Question 3

(a) A company that processes and distributes fruits and vegetables has just received a delivery of carrots. The production manager knows that carrots more than 28 cm in length will not fit the packaging of their supermarket client. The client has also specified that carrots less than 12 cm be rejected as these are deemed undesirable for their consumer market.

If the manager assumes that carrot length is normally distributed and knows from batch testing that mean carrot length is 21.6 cm, with a standard deviation of 3.2 cm, find the proportion of carrots that will fit the specifications for this product. Please use the Empirical rule to answer this question.

(b) The production manager selects a number of carrots at random from the delivery and arranges them in batches of 10 carrots. The average weight of these batches is found to be 660 g. If the delivery was 4.5 tonne of carrots, approximately how many carrots will be rejected for the product outlined in part (a) above?

Question 4

A marine biologist based in Coffs Harbour, NSW, has been researching the breeding location of the endangered Australian sea lion. After counting the number of pups born at four locations in the Coffs Harbour area over a five-year period (Jan 2014 - Dec 2018) the proportion of pups born per breeding location could be calculated. This is shown in Table 1.

In 2021 an environment committee of the Coffs Harbour local government was interested in using this historical data to help determine if a recent large-scale development of the marina had changed the proportion of Australian sea lion pups born across the known breeding locations. The committee commissioned a study that counted the sea lion pups born in 2021 across the same areas. Observations are shown in Table 2.

Consider the information above and answer the following.

(a) Write a null hypothesis appropriate for a chi-squared goodness of fit test using the data provided.

(b) What are the degrees of freedom for this Chi-square goodness of fit test?

(c) Calculate the x2 test statistic.

(d) Calculate the Critical Chi-squared value for a significance level of 0.05 (please state the Excel formula used) and make a decision based on statistical evidence for this test. What conclusion can be drawn from this?