Practice section 4 of 5

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]

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]

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]

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]

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

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

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]

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

(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]

A manufacturing process produces ball bearings with diameters with known standard deviation 0.04 cm. Under normal circumstances, the manufacturing process will produce ball bearings of mean diameter 0.5 cm.

(i) During a routine quality control check, a random sample of 25 ball bearings gives a mean of 0.51 cm. Is there evidence to believe that the manufacturing process is producing ball bearings of the stated diameter? Perform an appropriate test at 5% level of significance. State a necessary assumption for the test to be valid. [4]

An enhancement on the manufacturing process will ensure that the diameters of the ball bearings produced are less variable.

(ii) Measurements of a sample of 100 ball bearings give the following summary statistics:

$\sum x=50.6,\qquad\sum(x-0.5)^2=0.08345.$

Show that the unbiased estimate of the population variance is $8.07\times 10^{-4}$.

Is there evidence at the 5% level of significance that after the enhancement, the manufacturing process is producing oversized ball bearings? [4]

(iii) Another sample of 100 ball bearings yield the same summary statistics as the previous sample in (ii). Explain, with justification, whether the combined sample will give a different conclusion to (ii). [4]

(a) College students intending to further their studies overseas have to sit for a mandatory Overseas Universities Test (OUT). Researcher Mr Anand wishes to find out if male college students tend to score higher for OUT compared to female college students.

Mr Anand’s colleague randomly selects 150 male and 150 female students from the combined student population of three particular colleges near his home to form a sample of 300 college students for the research.

Explain whether this sample is a random sample. [2]

(b) The mean OUT score for all college students in 2016 is 66. Mr Anand randomly selects 240 college students taking OUT in 2017 and their scores, $x$, are summarised in the following table:

(i) Write down the unbiased estimates of the population mean and variance of the OUT scores for the college students in 2017. [1]

(ii) Test, at the 10% level of significance, whether the mean OUT score for all college students in 2017 is higher than the mean score attained in 2016. [4]

(iii) Explain what is meant by the phrase “10% level of significance” in this context. [1]

(iv) Mr Anand draws a new sample of 240 male college students. Using the unbiased estimate for the population variance computed in (i), find the range of values for the sample mean $\bar x$ that is required for this new sample to achieve a different conclusion from that in (ii). [4]