evidence to support the headteacher's suspicion.

It was later discovered that the 10 students used in the sample were all in the same advanced class.

Comment on the validity of the conclusion of the test.

The mean annual salary of the employees of a large company is $32 500. The company opens a new branch, and the annual salaries of the employees at the new branch are normally distributed with standard deviation $6100.

A manager wishes to test whether the mean annual salary at the new branch is different from $32 500. She takes a random sample of 20 employees at the new branch.

State suitable null and alternative hypotheses for the test.

Find the critical region for the sample mean salary, $\overline{X}$ dollars, for a test at the 5% significance level.

The total of the annual salaries of the 20 employees in the sample is $602 000.

State, with a reason, the conclusion of the test.

State, with a reason, whether the conclusion of the test would be different if a 10% significance level had been used.

The lengths, in millimetres, of components made by a machine are normally distributed with mean $\mu$ and standard deviation 10. In the past the value of $\mu$ was 30.

After the machine is serviced, an engineer tests, at the 5% significance level, whether the mean length has decreased. She uses the mean length of a random sample of $n$ components. You should assume that the standard deviation is unchanged.

The sample mean length is 28 mm, and the result of the test is that the null hypothesis is not rejected.

(i) Show that $\sqrt{n}<8.225$.

(ii) Find the largest possible value of $n$.

The lifetimes, in hours, of a type of light bulb are normally distributed with mean $\mu$ and standard deviation 60. The mean lifetime, $\overline{X}$ hours, of a random sample of 25 bulbs is used to test the null hypothesis ${\text{H}}_0:\mu =400$.

A manager suspects that the mean lifetime is greater than 400 hours.

Find the rejection region for a test of ${\text{H}}_0$ against ${\text{H}}_1:\mu >400$ at the 5% significance level.

A second manager suspects that the mean lifetime is less than 400 hours. She decides to reject ${\text{H}}_0$ if the sample mean lifetime is less than 382 hours.

Find the probability of a Type I error for this test.

A third manager tests, at the 5% significance level, whether the mean lifetime is different from 400 hours. The mean lifetime of the bulbs in the sample is 379 hours.

Carry out the test.

The time that teenagers in a certain country spend on social media has mean 132 minutes per day and standard deviation 24 minutes per day.

Mr Headnovel, a teacher, claims that the students at his school spend more time on social media, on average, than teenagers in the country as a whole. He takes a random sample of 15 students at his school and finds that their mean time spent on social media is 144 minutes per day.

Stating your hypotheses clearly, test Mr Headnovel's claim at the 5% significance level.

State two assumptions that are necessary for the test in part (a).

Adrenaline is a new rollercoaster at a theme park. The time, in minutes, that a customer spends queuing for Adrenaline is normally distributed with variance 52. The mean queuing time for the other rollercoasters at the park is 41 minutes.

The manager wishes to test, at the 10% significance level, whether the mean queuing time for Adrenaline is different from 41 minutes. She takes a sample of 10 customers over a period of several days and records their queuing times.

State suitable null and alternative hypotheses and find the critical region for the test.

The queuing times, in minutes, of the 10 customers are as follows.

38, 49, 40, 39, 49, 39, 59, 32, 55, 41

State, with a reason, the conclusion of the test.

It is later discovered that the manager always took her sample during the first hour after the theme park opened.

Explain how this might affect the validity of the conclusion in part (b).

The lengths of snails bred in captivity have mean 60 mm and standard deviation 7.2 mm.

Margot, a biologist, breeds snails on a new diet. She claims that the mean length of snails bred on the new diet is less than 60 mm. She measures a random sample of 20 snails bred on the new diet and finds that their mean length is 56.1 mm.

Stating your hypotheses clearly, test Margot's claim at the 1% significance level.

State two assumptions that are necessary for the test in part (a).

State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).

Margot repeats the same test, at the 1% significance level, using a different random sample of 20 snails bred on the new diet.

State the probability of a Type I error.

Ashleigh completes a crossword in a mean time of 28 minutes. Her brother Carl claims that his mean time, $\mu$ minutes, to complete a crossword is different from 28 minutes.

State why a two-tailed test is required, and state suitable null and alternative hypotheses for a test of Carl's claim.

Carl decides that if the mean time of a random sample of 40 crosswords is less than 27 minutes or greater than 29 minutes, he will conclude that his mean time is different from 28 minutes.

He records the times, $x$ minutes, for a random sample of 40 crosswords. His results are summarised below.

$n=40$, $\sum x=1103$, $\sum x^2=30870$

Calculate unbiased estimates of the population mean and variance of Carl's times.

Explain why Carl's test cannot have resulted in a Type I error.

Given that the true value of $\mu$ is 27.4, find the probability of a Type II error in Carl's test.

State, with a reason, whether the Central Limit Theorem was needed in your answer to part (d).

The weights of Crown pizzas made at a restaurant are normally distributed with standard deviation 42 g. It is thought that the mean weight, $\mu$ g, is 350.

A restaurant inspector believes that the mean weight is less than 350 g. She weighs a random sample of five Crown pizzas and carries out a test of ${\text{H}}_0:\mu =350$ against ${\text{H}}_1:\mu <350$ at the 5% significance level.

When she writes up her report, she can find only four of the five weights, in grams. These are shown below.

325.2, 356.1, 319.7, 300.5

The fifth weight is $w$ g, recorded to 1 decimal place.

Given that the result of the test is that ${\text{H}}_0$ is not rejected, find the smallest possible value of $w$.

The inspector's assistant says that, if the test had been carried out at the 10% significance level, the null hypothesis would have been rejected.

Given that the assistant is correct, find the set of possible values of $w$.

The lengths, in centimetres, of the leaves of a certain type of plant are normally distributed with mean $\mu$ and standard deviation 5.

A botanist believes that the mean length is greater than 20 cm. She tests, at the 10% significance level, the null hypothesis ${\text{H}}_0:\mu =20$ against the alternative hypothesis ${\text{H}}_1:\mu >20$, using the mean length of a random sample of $n$ leaves.

The sample mean length is 22 cm, and the result of the test is to reject the null hypothesis.

Find the smallest possible value of $n$.

A conker is the seed of a horse chestnut tree. A teacher claims that the mean mass of conkers is 14.5 g. Camilla believes that the mean mass is greater than this.

To test her belief, Camilla takes a random sample of 100 conkers and measures their masses, $m$ grams. Her results are summarised below.

$n=100$, $\sum m=1490$, $\sum m^2=22498$

Calculate unbiased estimates of the population mean and variance of the masses of conkers.

Test, at the 5% significance level, whether there is evidence to support Camilla's belief. State your hypotheses clearly.

A friend says, "This test is invalid because it uses the normal distribution, but we do not know whether the masses of conkers are normally distributed."

Explain whether the friend is correct.

State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (b).

Camilla carries out the same test, at the 5% significance level, using a different random sample of 100 conkers. You should assume that the unbiased estimate of the variance is still 3.

Given that the population mean mass is actually 15.3 g, find the probability of a Type II error.

Lucy leads a team of question writers. The number of questions that the team writes has a Poisson distribution with mean 25 per day.

After the team returns from a one-week holiday, Lucy believes that the mean number of questions written per day has changed. In a randomly chosen 5-day period after the holiday, the team writes 148 questions.

Use a suitable approximating distribution to test, at the 5% significance level, whether there is evidence to support Lucy's belief.

Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).

The population mean of the random variable $X~\text{N}\left(\mu ,5^2\right)$ is being tested using the null hypothesis ${\text{H}}_0:\mu =p$ against the alternative hypothesis ${\text{H}}_1:\mu \ne p$. A random sample of 16 observations is taken from the population and the sample mean is calculated as $\overline{x}=s$. There is insufficient evidence to reject the null hypothesis using a 5% level of significance.

When $p=30$, find the range of values for $s$.

When $s=25$, find the range of values for $p$.

The weights of adult pigs are normally distributed with mean 255 kg and variance 2000. An adult pig is described as supersized if it weighs more than 350 kg.

Find the probability that a randomly chosen adult pig is supersized.

Ramon, a farmer, believes that the probability that one of his adult pigs is supersized is greater than the value found in part (a). He chooses a random sample of 12 of his adult pigs and finds that 2 of them are supersized.

Stating your hypotheses clearly, test Ramon's belief at the 5% significance level.

Ramon also claims that the mean weight of his adult pigs is greater than 255 kg. The mean weight of the 12 pigs in his sample is 273 kg. You should assume that the weights of Ramon's adult pigs are normally distributed with variance 2000.

Stating your hypotheses clearly, test Ramon's claim at the 5% significance level.

Dr Yassin is a newly qualified dentist. The time, in seconds, that he takes to perform a routine tooth extraction is normally distributed with standard deviation 41. The mean time should be 420 seconds.

His supervisor, Dr Holden, records the times for a random sample of six extractions. Five of the times, in seconds, are shown below.

433, 381, 498, 363, 419

The sixth time is $t$ seconds.

Dr Holden carries out a test of ${\text{H}}_0:\mu =420$ against ${\text{H}}_1:\mu \ne 420$ at the 5% significance level.

Given that the result of the test is that ${\text{H}}_0$ is not rejected, find the set of possible values of $t$.

A colleague of Dr Holden points out that, using the same six times, a test of ${\text{H}}_0:\mu =420$ against ${\text{H}}_1:\mu <420$ at the 5% significance level would have rejected ${\text{H}}_0$.

Using this information, find an improved inequality for $t$.

The length, $L$ minutes, of a jazz song is normally distributed with standard deviation 0.71. A headline states that the mean length of a jazz song is 4 minutes.

Cyd claims that the mean length of a jazz song is less than 4 minutes. To test her claim, she takes a random sample of 40 jazz songs and finds the mean length, $\overline{L}$ minutes.

Stating your hypotheses clearly, find the rejection region for a test of Cyd's claim at the 5% significance level.

Cyd decides to include more songs in her sample. State, with a reason, the effect that this will have on the rejection region.

Cyd's new sample has $n$ songs, and its mean length is 3.95 minutes. This sample mean is in the rejection region for a test at the 5% significance level.

Find the smallest possible value of $n$.

Explain whether it was necessary to use the Central Limit Theorem in any part of this question.

The proportion of dalmatians that have no black spots on their faces is denoted by $p$. It is claimed that $p=0.2$, but Ella suspects that the true value is lower.

Ella takes a random sample of 101 dalmatians and finds that 88 of them have black spots on their faces.

Use a suitable approximating distribution to test Ella's suspicion at the 5% significance level.

Justify your choice of approximating distribution.

State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).

The marks of students on a statistics exam at a university are normally distributed with mean $\mu$ and standard deviation 12.

A lecturer tests the null hypothesis ${\text{H}}_0:\mu =55$ against the alternative hypothesis ${\text{H}}_1:\mu >55$, using the mean mark, $\overline{M}$, of a random sample of 25 students. She rejects ${\text{H}}_0$ if $\overline{M}>c$.

Given that the true value of $\mu$ is 59, the probability of a Type II error is 0.325.

(i) Find the value of $c$.

(ii) Find the significance level of the test.

(iii) State what is meant by a Type II error in this context.