Q18 · 2025 · Technology-activeComplex unfamiliar5 marks
QUESTION 18 (5 marks)
The results of an employee satisfaction survey of 500 employees at a large company are presented to board members.
The results include a 95% confidence interval for the proportion of satisfied employees. The lower end of the confidence interval is 0.648.
A board member would like to use the survey results to make the claim that the proportion of the satisfied employees in the entire company is larger than 75%.
Evaluate the reasonableness of the claim.
WORKED SOLUTION
5 marksQCAA guide · typeset solution
ANSWER
The claim is not reasonable; the 95% confidence interval is approximately .
Worked solution
Set up the 95% confidence interval
For a 95% central area, .
The lower cut-off of the 75% CI is given by
Solving for using a GDC gives
Substituting into the upper confidence-limit formula gives
The 95% confidence interval is therefore
The survey suggests we can be 95% confident the population proportion is between 64.8% and 72.9%. The claim of 75% is outside this interval, so the claim is not reasonable.
correctly determines the z-score for a 95% confidence interval
determines an equation involving the lower cut-off of the confidence-interval formula
determines the sample proportion for the survey results
determines the upper end of the 95% confidence interval
determines if the claim is reasonable
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