Proportion confidence interval

5 Each person in a random sample of 1200 people was asked whether he or she approved of certain proposals to reduce atmospheric pollution. It was found that 978 people approved. The proportion of people in the whole population who would approve is denoted by \(p\).

1. Write down an estimate \(\hat { p }\) of \(p\).
2. Find a 90\% confidence interval for \(p\).
3. Explain, in the context of the question, the meaning of a \(90 \%\) confidence interval.
4. Estimate the sample size that would give a value for \(\hat { p }\) that differs from the value of \(p\) by less than 0.01 with probability \(90 \%\).

5 An experiment with hybrid corn resulted in yellow kernels and purple kernels. Of a random sample of 90 kernels, 18 were yellow and 72 were purple.

1. Calculate an approximate \(90 \%\) confidence interval for the proportion of yellow kernels produced in all such experiments.
2. Deduce an approximate \(90 \%\) confidence interval for the proportion of purple kernels produced in all such experiments.
3. Explain what is meant by a \(90 \%\) confidence interval for a population proportion.
4. Mendel's theory of inheritance predicts that \(25 \%\) of all such kernels will be yellow. State, giving a reason, whether or not your calculations support the theory.

2 The population proportion of all men with red-green colour blindness is denoted by \(p\). Each of a random sample of 80 men was tested and it was found that 6 had red-green colour blindness.

1. Calculate an approximate \(95 \%\) confidence interval for \(p\).
2. For a different random sample of men, the proportion with red-green colour blindness is denoted by \(p _ { s }\). Estimate the sample size required in order that \(\left| p _ { s } - p \right| \leqslant 0.05\) with probability \(95 \%\).
3. Give one reason why the calculated sample size is an estimate.

5 The day before the 1992 General Election, an opinion poll showed that \(37.6 \%\) of a random sample of 1731 voters intended to vote for the Conservative party.

1. Calculate an approximate \(99.9 \%\) confidence interval for the proportion of voters intending to vote Conservative. The actual proportion voting Conservative was above the upper limit of the confidence interval.
2. Give two possible reasons for this occurrence.
3. What sample size would be required to produce a \(99.9 \%\) confidence interval of width 0.05 ?

3 In a random sample of credit card holders, it was found that \(28 \%\) of them used their card for internet purchases.

1. Given that the sample size is 1200 , find a \(98 \%\) confidence interval for the percentage of all credit card holders who use their card for internet purchases.
2. Estimate the smallest sample size for which a \(98 \%\) confidence interval would have a width of at most \(5 \%\), and state why the value found is only an estimate.

2 In order to estimate the total number of rabbits in a certain region, a random sample of 500 rabbits is captured, marked and released. After two days a random sample of 250 rabbits is captured and 24 are found to be marked. It may be assumed that there is no change in the population during the two days.

1. Estimate the total number of rabbits in the region.
2. Calculate an approximate \(95 \%\) confidence interval for the population proportion of marked rabbits.
3. Using your answer to part (ii), estimate a 95\% confidence interval for the total number of rabbits in the region.

2 A survey of a random sample of 200 passengers on UK internal flights revealed that 132 of them were on business trips.

1. Construct an approximate \(98 \%\) confidence interval for the proportion of passengers on UK internal flights that are on business trips.
2. Hence comment on the claim that more than 60 per cent of passengers on UK internal flights are on business trips.