Question 1

Blue sea Seafoods are a supplier of pre-packed seafood to the two major supermarket chains. Their Quality Assurance Officer is responsible for ensuring the pre-packaged items comply with their marked weights. They routinely inspect their smoked salmon product to determine if the contents of these packets weight at least the 180 grams advertised. A random sample of 55 packets of smoked salmon, whose advertised weight claims to be 180 grams, yielded the following statistics.

           X¯=181X¯=181 grams

           S=5.5S=5.5 grams

(a) Construct a 95% confidence interval estimate for the mean weight of the pre-packaged smoked salmon with an advertised weight of 180 grams. Hint: clearly justify the choice of the sampling distribution of the sample mean. Include the steps of the confidence interval estimation and appropriate critical values. 

(b) Interpret the estimated confidence interval obtained in (a).

Question 2

A Parking Enforcement Officer in Toowoomba is conducting an analysis on the amount of time left on parking meters in the CBD. A quick survey of 14 metered parking spaces where cars had just left provided the following times (in minutes).

(a) What assumptions must you make to obtain a confidence interval for the mean amount of time left for all vacant meters?

(b) Construct a 95% confidence interval estimate for the mean amount of time left for all vacant meters. 

(c) Interpret the confidence interval obtained in (b). 

(d) The Policy Manager is not totally convinced with the confidence interval estimated in (b), because it is too wide. They would like to be 99% sure that the sample mean is within 5 minutes of the true mean. What sample size is required to meet this requirement? Assume the population standard deviation is equal to the one identified for the sample of the 14 parking spaces.

Question 3

A survey of the audience at a popular comedian's show at the Sydney Opera House has been conducted and found that 69 of the 325 members of the audience who participated in the survey are visitors to Sydney.

(a) Calculate a 90% confidence interval for the proportion of audience members at the comedian's show who are visitors to Sydney. Present your work. 

(b) Interpret the confidence interval.

(c) To conduct a new follow-up study that would provide 95% confidence that the point estimate is correct to within ±± 0.05 of the population proportion, how large a sample size is required?

(d) To conduct a a new follow-up study that would provide 99% confidence that the point estimate is correct to within ±± 0.05 of the population proportion, how large a sample size is required?

Question 4

The treadwear index provided on car tyres helps prospective buyers make their purchasing decisions by indicating a tyre’s resistance to tread wear. A tyre with a treadwear grade of 200 should last twice as long, on average, as a tyre with a grade of 100. A consumer advocacy organisation wishes to test the validity of a popular branded tyre that claims a treadwear grade of 200. A random sample of 24 tyres indicates a sample mean treadwear index of 194.1 and a sample standard deviation of 19.8.  

(a) Using 0.05 level of significance, is their evidence to conclude that the tyres are not meeting the expectation of lasting at least twice as long as a tyre graded at 100? Show all your workings. 

(b) What assumptions are made in order to conduct the hypothesis test in (a)?

Question 5

A consultant is tasked with studying the purchasing habits of customers in an Indonesian coffee shop which specialises in Kopi Luwak. They select a random sample of 62 customers during a certain week and find that the mean amount spent was 287,000 rupiah, with a standard deviation of 67,000 rupiah. They also found that 33 customers would ‘strongly recommend’ the coffee shop to their family and friends. 

(a) At the 0.05 level of significance, is their evidence that the population mean amount spent was different from 265,000 rupiah? Show all your working and clearly indicate the assumptions used. 

(b) At the 0.05 level of significance, is there evidence that more than 50% of all the customers say they would ‘strongly recommend’ the coffee shop to their family and friends? Show all your workings and assumptions used.