Model and solve problems that involve interval estimates for proportions, with and without technology.

A market research company surveyed a random sample of 80 customers at a shopping centre. Of these customers, 28 indicated they shop online at least once per week. Determine the approximate 95% confidence interval for the proportion of all customers at the shopping centre who shop online at least once per week. Use \( z = 2 \) and express your answer correct to two decimal places.

A regional health authority conducts a telephone survey of \(n = 800\) randomly selected residents to assess awareness of a new public health campaign. Of those surveyed, 528 residents reported being aware of the campaign. (a) Calculate the sample proportion of residents aware of the campaign. [1 mark] (b) Determine the \(90\%\) confidence interval for the true proportion of all residents in the region who are aware of the campaign. Express your answer with endpoints correct to three decimal places. [2 marks] (c) A government advisor claims that the proportion of aware residents in the entire region exceeds \(0.70\). Based on your confidence interval from part (b), state whether this claim is reasonable and justify your answer. [1 mark]