Confidence interval for single proportion

1 Anita carried out a survey of 140 randomly selected students at her college. She found that 49 of these students watched a TV programme called Bunch.

1. Calculate an approximate \(98 \%\) confidence interval for the proportion, \(p\), of students at Anita's college who watch Bunch.
   Carlos says that the confidence interval found in (a) is not useful because it is too wide.
2. Without calculation, explain briefly how Carlos can use the results of Anita's survey to find a narrower confidence interval for \(p\).

2 A random sample of \(n\) people were questioned about their internet use. 87 of them had a high-speed internet connection. A confidence interval for the population proportion having a high-speed internet connection is \(0.1129 < p < 0.1771\).

1. Write down the mid-point of this confidence interval and hence find the value of \(n\).
2. This interval is an \(\alpha \%\) confidence interval. Find \(\alpha\).

5 A constitutional change was proposed for a Golf Club with a large membership. This was to be voted on at the Annual General Meeting. A month before this meeting the secretary asked a random sample of 50 members for their opinions. Out of the 50 members \(70 \%\) said they approved.

1. Calculate an approximate \(90 \%\) confidence interval for the proportion \(p\) of all members who would approve the proposal.
2. Explain what is meant by a \(90 \%\) confidence interval in this context.
3. Nearer the date of the meeting the secretary asked a random sample of \(n\) members, and, as before, \(70 \%\) said they approved. This time the secretary calculated an approximate \(99 \%\) confidence interval for \(p\). It is given that the confidence interval does not include 0.85 . Find the smallest possible value of \(n\).

1 A machine fills packets of flour whose nominal weights are 500 g . Each of a random sample of 100 packets was weighed and 14 packets weighed less than 500 g . The population proportion of packets that weigh less than 500 g is denoted by \(p\).

1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
2. The weights of the packets, in grams, are normally distributed with mean \(\mu\) and variance 50 . Assuming that \(p = 0.14\), calculate the value of \(\mu\).

5 The daily number of emergency calls received from district A may be modelled by a Poisson distribution with a mean of \(\lambda _ { \mathrm { A } }\). The daily number of emergency calls received from district B may be modelled by a Poisson distribution with a mean of \(\lambda _ { \mathrm { B } }\). During a period of 184 days, the number of emergency calls received from district A was 3312, whilst the number received from district B was 2760.

1. Construct an approximate \(95 \%\) confidence interval for \(\lambda _ { \mathrm { A } } - \lambda _ { \mathrm { B } }\).
2. State one assumption that is necessary in order to construct the confidence interval in part (a).

1 A hotel's management is concerned about the quality of the free pens that it provides in its meeting rooms. The hotel's assistant manager tests a random sample of 200 such pens and finds that 23 of them fail to write immediately.

1. Calculate an approximate \(\mathbf { 9 6 \% }\) confidence interval for the proportion of pens that fail to write immediately.
2. The supplier of the pens to the hotel claims that at most 2 in 50 pens fail to write immediately. Comment, with numerical justification, on the supplier's claim.
   [0pt] [2 marks] QUESTION
   PART Answer space for question 1

2 Emilia runs an online perfume business from home. She believes that she receives more orders on Mondays than on Fridays. She checked this during a period of 26 weeks and found that she received a total of 507 orders on the Mondays and a total of 416 orders on the Fridays. The daily numbers of orders that Emilia receives may be modelled by independent Poisson distributions with means \(\lambda _ { \mathrm { M } }\) for Mondays and \(\lambda _ { \mathrm { F } }\) for Fridays.

1. Construct an approximate \(99 \%\) confidence interval for \(\lambda _ { \mathrm { M } } - \lambda _ { \mathrm { F } }\).
2. Hence comment on Emilia's belief.

4. The Department of Health recommends that adults aged 18 to 65 should take part in at least 150 minutes of aerobic exercise per week. The results of a survey show that 940 out of 2000 randomly selected adults aged 18 to 65 in Wales take part in at least 150 minutes of aerobic exercise per week.

1. Calculate an approximate \(95 \%\) confidence interval for the proportion of adults aged 18 to 65 in Wales who take part in at least 150 minutes of aerobic exercise per week.
2. Give two reasons why the interval is approximate.
3. Suppose that a \(99 \%\) confidence interval is required, and that the width of the interval is to be no greater than \(0 \cdot 04\). Estimate the minimum additional number of adults to be surveyed to satisfy this requirement.