CAIE A-Level Mathematics 6.5 Hypothesis Tests Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise formulating hypotheses, selecting one- or two-tailed tests, calculating critical regions or p-values and stating contextual conclusions with Type I or II errors.

Exam points

define H₀ and H₁ in the population parameter and choose tails from the claimed change
compare a direct or approximated probability with the significance level, or use a critical region
state the conclusion as evidence about the contextual claim and distinguish Type I from Type II error

Covered syllabus points

6.5.1—Hypothesis tests 6.5.2—Normal approximation 6.5.3—Hypothesis tests 6.5.4—Hypothesis tests 6.5.5—Normal distribution

[Maximum number: 2]

A researcher is investigating whether the proportion of families who do not own a car in his town is different from the proportion of the population in the whole country, which is 10.1%. He takes a large random sample of families in his town and finds the proportion of families that do not own a car.

(a)

Explain why a two-tailed test is appropriate in this context.

[ 1 ]

(b)

State suitable null and alternative hypotheses for the test.

The researcher calculates the value of the test statistic z and finds that z=1.82. He carries out the test at the 5\% significance level.

[ 1 ]

[Maximum number: 8]

The mean mass of packets of Trueleaf tea is supposed to be 500 grams. An inspector wishes to test whether this value is correct. He weighs 60 randomly chosen packets and notes the mass, x grams, of each packet. The results are summarised as follows.

n=60 \quad \sum x=29970 \quad \sum x^{2}=14970300

Test, at the 5% significance level, whether the population mean mass is 500 grams.

[Maximum number: 8]

A certain website receives an average of $\mu$ hits per hour. In the past the value of $\mu$ was 14.4. After making some improvements, the owner of the website wishes to test whether the value of $\mu$ has increased. He chooses a 10 -minute period at random and finds that there were 6 hits during this period. You may assume that the number of hits the website receives in any given time period follows a Poisson distribution.

(a)

Carry out the test at the 2.5% significance level.

[ 6 ]

(b)

Explain whether it is possible that a Type I error or a Type II error or both may have been made in carrying out the test.

[ 2 ]

[Maximum number: 6]

It is known that 20% of households in a certain country contain more than 4 people. Laxmi believes that, in her town, the percentage is lower than 20%. She chooses a random sample of 40 households in her town and notes the number which contain more than 4 people. She then carries out a test at the 2.5% significance level using a binomial distribution.

(a)

Find the probability of a Type I error.

[ 4 ]

(b)

State the rejection region for the test.

Laxmi finds that exactly 2 households in her sample contain more than 4 people.

[ 1 ]

(c)

Explain why it is impossible for Laxmi to make a Type II error.

[ 1 ]

[Maximum number: 12]

The weekly profit, in dollars, made by a certain firm has a normal distribution. In the past, the weekly profit had the distribution $\mathrm{N}\left(736,26^{2}\right)$ . Following a change in management, the mean weekly profit for 35 randomly chosen weeks is $ 725.

(a)

Stating a necessary assumption, test at the 2% significance level whether the mean weekly profit has decreased.

The mean weekly profit for another random sample of 35 weeks is found and a similar test is carried out at the 2% significance level.

[ 6 ]

(b)

State the probability of a Type I error.

[ 1 ]

(c)

Given that the mean weekly profit is now in fact $ 718, find the probability of a Type II error.

[ 5 ]

Exam points

Covered syllabus points

Question 2

Question 2(a)

Question 2(b)

Question 2

Question 3

Question 3(a)

Question 3(b)

Question 5

Question 5(a)

Question 5(b)

Question 5(c)

Question 6

Question 6(a)

Question 6(b)

Question 6(c)