Confidence interval for difference of proportions

1 An analysis of a random sample of 150 urban dwellings for sale showed that 102 are semi-detached. An analysis of an independent random sample of 80 rural dwellings for sale showed that 36 are semi-detached.

1. Construct an approximate \(99 \%\) confidence interval for the difference between the proportion of urban dwellings for sale that are semi-detached and the proportion of rural dwellings for sale that are semi-detached.
2. Hence comment on the claim that there is no difference between these two proportions.

5
An examination of 160 e-mails received by Gopal showed that 72 had attachments. An examination of 250 e-mails received by Haley showed that 102 had attachments.
Stating two necessary assumptions about the selection of e-mails, construct an approximate \(99 \%\) confidence interval for the difference between the proportion of e-mails received by Gopal that have attachments and the proportion of e-mails received by Haley that have attachments.
(8 marks)
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1 In advance of a referendum on independence, the regional assembly of an eastern province of a particular country carried out an opinion poll to assess the strength of the 'Yes' vote. Of the 480 men polled, 264 indicated that they intended to vote 'Yes', and of the 500 women polled, 220 indicated that they intended to vote 'Yes'.

1. Construct an approximate 95\% confidence interval for the difference between the proportion of men who intend to vote 'Yes' and the proportion of women who intend to vote 'Yes'.
2. Comment on a claim that, in the forthcoming referendum, the percentage of men voting 'Yes' will exceed the percentage of women voting 'Yes' by at least 2.5 per cent. Justify your answer.
   [0pt] [2 marks]
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