Confidence interval for proportion

5 Sunita has a six-sided die with faces marked \(1,2,3,4,5,6\). The probability that the die shows a six on any throw is \(p\). Sunita throws the die 500 times and finds that it shows a six 70 times.

1. Calculate an approximate \(99 \%\) confidence interval for \(p\).
2. Sunita believes that the die is fair. Use your answer to part (a) to comment on her belief.
3. Sunita uses the result of her 500 throws to calculate an \(\alpha \%\) confidence interval for \(p\). This interval has width 0.04 . Find the value of \(\alpha\).

1 In a survey of 200 randomly chosen students from a certain college, 23\% of the students said that they owned a car. Calculate an approximate \(93 \%\) confidence interval for the proportion of students from the college who own a car.

5 Mahmoud throws a coin 400 times and finds that it shows heads 184 times. The probability that the coin shows heads on any throw is denoted by \(p\).

1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
2. Mahmoud claims that the coin is not fair. Use your answer to part (i) to comment on this claim.
3. Mahmoud's result of 184 heads in 400 throws gives an \(\alpha \%\) confidence interval for \(p\) with width 0.1 . Calculate the value of \(\alpha\).

3 A die is biased so that the probability that it shows a six on any throw is \(p\).

1. In an experiment, the die shows a six on 22 out of 100 throws. Find an approximate \(97 \%\) confidence interval for \(p\).
2. The experiment is repeated and another \(97 \%\) confidence interval is found. Find the probability that exactly one of the two confidence intervals includes the true value of \(p\).

1 In a survey of 2000 randomly chosen adults, 1602 said that they owned a smartphone. Calculate an approximate \(95 \%\) confidence interval for the proportion of adults in the whole population who own a smartphone.

3 The probability that a certain spinner lands on red on any spin is \(p\). The spinner is spun 140 times and it lands on red 35 times.

1. Find an approximate \(96 \%\) confidence interval for \(p\).
   From three further experiments, Jack finds a 90\% confidence interval, a 95\% confidence interval and a 99\% confidence interval for \(p\).
2. Find the probability that exactly two of these confidence intervals contain the true value of \(p\).

2 In a random sample of 70 bars of Luxcleanse soap, 18 were found to be undersized.

1. Calculate an approximate \(90 \%\) confidence interval for the proportion of all bars of Luxcleanse soap that are undersized.
2. Give a reason why your interval is only approximate.

3 In a sample of 50 students at Batlin college, 18 support the football club Real Madrid.

1. Calculate an approximate \(98 \%\) confidence interval for the proportion of students at Batlin college who support Real Madrid.
2. Give one condition for this to be a reliable result.

2 A six-sided die is suspected of bias. The die is thrown 100 times and it is found that the score is 2 on 20 throws. It is given that the probability of obtaining a score of 2 on any throw is \(p\).

1. Find an approximate \(94 \%\) confidence interval for \(p\).
2. Use your answer to part (i) to comment on whether the die may be biased.

1 A coin is thrown 100 times and it shows heads 60 times. Calculate an approximate \(98 \%\) confidence interval for the probability, \(p\), that the coin shows heads on any throw.

1 In a survey, 36 out of 120 randomly selected voters in Hungton said that if there were an election next week they would vote for the Alpha party. Calculate an approximate \(90 \%\) confidence interval for the proportion of voters in Hungton who would vote for the Alpha party.

2 In a survey, a random sample of 250 adults in Fromleigh were asked to fill in a questionnaire about their travel.

1. It was found that 102 adults in the sample travel by bus. Find an approximate \(90 \%\) confidence interval for the proportion of all the adults in Fromleigh who travel by bus.
2. The survey included a question about the amount, \(x\) dollars, spent on travel per year. The results are summarised as follows. $$n = 250 \quad \Sigma x = 50460 \quad \Sigma x ^ { 2 } = 19854200$$ Find unbiased estimates of the population mean and variance of the amount spent per year on travel.
   A councillor wanted to select a random sample of houses in Fromleigh. He planned to select the first house on each of the 143 streets in Fromleigh.
3. Explain why this would not provide a random sample.

3 A survey was conducted to find the proportion of people owning DVD players. It was found that 203 out of a random sample of 278 people owned a DVD player.

1. Calculate a \(97 \%\) confidence interval for the true proportion of people who own a DVD player. A second survey to find the proportion of people owning DVD players was conducted at 10 o'clock on a Thursday morning in a shopping centre.
2. Give one reason why this is not a satisfactory sample.

3

1. Explain what is meant by the term 'random sample'. In a random sample of 350 food shops it was found that 130 of them had Special Offers.
2. Calculate an approximate \(95 \%\) confidence interval for the proportion of all food shops with Special Offers.
3. Estimate the size of a random sample required for an approximate \(95 \%\) confidence interval for this proportion to have a width of 0.04 .

1 In a survey of 1000 randomly chosen adults, 605 said that they used email. Calculate a \(90 \%\) confidence interval for the proportion of adults in the whole population who use email.

6 In order to obtain a random sample of people who live in her town, Jane chooses people at random from the telephone directory for her town.

1. Give a reason why Jane's method will not give a random sample of people who live in the town. Jane now uses a valid method to choose a random sample of 200 people from her town and finds that 38 live in apartments.
2. Calculate an approximate \(99 \%\) confidence interval for the proportion of all people in Jane's town who live in apartments.
3. Jane uses the same sample to give a confidence interval of width 0.1 for this proportion. This interval is an \(x \%\) confidence interval. Find the value of \(x\).

4 In a survey a random sample of 150 households in Nantville were asked to fill in a questionnaire about household budgeting.

1. The results showed that 33 households owned more than one car. Find an approximate \(99 \%\) confidence interval for the proportion of all households in Nantville with more than one car. [4]
2. The results also included the weekly expenditure on food, \(x\) dollars, of the households. These were summarised as follows. $$n = 150 \quad \Sigma x = 19035 \quad \Sigma x ^ { 2 } = 4054716$$ Find unbiased estimates of the mean and variance of the weekly expenditure on food of all households in Nantville.
3. The government has a list of all the households in Nantville numbered from 1 to 9526. Describe briefly how to use random numbers to select a sample of 150 households from this list.

1 In order to judge the support for a new method of collecting household waste, a city council arranged a survey of 400 householders selected at random. The results showed that 186 householders were in favour of the new method.

1. Calculate a 95\% confidence interval for the proportion of all householders who are in favour of the new method. A city councillor said he believed that as many householders were in favour of the new method as were against it.
2. Comment on the councillor's belief.

3

1. A company packages butter. Of 50 randomly selected packs, 8 were found to have damaged wrappers. Find an approximate \(95 \%\) confidence interval for the proportion of packs with damaged wrappers.
2. The mass of a pack has a normal distribution with standard deviation 8.5 g . In a random sample of 10 packs the masses, in g , are as follows. $$\begin{array} { l l l l l l l l l l } 220 & 225 & 218 & 223 & 224 & 220 & 229 & 228 & 226 & 228 \end{array}$$ Find a 99\% confidence interval for the mean mass of a pack.

3. Tony runs a pie stand that sells two types of pie outside a football ground. He wants to estimate the probability that a customer will buy a steak pie rather than a vegetable pie. He conducts a survey by randomly selecting customers and recording their choice of pie. When he feels he has enough data, he notes that 55 customers bought steak pies and 25 bought vegetable pies.

1. Calculate an approximate \(90 \%\) confidence interval for \(p\), the probability that a randomly selected customer buys a steak pie.
2. Suppose that Tony carries out 50 such surveys and calculates \(90 \%\) confidence intervals for each survey. Determine the expected number of these confidence intervals that would contain the true value of \(p\).

4. (a) In an opinion poll of 1800 people, 1242 said that they preferred red wine to white wine. Calculate a 95\% confidence interval for the proportion of people in the population who prefer red wine to white wine.
(b) In another opinion poll of 1000 people on the same subject, the following confidence interval was calculated.
[0pt] [0.672, 0.732]. Determine

1. the number of people in the sample who stated that they prefer red wine to white wine,
2. the confidence level of the confidence interval, giving your answer as a percentage correct to three significant figures.