Q13 · 2020 · Technology-activeSimple familiar6 marks
QUESTION 13 (6 marks)
An online retailer claims that 90% of all orders are shipped within 12 hours of being received. On a particular day, 121 orders were received and 102 orders were shipped within 12 hours. The distribution of the sample proportion of all orders that are shipped within 12 hours of being received on any day is approximately normal.
a)[1 mark]
State the sample proportion of orders shipped within 12 hours.
b)[3 marks]
Assuming the online retailer’s claim is true, find the probability that, in a random sample of 121, less than 85% of all orders are shipped within 12 hours.
c)[2 marks]
Use the result from 13b) to evaluate the reasonableness of the online retailer’s claim.
WORKED SOLUTION
6 marksQCAA guide · typeset solution
ANSWER
a) ; b) 0.0334; c) the claim is dubious
Worked solution
a) .
b) Assuming , the sampling distribution has and Using a GDC, .
c) There is only a 3.34% chance of obtaining a sample proportion below 0.85 if the true population proportion is 0.90. A result as low as the observed sample proportion would therefore be unusual if the claim were true, so the claim is dubious.
correctly determines the sample proportion
correctly identifies the mean of the population
determines expression for standard deviation of the normal distribution
determines probability of producing sample proportion or less
identifies meaning of the probability of producing the sample proportion
evaluates reasonableness of the retailer’s claim
One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.
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