Page 68 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing
21. A large chain of telecommunication service provider introduces a training programme for counter staff
to increase their productivity and morale. The management believes that the mean time for collecting
customers’ phone-bill payment should be 120 seconds. After the training programme, the times to
collect 90 bills are found to have a mean of 113.5 seconds and a standard deviation of 35 seconds.
(a) Determine, at the 5% significance level, whether, after the training programme, the mean time
to collect a bill is less than 120 seconds.
(b) What assumption is necessary for the test in part (a) to be justifiable?

22. The proportion of members of a certain badminton club who are able to explain the rules of badminton
game correctly is p. A random sample of 10 members of the badminton club is selected and 4 members
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are able to explain the rules correctly. Test the null hypothesis, H : p = 0.7 against the alternative
0
hypothesis H : p , 0.7 at the 10% significance level.
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23. For a binomial distribution B(15, p), it is to test H : p = 0.25 versus H : p . 0.25.
0
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(a) Using a significance level of 5%, determine the critical region for this test. The area of the critical
region at the left end should be as close as possible to 0.05.
(b) Find the actual significance level of this test and the critical value.
(c) Draw a conclusion from this test.
24. In a class, 25 students are selected randomly and their reaction times, in seconds, to a particular
experiment, are measured. The mean reaction time for the students is 6.2 seconds with a variance of
2
4 s .
(a) Find the 95% confidence interval for the mean reaction time for all students.
(b) State any assumptions necessary to make this valid estimate.
(c) Based on your answer in (a), would the null hypothesis that the true mean is 7.0 seconds be
rejected at the 5% significance level? Why?
25. A peanut factory claims that at most 6% of the peanut shells contain no nuts. 100 peanuts are selected
at random and it is found that 9 of them were empty. Test at the 5% level of significance whether or
not the claim made by the factory is true.
26. Scores from a standardised memory test of all the secondary students in a school are normally distributed
with a mean μ and a standard deviation σ = 18. The score of a random sample of 60 such students has
a mean of 76. Perform a test to determine whether μ is less than 80, using the 5% significance level.
27. A random sample of 65 bags of groundnut which are labelled as 100 grams, are weighed. The mean
weight of the bags is 99 grams with an estimate of the standard deviation of 4.21 grams. Test whether
the mean weight of all bags of groundnut packed is less than 100 grams. Use the 5% level of significance.

28. A machine fills bottles with cooking oil. Prior to maintenance of the machine, the volume of cooking
oil in a bottle could be modelled by a normal distribution with mean 554 ml and standard deviation 5
3.5 ml. Following this new setting of the machine, the mean volume of cooking oil of a random sample
of 12 bottles is 555.1 ml.
(a) Carry out a test, at the 10% significance level, to decide whether the mean volume of cooking oil
filled by the machine has changed. Assume that the distribution of volume is still normal with
the variance remain unchanged.
(b) Find the largest significance level such that there is evidence that the mean volume of the cooking
oil has increased.
29. Steel rods produced on a production line are supposed to average 25.2 mm in diameter. Assume the
diameters of the steel rods are normally distributed with standard deviation 0.11 mm. It is desired to
check that the mean diameter of the rods is under control within certain limits. Suppose a random
sample of 80 steel rods is taken and the diameter of each rod is measured. Test at the 2% level to
determine the limits of the mean diameter such that the steel rods produced would be acceptable.

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05 STPM Math(T) T3.indd 259 28/10/2021 10:24 AM