Milk cartons are advertised to contain one litre of milk. From a randomly sample of 18 one-litre cartons whose contents were accurately measured by the South African Bureau of Standards (SABS), it was found that the average fill was 0.98 litres with a sample standard deviation of 0.052 litres. Assume that the fill of one-litre milk cartons is normally distributed.

a) Using a 99% confidence level, estimate the actual mean fill of all one-litre milk cartons. Interpret your findings and comment on the question : Do one-litre cartons of milk cartons contain one litre of milk on average?

b) Repeat the exercise in (a) above, but use 95% confidence level. Explain any differences in the conclusions.

Briefly discuss the importance of confidence level application in manufacturing industry. In your discussion outline the benefits to the company and consumer. Cite practical examples to support your answer.

QUESTION TWO (25 MARKS)

A transport sub-committe of the Cape Town Council wanted to know if there is any difference in the mean commuting time to work between bus and train commuters and the following data was collected for each sample of commuters.

Bus commuters Train commuters

Sample mean (minutes) 35.3 31.8

Sample standard deviation 7.8 4.6

Sample size 22 36

The sub-committee would like to know if it takes bus commuters longer, on average, to get to work than train commuters.

(a) Name the two populations and then formulate the null and alternative hypothesis to address the sub-committees management question. (6) (b) Test the null at 1% significance level. (15) (c) Based on the statistics finding in (b) , which of the two public transport facilities is (bus or train) should the city council prioritise for upgrading to reduce the average commuting time for workers? (4)

QUESTION THREE (15 MARKS)

A meat packer is investigating the marked mass shown on Vienna sausages. A pilot study showed a mean of 11.8kg per pack and a standard deviation of 0.7kg.

How many packs should be sampled in order to be 98% confident that the sample mean will differ by at most 0.2kg?

QUESTION FOUR (30 MARKS)

A manufacturer of soap wishes to know if customers have a preference for a bath soap fragrances. To answer their question, a random sample of 200 adult shoppers is offered a free bar of soap .The recipients may choose from among the four flavours. The choices are as follows:

Rose Lavender Sandalwood Lemon

66 53 45 36

Based on the above information, conduct a Chi-squared Goodness-of-Fit Test at 1% significance level