Q14 · 2022 · Technology-activeSimple familiar8 marks
QUESTION 14 (8 marks)
Ravi randomly sampled 200 different pet owners in Brisbane and found that 50 celebrate their pet’s birthday.
a)[2 marks]
Determine an approximate 95% confidence interval for the proportion of Brisbane pet owners who celebrate their pet’s birthday.
b)
[2 marks]
Two of Ravi’s friends also randomly sampled Brisbane pet owners. The results are shown in the table.
| Friend’s name | Number sampled | Number who celebrate their pet’s birthday |
|---|---|---|
| Khadija | 100 | 26 |
| Tim | 150 | 34 |
Khadija suggested a more precise estimate for the proportion of Brisbane pet owners who celebrate their pet’s birthday could be obtained by combining their results. Using all available data, determine an approximate 95% confidence interval for the proportion of Brisbane pet owners who celebrate their pet’s birthday.
c)[2 marks]
Use the results from Questions 14a) and 14b) to evaluate the reasonableness of Khadija’s suggestion.
d)[2 marks]
The proportion of all Brisbane pet owners who celebrate their pet’s birthday is 0.24. Using the normal approximation, determine the probability that in a randomly selected sample of size 200, more than 30% of pet owners celebrate their pet’s birthday.
WORKED SOLUTION
8 marksQCAA guide · typeset solution
ANSWER
a) ; b) ; c) the combined sample gives a narrower interval and hence a more precise estimate; d) .
Worked solution
a) Using , and , the approximate confidence interval is
b) Combining all samples gives and . Using a GDC gives
c) By combining the results, the sample size is increased and the confidence interval width is reduced. The new sample statistic provides a better estimate for the population parameter.
d) Using the normal approximation, the mean is and Using a GDC,
correctly identifies all of the information required to establish the confidence interval
correctly determines the confidence interval
correctly determines and for the combined sample
determines confidence interval
identifies changed width of confidence interval
evaluates the reasonableness of Khadija’s suggestion
correctly determines the mean and standard deviation of the normal distribution
determines the probability
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