Paired sample confidence interval

A question is this type if and only if it requires constructing a confidence interval for the mean difference using paired data, applying the t-distribution to the computed differences.

2 questions · Standard +0.3

5.05d Confidence intervals: using normal distribution
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CAIE FP2 2009 June Q8
8 marks Standard +0.3
8 Part of a research study of identical twins who had been separated at birth involved a random sample of 9 pairs, in which one twin had been raised by the natural parents and the other by adoptive parents. The IQ scores of these twins were measured, with the following results.
Twin pair123456789
IQ of twin raised by natural parents8292115132889511283123
IQ of twin raised by adoptive parents9288115134979410788130
It may be assumed that the difference in IQ scores has a normal distribution. The mean IQ scores of separated twins raised by natural parents and by adoptive parents are denoted by \(\mu _ { N }\) and \(\mu _ { A }\) respectively. Obtain a \(90 \%\) confidence interval for \(\mu _ { N } - \mu _ { A }\). One of the researchers claimed that there was no evidence of a difference between the two population means. State, giving a reason, whether the confidence interval supports this claim.
CAIE FP2 2011 June Q8
11 marks Standard +0.3
8 A company decides that its employees should follow an exercise programme for 30 minutes each day, with the aim that they lose weight and increase productivity. The weights, in kg , of a random sample of 8 employees at the start of the programme and after following the programme for 6 weeks are shown in the table.
Employee\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Weight before \(( \mathrm { kg } )\)98.687.390.485.2100.592.489.991.3
Weight after \(( \mathrm { kg } )\)93.585.288.284.695.489.386.087.6
Assuming that loss in weight is normally distributed, find a 95\% confidence interval for the mean loss in weight of the company's employees. Test at the \(5 \%\) significance level whether, after the exercise programme, there is a reduction of more than 2.5 kg in the population mean weight.