Review practice · Section 1 of 2

A teacher claims that the average time a student spends on mathematics revision per day is 120 minutes. A random sample of 150 students is observed and the time, $x$ min, that each student spends on mathematics revision per day noted. It is found that

$\sum(x-120)=-1500,\qquad\sum(x-120)^2=462\,000.$

(i) Test, at the 1% level of significance, whether the teacher has overstated the average time spent on mathematics revision per day by a student.4 marks

(ii) Using the unbiased estimate of the population variance in (i), what is the least number of students that is required in a sample whose mean is 110 hours to arrive at a different conclusion from that in part (i)?2 marks

A random sample of 50 packets of ‘Nilo Bar’ snacks is weighed and the mass, $x$ grams, is recorded. The results are summarised by

$\sum(x-150)=-250,\qquad\sum(x-150)^2=3000.$

The population mean mass is $\mu$ grams. A test is carried out at the 1% significance level with the following hypotheses:

$H_0:\mu=\mu_0,\qquad H_1:\mu\ne\mu_0.$

Given that $H_0$ is rejected in favour of $H_1$, find the set of possible values of $\mu_0$.5 marks

In year 2008, a survey was done by the Ministry of Manpower (MOM) to determine the amount of daily wages, denoted as $X$, that part-time workers were receiving. Past records show that on average, the wages that each worker received was $30.

A year later, MOM wants to find out if there is a change in the mean wage. Data from 90 workers are collected and their wages are summarized as shown below:

$\bar x=31,\qquad\sum(x-31)^2=1947.$

Calculate the unbiased estimate of the variance of $X$.1 mark

MOM decides to do a test on the collected data. Find the range of values of $\alpha$ if MOM were to conclude that there is a change in the mean wage at $\alpha\%$ level of significance.3 marks

A machine is set to fill bottles with 1000 ml of soy milk. A random sample of 80 filled bottles was taken and the volume of soy milk in each bottle was measured. The sample mean and standard deviation obtained was 997.7 ml and 11.23 ml respectively. Test, at the 4% level of significance, whether the mean volume of soy milk is 1000 ml.5 marks

Based on past data, the mean number of dengue cases per day was reported as 30. The authorities suspect that the recent hot and wet weather has increased the mean number of dengue cases per day. A random sample of 60 days is taken and the number of dengue cases per day, $X$, is summarised by

$\sum(x-8)=1400,\qquad\sum(x-8)^2=34800.$

(i) Calculate unbiased estimates of the mean and variance of $X$.2 marks

(ii) Test, at the 5% significance level, whether the mean number of dengue cases per day has increased.4 marks

Explain, in the context of the question, the meaning of the p-value.1 mark

The company Snatch provides a ride-hailing service comprising taxis and private cars in Singapore. Snatch claims that the mean waiting time for a passenger from the booking time to the time of the vehicle’s arrival is 7 minutes.

To test whether the claim is true, a random sample of 30 passengers’ waiting times is obtained. The standard deviation of the sample is 2 minutes. A hypothesis test conducted concludes that there is sufficient evidence at the 1% significance level to reject the claim.

(i) State appropriate hypotheses and the distribution of the test statistic used.3 marks

(ii) Find the range of values of the sample mean waiting time, $\bar t$.3 marks

(iii) A hypothesis test is conducted at the 1% significance level whether the mean waiting time of passengers is more than 7 minutes. Using the existing sample, deduce the conclusion of this test if the sample mean waiting time is more than 7 minutes.2 marks